Research Article Classical Analog of Extended Phase Space SUSY...

Hindawi Publishing CorporationISRNMathematical PhysicsVolume 2013 Article ID 639424 10 pageshttpdxdoiorg1011552013639424

Research ArticleClassical Analog of Extended Phase Space SUSY and Its Breaking

Gagik Ter-Kazarian

Byurakan Astrophysical Observatory Aragatsotn District 378433 Byurakan Armenia

Correspondence should be addressed to Gagik Ter-Kazarian gago 50yahoocom

Received 21 April 2013 Accepted 12 May 2013

Academic Editors Z Qiao and P Roy

Copyright copy 2013 Gagik Ter-KazarianThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We derive the classical analog of the extended phase space quantum mechanics of the particle with odd degrees of freedom whichgives rise to (119873 = 2)-realization of supersymmetry (SUSY) algebra By means of an iterative procedure we find the approximateground state solutions to the extended Schrodinger-like equation and use these solutions further to calculate the parameters whichmeasure the breaking of extended SUSY such as the ground state energy Consequently we calculate a more practical measure forthe SUSY breaking which is the expectation value of an auxiliary field We analyze nonperturbative mechanism for extended phasespace SUSY breaking in the instanton picture and show that this has resulted from tunneling between the classical vacua of thetheory Particular attention is given to the algebraic properties of shape invariance and spectrum generating algebra

1 Introduction

An interesting question of keeping the symmetry betweencanonical coordinates and momenta in the process ofquantization deserves an investigation From its historicaldevelopment this aspect of statistical quantum mechanicsunfortunately has attracted little attention However muchuse has been made of the technique of ordering of canonicalcoordinates (119902) and momenta (119901) in quantum mechanics[1 2] It was observed that the concept of an extendedLagrangian L(119901 119902 119902) in phase space allows a subsequentextension of Hamiltonrsquos principle to actions minimum alongthe actual trajectories in (119901 119902)- rather than in 119902-space Thisleads to the phase space formulation of quantum mechanicsConsequently this formalism was developed further in [3]by addressing the extended phase space stochastic quanti-zation of Hamiltonian systems with first class holonomicconstraints This in a natural way results in the Faddeev-Popov conventional path-integral measure for gauge systemsContinuing along this line in the present paper we addressthe classical analog of the extended phase space (119873 = 2)-SUSY quantum mechanics [4] of the particles which haveboth bosonic and fermionic degrees of freedom that isthe quantum field theory in (0 + 1)-dimensions in (119902 119901)-space exhibiting supersymmetry (for conventional SUSY

quantummechanics see [5ndash12])We analyze in detail the non-perturbative mechanism for supersymmetry breaking in theinstanton picture ([13]) This paper has been organized asfollows In the first part (Sections 2 and 3) we derive theclassical analog of the extended phase space SUSY quantummechanics and obtain the integrals of motion Consequentlywe describe the extended phase space (119873 = 2)-SUSY algebraIn the second part (Sections 4 and 5) by means of an iterativescheme first we find the approximate ground state solutionsto the extended Schrodinger-like equation and then calculatethe parameters which measure the breaking of extendedSUSY such as the ground state energy We calculate a morepractical measure for the SUSY breaking in particular infield theories which is the expectation value of an auxiliaryfield We analyze nonsperturbative mechanism for extendedphase space SUSY breaking in the instanton picture and showthat this has resulted from tunneling between the classicalvacua of the theory Section 6 deals with the independentgroup theoretical methods with nonlinear extensions of Liealgebras from the perspective of extended phase space SUSYquantum mechanics and further shows how it can be usefulfor spectrum generating algebra The concluding remarksare given in Section 7 Unless otherwise stated we take thegeometrized units (ℎ = 119888 = 1) Also an implicit summationon repeated indices is assumed throughout this paper

2 ISRNMathematical Physics

2 The Integrals of Motion

For the benefit of those not familiar with the framework ofextended phase space quantization enough details are givenin the following tomake the rest of the paper understandableThe interested reader is invited to consult the original papers[1 2] for further details In the framework of the proposedformalism the extended Lagrangian can be written as

Lext (119901 119902 119902) = minus 119902119894119901119894minus 119902

119894119894+L

119902+L

119901 (1)

where a dynamical system with 119873 degrees of freedomdescribed by the 2119873 independent coordinates 119902= (119902

1 119902

119873)

and momenta 119901 = (1199011 119901

119873) which are not in gen-

eral canonical pairs A Lagrangian L119902(119902 119902) is given in

119902-representation and the corresponding L119901(119901 ) in 119901-

representation Adotwill indicate differentiationwith respectto 119905 The independent nature of 119901 and 119902 gives the freedom ofintroducing a second set of canonical momenta for both 119901and 119902 through the extended Lagrangian

120587119902119894=120597Lext120597 119902

119894

=120597L119902

120597 119902119894

minus 119901119894

120587119901119894=120597Lext120597

119894

=120597L119901

120597119894

minus 119902119894

(2)

One may now define an extended Hamiltonian

119867ext (119901 119902 120587119901 120587119902) = 120587119902119894 119902119894 + 120587119901119894 119894 minusLext (119901 119902 119902)

= 119867 (119901 + 120587119902 119902) minus 119867 (119901 119902 + 120587

119901)

(3)

where 119867(119901 119902) = 119901119894119902119894minusL119902

= 119902119894119894minusL119901 is the conventional

Hamiltonian of the system In particular vanishing of 120587119902or

120587119901is the condition for 119901 and 119902 to constitute a canonical

pair In the language of statistical quantum mechanics thischoice picks up a pure state (actual path) Otherwise one isdealing with a mixed state (virtual path) One may howeverenvisage that the full machinery of the conventional quantummechanical dynamics is extendible to the extended dynamicsas alluded to the above Here 119901 and 119902 will be considered asindependent 119888-number operators on the integrable complexfunction 120594(119902 119901) One of the key assumptions of extendedphase space quantization [1 2] is the differential operatorsand commutation brackets for 120587

119901and 120587

119902borrowed from the

conventional quantum mechanics as follows

120587119902119894= minus119894

120597

120597119902119894

[120587119902119894 119902119895] = minus119894120575

119894119895

120587119901119894= minus119894

120597

120597119901119894

[120587119901119894 119901

119895] = minus119894120575

119894119895

(4)

Note also the following

[119901119894 119902119895] = [119901

119894 119901

119895] = [119902

119894 119902119895] = [120587

119901119894 120587

119902119895]

= [120587119901119894 120587

119901119895] = [120587

119902119894 120587

119902119895] = 0

(5)

By the virtue of (4) and (5) 119867ext is now an operator on 120594Along the trajectories in (119901 119902)-space however it producesthe state functions 120594(119901 119902 119905) via the following Schrodinger-like equation

119894120597

120597119905120594 = 119867ext 120594 (6)

Solutions of (6) are

120594 (119902 119901 119905) = 120594119903(119902 119901 119905) 119890

minus119894119901119902

= 119886120572120573120595120572(119902 119905) 120601

lowast

120573(119901 119905) 119890

minus119894119901119902

(7)

where 119886 = 119886dagger positive definite tr 119886 = 1 and 120595

120572and

120601lowast

120572are solutions of the conventional Schrodinger equation

in 119902- and 119901-representations respectively They are mutuallyFourier transforms Note that the 120572 and 120573 are not in generaleigenindices The normalizable 120594 (int120594119889119901119889119902 = tr(119886) = 1)is a physically acceptable solution The exponential factor isa consequence of the total time derivative minus119889(119902119901)119889119905 in (1)which can be eliminated Actually it is easily verified that

(119901 + 120587119902) 120594

119903(119902 119901 119905) = (120587

119902120594119903(119902 119901 119905)) 119890

minus119894119901119902 (8)

and so on Substitution of (8) in (6) gives

119894120597

120597119905120594 (119902 119901 119905) = 119867ext 120594119903 (119902 119901 119905) (9)

provided by the reducedHamiltonian119867ext From now onwereplace119867ext by119867ext and 120594119903(119902 119901 119905) by 120594119903(119902 119901 119905) respectivelyand retain former notational conventions

It is certainly desirable to derive the classical analog of theextended phase space quantummechanics of the particlewithodd degrees of freedom directly from what may be taken asthe first principleTherefore following [10ndash12] let us considera nonrelativistic particle of unit mass with two (120572 = 1 2)odd (Grassmann) degrees of freedomThe classical extendedLagrangian equation (1) can be written

Lext (119901 119902 119902) = minus 119902119901 minus 119902 +1

21199022minus 119865 (119902) +

1

22

minus 119866 (119901) minus 119877 (119902 119901)119873 +1

2120595120572120572

(10)

provided by 119873 = 12059511205952= minus119894120595

+120595minus Here 119865(119902) R rarr

R 119866(119901) R rarr R and 119877(119902 119901) R rarr R are arbitrarypiecewise continuously differentiable functions given overthe 1-dimensional Euclidean space R The 120595

120572are two odd

(Grassmann) degrees of freedom The nontrivial Poisson-Dirac brackets of the system (10) are

119902 120587119902 = 1 119901 120587

119901 = 1 120595

120572 120595

120573 = 120575

120572120573

120595+ 120595

minus = 1 120595

2

plusmn= 0 120595

plusmn=1

radic2(120595

1plusmn 119894120595

2)

(11)

The extended Hamiltonian119867ext (3) reads

119867ext (119901 119902 120587119901 120587119902) =1

2(119901 + 120587

119902)2

+ 1198652(119902) minus

1

2(119902 + 120587

119901)2

minus 1198662(119901) + 119877 (119902 119901)119873

(12)

ISRNMathematical Physics 3

which according to (8) reduces to

119867ext (119901 119902 120587119901 120587119902) =1

21205872

119902+ 119865

2(119902) minus

1

21205872

119901

minus 1198662(119901) + 119877 (119902 119901)119873

(13)

TheHamiltonian equation (13) yields the following equationsof motion

119902 = 120587119902 = 120587

119901

119902= minus119865

1015840

119902(119902) minus 119877

1015840

119902(119902 119901)119873

119901= minus119866

1015840

119901(119901) + 119877

1015840

119901(119902 119901)119873

plusmn= plusmn119894119877 (119902 119901) 120595

plusmn

(14)

Aprimewill indicate differentiationwith respect either to 119902 or119901 Thus119873 is the integral of motion additional to119867ext Alongthe trajectories 119902(119905) and 119901(119905) in (119901 119902)-spaces the solution toequations of motion for odd variables is

120595plusmn (119905) = 120595plusmn (1199050) exp [plusmn119894 int

119905

1199050

119877 (119902 (120591) 119901 (120591)) 119889120591] (15)

Hence the odd quantities

120579plusmn= 120579

plusmn (119905) exp [∓119894 int119905

1199050

119877 (119902 (120591) 119901 (120591)) 119889120591] (16)

are nonlocal in time integrals of motion In trivial case 119877 =

0 we have plusmn= 0 and 120579

plusmn= 120579

plusmn Suppose that the system

has even complex conjugate quantities119861119902119901plusmn

(119861119902119901+

)lowast= 119861

119902119901minus

whose evolution looks up to the term proportional to119873 likethe evolution of odd variables in (14)Then local odd integralsof motion could be constructed in the form

119876119902119901plusmn

= 119861119902119901∓

120595plusmn (17)

Let us introduce the oscillator-like bosonic variables 119861119902119901plusmn

in119902- and 119901-representations as follows

119861119902∓L

2(R) 997888rarrL

2(R) 119861

119902∓= [119901 + 120587

119902plusmn 119894119882 (119902)]

119861119901∓L

2(R) 997888rarrL

2(R) 119861

119901∓= [119902 + 120587

119901plusmn 119894119881 (119901)]

(18)

In the expressions (18)119882(119902) R rarr R and119881(119901) R rarr Rare the piecewise continuously differentiable functions calledSUSY potentials In particular case if119877(119902 119901) = 119877

119902(119902)minus119877

119901(119901)

for the evolution of 119861119902119901plusmn

we obtain

119902∓= [minus (119865

1015840

119902+ 119877

1015840

119902119873) plusmn 119894119882

1015840

119902(119902) (119901 + 120587

119902)]

119901∓= [minus (119866

1015840

119901+ 119877

1015840

119901119873) plusmn 119894119881

1015840

119901(119901) (119902 + 120587

119901)]

(19)

Consequently

119902plusmn= plusmn119894 [(119882

1015840

119902minus 119877

1015840

119902plusmn) plusmn 119894 (119865

1015840

119902minus119882119882

1015840

119902) 120595

∓]

119901plusmn= plusmn119894 [(119881

1015840

119901minus 119877

1015840

119901plusmn) plusmn 119894 (119866

1015840

119901minus 119881119881

1015840

119901) 120595

∓]

(20)

This shows that either 119902plusmn= 0 or

119901plusmn= 0 when 1198821015840

119902(119902) =

1198771015840

119902plusmn(119902) and 119865

1015840

119902= (12)(119882

2)1015840

119902or 1198811015840

119901(119901) = 119877

1015840

119901plusmn(119901) and

1198661015840

119901= (12)(119881

2)1015840

119901 respectively Therefore when the functions

119877119902119901

and 119865(119902) 119866(119901) are related as

1198771015840

119902plusmn(119902) = 119882

1015840

119902(119902) 119865

119902=1

2(119882

2) + 119862

119902

1198771015840

119901plusmn(119901) = 119881

1015840

119901(119901) 119865

119901=1

2(119881

2) + 119862

119901

(21)

where 119862119902119901

are constants then odd quantities 119876119902119901plusmn

areintegrals of motion in addition to 119867ext and 119873 According to(3) and (13) let us present119867ext in the form119867ext = 119867119902

minus 119867119901

where

119867119902=1

21205872

119902+ 119865

2(119902) + 119877

119902119873

119867119901=1

21205872

119901+ 119866

2(119901) + 119877

119901119873

(22)

Then 119876119902119901plusmn

and 119873 together with the 119867119902and 119867

119901form the

classical analog of the extended phase space SUSY algebra asfollows

119876119902119901+

119876119902119901minus

= minus119894 (119867119902119901minus 119862

119902119901)

119867119902119901 119876

119902119901plusmn = 119876

119902119901plusmn 119876

119902119901plusmn = 0

119873119876119902119901plusmn

= plusmn119894119876119902119901plusmn

119873119867119902119901 = 0

(23)

with constants119862119902119901

playing a role of a central charges in (119902 119901)-spaces 119873 is classical analog of the grading operator Putting119862119902= 119862

119901= 0 we arrive at the classical analog of the extended

phase space SUSY quantummechanics given by the extendedLagrangian

Lext (119901 119902 119902) =1

21205872

119902minus1

2119882

2(119902) +

1

21205872

119901minus1

21198812(119901)

+ 12059511205952(119882

1015840

119902+ 119881

1015840

119901) +

1

2120595120572120572

(24)

We conclude that the classical system (10) is characterizedby the presence of two additional local in time odd integralsof motion equation (17) being supersymmetry generatorsAlong the actual trajectories in 119902-space (24) reproduces theresults obtained in [6]

3 The Path Integral Formulation

In the matrix formulation of extended phase space (119873 = 2)-SUSY quantum mechanics the

plusmnwill be two real fermionic

creation and annihilation nilpotent operators describingthe fermionic variables The

plusmn having anticommuting 119888-

number eigenvalues imply

plusmn= radic

1

2(

1plusmn 119894

2)

120572

120573 = 120575

120572120573

+

minus = 1

2

plusmn= 0

(25)

They can be represented by finite dimensional matrices plusmn=

120590plusmn where 120590plusmn = (120590

1plusmn120590

2)2 are the usual raising and lowering


operators for the eigenvalues of 1205903which is the diagonal Pauli

matrix The fermionic operator 119891 reads 119891 C2rarr C2

119891 =

(12)[+

minus] which commutes with the119867ext and is diagonal

in this representation with conserved eigenvalues plusmn12 Dueto it the wave functions become two-component objects asfollows

120594 (119902 119901) = (120594+12

(119902 119901)

120594minus12

(119902 119901)) = (

1205941(119902 119901)

1205942(119902 119901)

) = (

1205951(119902) 120601

1(119901)

1205952(119902) 120601

2(119901))

(26)

where the states 12059512(119902) 120601

12(119901) correspond to fermionic

quantum number 119891 = plusmn12 respectively in 119902- and 119901-spacesThey belong to Hilbert space H = H

0otimes C2

= [L2(R) otimes

L2(R)]otimesC2 Hence theHamiltonian119867ext of extended phase

space (119873 = 2)-SUSY quantummechanical system becomes a2 times 2matrix as follows

119867ext = (119867+

0

0 119867minus

) =1

2(

2

119902+119882

2(119902) + 119894119882

1015840

119902(119902) [

1

2])

minus1

2(

2

119901+ 119881

2(119901) + 119894119881

1015840

119902(119901) [

1

2])

(27)

To infer the extendedHamiltonian equation (27) equivalentlyone may start from the 119888-number extended Lagrangian ofextended phase space quantum field theory in (0 + 1)-dimensions in 119902- and 119901-spaces as follows

Lext (119901 119902 119902) = minus 119902119901 minus 119902 +1

2[(119889119902

119889119905)

2

minus1198822(119902)]

+ 1198911198821015840

119902(119902) +

1

2[(119889119901

119889119905)

2

minus 1198812(119901)]

+ 1198911198811015840

119901(119901)

(28)

In dealing with abstract space of eigenstates of the conjugateoperator

plusmnwhich have anticommuting 119888-number eigenval-

ues suppose that |00minus⟩ is the normalized zero-eigenstate of 119902and

minusas follows

119902 |00minus⟩ = 0 minus |00minus⟩ = 0 (29)

The state |00+⟩ is defined by

|00+⟩ = + |00minus⟩ = 0 (30)

then +|00+⟩ = 0

minus|00+⟩ = |00minus⟩ Taking into account

that daggerplusmn=

∓ we get

⟨∓00| plusmn = 0 ⟨∓00| ∓ = ⟨plusmn00| (31)

Now we may introduce the notation 120572 120573 for the anticom-muting eigenvalues of

plusmn Consistency requires

120572plusmn= minus

plusmn120572 120572 |00plusmn⟩ = plusmn |00plusmn⟩ 120572 (32)

The eigenstates of 119902 minuscan be constructed as

1003816100381610038161003816119902120572minus⟩ = 119890minus119894119902119901minus120572+

|00minus⟩ (33)

and thus

1199021003816100381610038161003816119902120572minus⟩ = 119902

1003816100381610038161003816119902120572minus⟩ minus

1003816100381610038161003816119902120572minus⟩ = 1205721003816100381610038161003816119902120572minus⟩ (34)

Then the 119902and

+eigenstates are obtained by Fourier

transformation as follows

1003816100381610038161003816 119902120573+⟩ = minusint119889120572119890120572120573 1003816100381610038161003816 119902120572minus⟩

10038161003816100381610038161003816120587119902120572plusmn⟩ = minusint119889119902119890

119894119902120587119902 1003816100381610038161003816 119902120572plusmn⟩

1003816100381610038161003816119901120573+⟩ = minusint119889120572119890120572120573 1003816100381610038161003816119901120572minus⟩

10038161003816100381610038161003816120587119901120572plusmn⟩ = minusint119889119901119890

119894119901120587119901 1003816100381610038161003816119901120572plusmn⟩

(35)

which gives

119902

10038161003816100381610038161003816120587119902120572plusmn⟩ = 120587

119902

10038161003816100381610038161003816120587119902120572plusmn⟩

+

10038161003816100381610038161003816(119902 120587

119902) 120573+⟩ = 120573

10038161003816100381610038161003816(119902 120587

119902) 120573+⟩

119901

10038161003816100381610038161003816120587119901120572plusmn⟩ = 120587

119901

10038161003816100381610038161003816120587119901120572plusmn⟩

+

10038161003816100381610038161003816(119901 120587

119901) 120573+⟩ = 120573

10038161003816100381610038161003816(119901 120587

119901) 120573+⟩

(36)

The following completeness relations hold

minus int1198891205721198891199021003816100381610038161003816 119902120572plusmn⟩⟨∓120572

lowast1199021003816100381610038161003816 = 1

minus int119889120572

119889120587119902

2120587

10038161003816100381610038161003816120587119902120572plusmn⟩⟨∓120572

lowast120587119902

10038161003816100381610038161003816= 1

minus int1198891205721198891199011003816100381610038161003816119901120572plusmn⟩⟨∓120572

lowast1199011003816100381610038161003816 = 1

minus int119889120572

119889120587119901

2120587

10038161003816100381610038161003816120587119901120572plusmn⟩⟨∓120572

lowast120587119901

10038161003816100381610038161003816= 1

(37)

The time evolution of the state |119905⟩ is now given as follows

120594minus(119902120572119901120573119905) = minusint119889120572

1015840119889119902

1015840119889120573

1015840119889119901

1015840119870(119902120572119901120573119905 | 119902

10158401205721015840119901101584012057310158401199051015840)

(38)

The kernel reads

119870(119902120572119901120573119905 | 11990210158401205721015840119901101584012057310158401199051015840)

= ⟨+119902120572lowast119901120573

lowast1003816100381610038161003816 119890minus119894119867ext(119905minus119905

1015840) 100381610038161003816100381610038161199021015840120572101584011990110158401205731015840⟩

(39)

which can be evaluated by the path integral Actually analternative approach to describe the state space and dynamicsof the extended phase space quantum system is by the pathintegral [3] which reads

K1198911198911015840 (119902119901119905 | 119902

101584011990110158401199051015840) = ⟨119902119901119891

1003816100381610038161003816 119890minus119894119867ext(119905minus119905

1015840) 10038161003816100381610038161003816119902101584011990110158401198911015840⟩ (40)

In the path integral equation (40) the individual states arecharacterized by the energy and the fermionic quantum


number 119891 With the Hamiltonian 119867ext the path integralequation (40) is diagonal as follows

K1198911198911015840 (119902119901119905 | 119902

101584011990110158401199051015840)

=K1198911198911015840 (119902119905 | 119902

10158401199051015840)K

1198911198911015840 (119901119905 | 119901

10158401199051015840)

= 1205751198911198911015840 int

119902

1199021015840

D119902int119901

1199011015840

D119901 exp(119894int119905

1199051015840

Lext (119901 119902 119902) 119889119905)

(41)

Knowing the path integral equation (41) it is sufficient tospecify the initial wave function 120594

119891(1199021015840 119901

1015840 1199051015840) to obtain all

possible information about the system at any later time 119905 by

120594119891(119902 119901 119905) = sum

1198911015840

int1198891199021015840119889119901

1015840K

1198911198911015840 (119902119901119905 | 119902

101584011990110158401199051015840) 120594

1198911015840 (119902

1015840 119901

1015840 1199051015840)

(42)

with 120594plusmn12

(119902 119901 119905) = 12059412(119902 119901 119905) (26) In terms of anti-

commuting 119888-number operators 120577 and 120578 defining 120595 =

radic12 (120578+120577

119894(120578minus120577)) the path integral equation (41) becomes

K (119902120572119901120573119905 | 11990210158401205721015840119901101584012057310158401199051015840) = int

119902120572119901120573

1199021015840120572101584011990110158401205731015840

D119902D119901D120577D120578

times exp(119894int119905

1199051015840

Lext (119901 119902 119902) 119889119905)

(43)

The functional integral is taken over all trajectories from 1199021015840 1205721015840to 119902 120572 and 1199011015840 1205731015840 to 119901 120573 between the times 1199051015840 and 119905

4 Solution of the Extended SchroumldingerEquation with Small Energy Eigenvalue 120576

Adopting the technique developed in [13] first we usethe iterative scheme to find the approximate ground statesolutions to the extended Schrodinger-like equation

119867ext120594 (119902 119901) = (119867119902minus 119867

119901) 120594 (119902 119901) = 120576120594 (119902 119901) (44)

with energy 120576Wewill then use these solutions to calculate theparameters which measure the breaking of extended SUSYsuch as the ground state energy The approximation whichwent into the derivation of solutions of (44) meets ourinterest that the ground state energy 120576 is supposedly smallAs we mentioned above the solutions for nonzero 120576 come inpairs of the form

120594uarr(119902 119901) = (

1205941(119902 119901)

0) or 120594

darr(119902 119901) = (

0

1205942(119902 119901)

)

(45)

related by supersymmetry where 12059412(119902 119901) = 120595

12(119902)120601

12(119901)

The state space of the system is defined by all the normalizablesolutions of (44) and the individual states are characterizedby the energies 120576

119902and the fermionic quantumnumber119891 One

of these solutions is acceptable only if119882(119902) and119881(119901) becomeinfinite at both 119902 rarr plusmninfin and 119901 rarr plusmninfin respectivelywith the same sign If this condition is not satisfied neitherof the solutions is normalizable nor they can represent theground state of the system Equation (44) yields the followingrelations between energy eigenstates with fermionic quantumnumber plusmn12

[(120597

120597119902+119882

119902(119902)) minus (

120597

120597119901+ 119881

119901(119901))]120595

1(119902) 120601

1(119901)

= radic21205761199021205952 (119902) 1206011 (119901) minus radic21205761199011205951 (119902) 1206012 (119901)

(46)

[(minus120597

120597119902+119882

119902(119902)) minus (minus

120597

120597119901+ 119881

119901(119901))]120595

2(119902) 120601

2(119901)

= radic21205761199021205951 (119902) 1206012 (119901) minus radic2120576119901 1205952 (119902) 1206011 (119901)

(47)

where 120576 = 120576119902minus120576

119901and 120576

119902and 120576

119901are the eigenvalues of119867

119902and

119867119901 respectively The technique now is to devise an iterative

approximation scheme to solve (46) and (47) by taking atrial wave function for 120594

2(119902 119901) substitute this into the first

equation (46) and integrate it to obtain an approximationfor 120594

1(119902 119901) This can be used as an ansatz in the second

equation (47) to find an improved solution for 1205942(119902 119901) and

so forth As it was shown in [13] the procedure converges forwell-behaved potentials with a judicious choice of initial trialfunction If the119882

119902and 119881

119901are odd then

1205951(minus119902) = 120595

2(119902) 120601

1(minus119901) = 120601

2(119901) (48)

since they satisfy the same eigenvalue equation It is straight-forward then for example to obtain

[(120597

120597119902+119882

119902(119902)) minus (

120597

120597119901+ 119881

119901(119901))]120595

1(119902) 120601

1(119901)

= radic21205761199021205951 (minus119902) 1206011 (119901) minus radic21205761199011205951 (119902) 1206011 (minus119901)

(49)

The independent nature of 119902 and119901 gives the freedomof taking119902 = 0 119901 = 0 which yield an expression for energies asfollows

radic2120576119902 = 119882(0) +1205951015840

1(0)

1205951 (0)

radic2120576119901 = 119881 (0) +1206011015840

1(0)

1206011 (0)

(50)

Suppose that the potentials 119882119902(119902) and 119881

119901(119901) have a maxi-

mum at 119902minusand 119901

minus and minimum at 119902

+and 119901

+ respectively

For the simplicity sake we choose the trial wave functions as

120595(0)

12(119902) = 120575 (119902 minus 119902

plusmn) 120601

(0)

12(119901) = 120575 (119901 minus 119901

plusmn) (51)


After one iteration we obtain

120594(1)

1(119902 119901) =

1

119873119902119873119901

120579 (119902 minus 119902minus) 120579 (119901 minus 119901

minus)

times 119890minusint119902

01198891199021015840119882(1199021015840)+int119901

01198891199011015840119881(1199011015840)

120594(1)

2(119902 119901) =

1

119873119902119873119901

120579 (119902+minus 119902) 120579 (119901

+minus 119901)

times 119890int119902

01198891199021015840119882(1199021015840)minusint119901

01198891199011015840119881(1199011015840)

(52)

where 119873119902and 119873

119901are the normalization factors The next

approximation leads to

120594(2)

1(119902 119901) =

1

1198731015840119890minusint119902

01198891199021015840119882(1199021015840)+int119901

01198891199011015840119881(1199011015840)

times int

infin

max(minus119902119902minus)119890minus2int1199021015840

011988911990210158401015840119882(11990210158401015840)119889119902

1015840

times int

infin

max(minus119901119901minus)1198902 int1199011015840

011988911990110158401015840119881(11990110158401015840)119889119901

1015840

120594(2)

2(119902 119901) =

1

1198731015840119890int119902

01198891199021015840119882(1199021015840)minusint119901

01198891199011015840119881(1199011015840)

times int

min(minus119902119902+)

infin

1198902 int1199021015840

011988911990210158401015840119882(11990210158401015840)119889119902

1015840

times int

min(minus119901119901+)

infin

119890minus2int1199011015840

011988911990110158401015840119881(11990110158401015840)119889119901

1015840

(53)

It can be easily verified that to this level of precision (52) isself-consistent solution Actually for example for 1199021015840 gt 119902

minusthe

exponential 119890minus2int1199021015840

0119882(11990210158401015840)11988911990210158401015840

will peak sharply around 119902+and

may be approximated by a 120575-function 119888120575(119902+minus 119902

1015840) similarly

1198902 int1199021015840

0119882(11990210158401015840)11988911990210158401015840

may be replaced approximately by 119888120575(119902minusminus 119902

1015840)

for 1199021015840 lt 119902+ The same arguments hold for the 119901-space

With these approximations equations (53) reduce to (52)Thenormalization constant1198731015840 is

1198731015840= (int

infin

119902minus

119889119902119890minus2int119902

0119882(1199021015840)1198891199021015840

int

infin

119901minus

119889119901119890minus2 int1199011015840

0119881(1199011015840)1198891199011015840

)

32

(54)

The energy expectation value

120576 = (1205941 119867ext1205941) (55)

gives the same result as that obtained for odd potentialsby means of (50) and (53) Assuming the exponentials119890minus2int0

119902minus119882(119902)119889119902 and 119890minus2int

0

119901minus119881(119901)119889119901 to be small which is correct to

the same approximations underlying (55) the difference isnegligible and the integrals in both cases may be replacedby Gaussians around 119902

+and 119901

+ respectively Hence it is

straightforward to obtain

120576 =ℎ119882

1015840(119902+)

2120587119890minus2Δ119882ℎ

minusℎ119881

1015840(119901

+)

2120587119890minus2Δ119881ℎ

(56)

which gives direct evidence for the SUSY breaking in theextended phase space quantum mechanical system Here wehave reinstated ℎ to show the order of adopted approxima-tion and its non-perturbative nature We also denoted

Δ119882 = int

119902minus

119902+

119882(119902) 119889119902 Δ119881 = int

119901minus

119901+

119881 (119901) 119889119901 (57)

However a more practical measure for the SUSY breakingin particular in field theories is the expectation value ofan auxiliary field which can be replaced by its equation ofmotion right from the start as follows

⟨119865⟩ = (120594uarr 119894 119876+ 120590minus120594uarr) (58)

Taking into account the relation 119876+120594uarr

= 0 with 119876+

commuting with 119867ext which means that the intermediatestate must have the same energy as 120594 (58) can be written interms of a complete set of states as

⟨119865⟩ = 119894 (120594uarr 119876+120594darr) (120594darr 120590minus 120594uarr) (59)

According to (55) we have

120576 = ⟨119867ext⟩ = (120594uarr 119867ext120594uarr) =1

2(120594

uarr 119876

+120594darr) (120594

darr 119876

minus120594uarr)

= 120576119902minus 120576

119901= ⟨119867

119902⟩ minus ⟨119867

119901⟩ = (120595

uarr 119867

119902120595uarr) minus (120601

uarr 119867

119901120601uarr)

=1

2(120595

uarr 119876

119902+120595darr) (120595

darr 119876

119902minus120595uarr)

minus1

2(120601

uarr 119876

119901+120601darr) (120601

darr 119876


(60)

where 120594uarrdarr= 120595

uarrdarr120601uarrdarr 120595

uarr= (

1205951

0) 120595

darr= (

0

1205952) 120601

uarr= (

1206011

0) and

120601darr= (

0

1206012) From SUSY algebra it follows immediately that

1

radic120576119876minus120594uarr= 120594

darr=

minus120594uarr

1

radic120576119902

119876119902minus120595uarr= 120595

darr=

minus120595uarr

1

radic120576119901

119876119901minus120601uarr= 120601

darr=

minus120601uarr

(61)

By virtue of (61) (60) reads

radic120576 (120594uarr 119876

+120594darr) (120594

darr

minus120594uarr)

= radic120576119902 (120595uarr 119876119902+120595darr) (120595darr minus120595uarr)

minus radic120576119901 (120601uarr 119876119901+120601darr) (120601darr minus120601uarr)

(62)


Using the matrix representations of119876119902+ 119876

119901+ and

minusand the

wave functions equation (53) one gets [13]

(120595darr

minus120595uarr) = radic

1198821015840(119902+)

120587119890minusΔ119882

Δ119902

(120601darr


1198811015840(119901

+)

120587119890minusΔ119881

Δ119901

(120595uarr 119876

119902+120595darr) = 119894radic

1198821015840(119902minus)

120587119890minusΔ119882

(120601uarr 119876

119901+120601darr) = 119894radic

1198811015840(119901

minus)

120587119890minusΔ119881

(63)

where Δ119902 = 119902+minus 119902

minusand Δ119901 = 119901

+minus 119901

minus Hence

(120594uarr 119876

+120594darr) (120594

darr

minus120594uarr)

= 2119894radic120576[

[

(

120576119902

120576)radic120576

119902radic119882

1015840(119902minus)

120587Δ119902

minus (

120576119901

120576)radic120576

119901radic1198811015840(119901

minus)

120587Δ119901]

]

(64)

⟨119865⟩ = minus2radic120576 [

[

(

120576119902

120576)radic

1198821015840(119902+)

120587119890minus2Δ119882

Δ119902

minus (

120576119901

120576)radic

1198811015840(119901)

120587119890minus2Δ119881

Δ119901]

]

(65)

Along the actual trajectories in 119902-space (65) reproduces theresults obtained in [13]

5 An Extended SUSY Breaking in theInstanton Picture

In this Section our goal is to show that the expressionsequations (64) and (65) can be obtained in the path integralformulation of the theory by calculating the matrix elementsthat is the effect of tunneling between two classical vacuaby using a one-instanton background That is the matrixelements of

plusmn 119876

119902plusmn and 119876

119901plusmncan be calculated in the

background of the classical solution 119902119888= minus119882

119888and

119888= minus119881

119888

In doing this we rewrite the matrix element equation (62) interms of eigenstates of the conjugate operator

plusmnas follows

radic120576 [⟨+0119902+119901+

1003816100381610038161003816 119890minus119894119867ext(119879minus119905)119876

+119890minus119894119867ext(119879+119905) 1003816100381610038161003816 119902minus119901minus0minus⟩⟨minus0119902minus119901minus

1003816100381610038161003816

times 119890minus119894119867ext(119879minus119905)

minus119890minus119894119867ext(119879+119905) 1003816100381610038161003816 119902+119901+0+⟩]119879rarrminus119894infin

= radic120576119902 [⟨+0119902+1003816100381610038161003816 119890

minus119894119867119902(119879minus119905)119876119902+119890minus119894119867119902(119879+119905) 1003816100381610038161003816 119902minus0minus⟩⟨minus0119902minus

1003816100381610038161003816

times 119890minus119894119867119902(119879minus119905)

minus

10038161003816100381610038161003816119890minus119894119867119902(119879+119905)119902

+0+⟩]

119879rarrminus119894infin

minus radic120576119901 [⟨+0119901+1003816100381610038161003816 119890minus119894119867119901(119879minus119905)119876

119901+119890minus119894119867119901(119879+119905)

times1003816100381610038161003816119901minus0minus⟩⟨minus0119901minus

1003816100381610038161003816 minus

10038161003816100381610038161003816119901+0+⟩ ]


(66)

in the limit 119879 rarr minus119894infin This reduces to

radic120576 ⟨+0119902+119901+

1003816100381610038161003816119876+1003816100381610038161003816 119902minus119901minus0minus⟩⟨minus0119902minus119901minus

1003816100381610038161003816 minus1003816100381610038161003816 119902+119901+0+⟩

= radic120576119902 ⟨+0119902+1003816100381610038161003816119876119902+

1003816100381610038161003816 119902minus0minus⟩⟨minus0119902minus1003816100381610038161003816 minus

1003816100381610038161003816 119902+0+⟩

minus radic120576119901 ⟨+0119901+1003816100381610038161003816119876119901+

1003816100381610038161003816119901minus0minus⟩⟨minus0119901minus1003816100381610038161003816 minus

1003816100381610038161003816119901+0+⟩

(67)

which in turn can be presented by path integrals definedin terms of anticommuting 119888-number operators 120577 and 120578

with Euclidean actions of the instantons in 119902- and 119901-spacesrespectively Following [13] these functional integrals includean integration over instanton time 120591

0which is due to the

problem of zero modes of the bilinear terms in Euclideanactions This arises from time-transformation of instantonsand SUSY transformations on them respectively The exis-tence of zero modes gives rise to non-Gaussian behaviour ofthe functional integral Due to it the matrix elements abovedo not receive any contributions from either no-instantonor anti-instanton configurations The zero mode problem issolved by introducing a collective coordinate 120591

0replacing the

bosonic zero mode [14] Whereas the functional integralsdepend only on the difference 120591 minus 120591

0 Note also that multi-

instanton configurations could contribute in principle pro-vided they have not more than one normalizable fermioniczero mode But as it was shown in [13] their contribution isclearly smaller with respect to radic2120576119902 and radic2120576119901 In the casewhen the SUSY potentials in 119902- and 119901-spaces have morethan two extrema 119902] and 119901120583 ] 120583 = 1 2 119873 one can putconditions on the SUSY potentials as follows

int

infin

0

119882(1199021015840) 119889119902

1015840997888rarr infin at 119902 997888rarr plusmninfin for 120595+

0

int

infin

0

119882(1199021015840) 119889119902

1015840997888rarr minusinfin at 119902 997888rarr plusmninfin for 120595minus

0

(68)

and similar to 119881(119901) that the extrema are well separated asfollows int119902]+1

119902]

119882(1199021015840)119889119902

1015840≫ 1 int119902120583+1

119901120583

119881(1199011015840)119889119901

1015840≫ 1 Around

each of the classical minima 119902] and119901120583 of the potentials1198822(119902)

and 1198812(119901) respectively one can approximate the theory

by a supersymmetric harmonic oscillator Then there are119873 ground states which have zero energy These states aredescribed by upper or lower component of the wave functiondepending on whether ] and 120583 are odd or even With thisprovision the functional integrals are calculated in [13] whichallow us consequently to write

⟨+0119902+

1003816100381610038161003816119876119902+1003816100381610038161003816 119902minus0minus⟩ = 119894

radic119882

1015840

119888(119902+)

120587119890minusΔ119882119888

⟨minus0119902minus

1003816100381610038161003816 minus1003816100381610038161003816 119902+0+⟩ =

radic119882

1015840

119888(119902+)

120587119890minusΔ119882119888Δ119902

119888

(69)


and so forth Inserting this in (67) we arrive at (64)

⟨+0119902+119901+


1003816100381610038161003816 minus1003816100381610038161003816 119902+119901+0+⟩

= 2119894radic120576[

[

(

120576119902

120576)radic120576119902

radic119882

1015840

119888(119902+)

120587Δ119902

119888

minus (

120576119901

120576)radic120576119901

radic1198811015840

119888(119901

+)

120587Δ119901

119888]

]

(70)

and thus of (65) as its inevitable corollary This proves thatthe extended SUSY breaking has resulted from tunnelingbetween the classical vacua of the theory The correctionsto this picture are due to higher-order terms and quantum-tunneling effects

6 Spectrum Generating Algebra

An extended Hamiltonian 119867ext equation (27) can be treatedas a set of two ordinary two-dimensional partner Hamiltoni-ans [4] as follows

119867plusmn=1

2[120587

2

119902minus 120587

2

119901+ 119880

plusmn(119902 119901)] (71)

provided by partner potentials

119880plusmn(119902 119901) = 119880

119902plusmn(119902) minus 119880

119901plusmn(119901)

119880119902plusmn(119902) = 119882

2(119902 119886

119902) ∓119882

1015840

119902(119902 119886

119902)

119880119901plusmn(119901) = 119881

2(119901 119886

119901) ∓ 119881

1015840

119901(119901 119886

119901)

(72)

A subset of the SUSY potentials for which the Schrodinger-like equations are exactly solvable share an integrabilityconditions of shape-invariance [15] as follows

119880+(1198860 119902 119901) = 119880

minus(1198861 119902 119901) + 119877 (119886

0) 119886

1= 119891 (119886

0)

(73)

where 1198860and 119886

1are a set of parameters that specify phase-

space-independent properties of the potentials and thereminder 119877(119886

0) is independent of (119902 119901)

61 Algebraic Properties of Shape Invariance Using the stan-dard technique we may construct a series of Hamiltonians119867119873 119873 = 0 1 2

119867119873=1

2[120587

2

119902minus 120587

2

119901+ 119880

minus(119886119873 119902 119901) +

119873

sum

119896=1

119877 (119886119896)] (74)

where 119886119873= 119891

(119873)(119886) (119873 is the number of iterations) From

(72) and (73) we obtain then 119873 = 119899 + 119898 coupled nonlineardifferential equations which are the two recurrence relationsof Riccati-type differential equations as follows

1198822

119899+1(119902) + 119882

1015840

119902(119899+1)(119902) = 119882

2

119899(119902) minus 119882

1015840

119902119899(119902) minus 120583

119899

1198812

119898+1(119901) + 119881

1015840

119901(119898+1)(119901) = 119881

2

119898(119901) minus 119881

1015840

119901119898(119901) minus ]

119898

(75)

where we denote 119882119899(119902) equiv 119882(119902 119886

119902119899) 119881

119898(119901) equiv 119881(119901 119886

119901119898)

120583119899equiv 119877

119902(119886119902119899) and ]

119898equiv 119877

119901(119886119901119898) Here we admit that

for unbroken SUSY the eigenstates of the potentials 119880119902119901

respectively are

119864(minus)

1199020= 0 119864

(minus)

119902119899=

119899minus1

sum

119894=0

120583119894 119864

(minus)

1199010= 0 119864

(minus)

119901119898=

119898minus1

sum

119895=0

]119895

(76)

that is the ground states are at zero energies characteristic ofunbroken supersymmetryThe differential equations (75) canbe investigated to find exactly solvable potentials The shapeinvariance condition equation (73) can be expressed in termsof bosonic operators as

119861+(119909 119886

0) 119861

minus(119909 119886

0) minus 119861

minus(119909 119886

1) 119861

+(119909 119886

1)

= 119877 (1198860) = 120583

0minus ]

0

119861aelig+ (aelig 119886aelig0) 119861aeligminus (aelig 119886aelig0) minus 119861aeligminus (aelig 119886aelig1) 119861aelig+ (aelig 119886aelig1)

= (1205830or ]

0)

(77)

where 119909(119902 119901)- is the coordinate in (119902 119901)-space aelig denotesconcisely either 119902- or 119901-representations (no summation onaelig is assumed) To classify algebras associated with the shapeinvariance following [16] we introduce an auxiliary variables120601(120601

119902 120601119901) and define the following creation and annihilation

operators

119869+= 119890

119894119896120601119861+(119909 120594 (119894120597

120601)) 119869

minus= 119861

minus(119909 120594 (119894120597

120601)) 119890

minus119894119896120601

(78)

where 119896(119896119902 119896119901) are an arbitrary real constants and 120594(120594

119902 120594119901)

are an arbitrary real functions Consequently the creationand annihilation operators in (119902 119901)-spaces can be written as

119869aelig+ = 119890119894119896aelig120601aelig119861aelig+ (aelig 120594aelig (119894120597120601aelig))

119869aeligminus = 119861aeligminus (aelig 120594aelig (119894120597120601aelig)) 119890minus119894119896aelig120601aelig

(79)

The operators 119861plusmn(119909 120594(119894120597

120601)) and 119861aeligplusmn(aelig 120594aelig(119894120597120601aelig )) are the

generalization of (77) where 1198860rarr 120594(119894120597

120601) and 119886aelig0 rarr

120594aelig(119894120597120601aelig ) One can easily prove the following relations

119890119894119896120601119861+(119909 120594 (119894120597

120601)) = 119861

+(119909 120594 (119894120597

120601+ 119896)) 119890

119894119896120601

119861minus(119909 120594 (119894120597

120601)) 119890

minus119894119896120601= 119890

minus119894119896120601119861minus(119909 120594 (119894120597

120601+ 119896))

(80)

and that

119890119894119896aelig120601aelig119861aelig+ (aelig 120594aelig (119894120597120601aelig)) = 119861aelig+ (aelig 120594aelig (119894120597120601aelig + 119896aelig)) 119890

119894119896aelig120601aelig

119861aeligminus (aelig 120594aelig (119894120597120601aelig)) 119890minus119894119896aelig120601aelig = 119890

minus119894119896aelig120601aelig119861aeligminus (aelig 120594aelig (119894120597120601aelig + 119896aelig)) (81)

If we choose a function 120594(119894120597120601) such that 120594(119894120597

120601+ 119896) =

119891[120594(119894120597120601)] then we have identified 119886

0rarr 120594(119894120597

120601) and

1198861= 119891(119886

0) rarr 119891[120594(119894120597

120601)] = 120594(119894120597

120601+ 119896) Similar relations can


be obtained for the aelig-representations From (77) we obtainthen

119861+(119909 120594 (119894120597

120601)) 119861

minus(119909 120594 (119894120597

120601))

minus 119861minus(119909 120594 (119894120597

120601+ 119896)) 119861

+(119909 120594 (119894120597

120601+ 119896)) = 119877 [120594 (119894120597

120601)]

119861aelig+ (aelig 120594aelig (119894120597120601aelig)) 119861aeligminus (aelig 120594 (119894120597120601aelig))

minus 119861aeligminus (aelig 120594aelig (119894120597120601aelig + 119896aelig)) 119861aelig+ (aelig 120594aelig (119894120597120601aelig + 119896aelig))

= 119877aelig [120594aelig (119894120597120601aelig)]

(82)

Introducing the operators 1198693= minus(119894119896)120597

120601and 119869aelig3 = minus(119894119896aelig)120597120601aelig

and combining (79) and (82) wemay arrive at a deformed Liealgebras as follows

[119869+ 119869minus] = [119869

119902+ 119869119902minus] minus [119869

119901+ 119869119901minus]

= 120585 (1198693) = 120585

119902(1198691199023) minus 120585

119901(1198691199013)

[1198693 119869plusmn] = plusmn119869

plusmn [119869aelig+ 119869aeligminus] = 120585aelig (119869aelig3)

[119869aelig3 119869aeligplusmn] = plusmn119869aeligplusmn

(83)

where 120585(1198693) equiv 119877[120594(119894120597

120601)] and 120585aelig(119869aelig3) equiv 119877aelig[120594aelig(119894120597120601aelig )] define

the deformations Different 120594 functions in (82) define differ-ent reparametrizations corresponding to several models Forexample

(1) the translational models (1198861= 119886

0+ 119896) correspond to

120594(119911) = 119911 If 119877 is a linear function of 1198693 the algebra

becomes SO(21) or SO(3) Similar in many respectsprediction is made in somewhat different method byBalantekin [17]

(2) the scaling models (1198861= 119890

1198961198860) correspond to 120594(119911) =

119890119911 and so forth

62 The Unitary Representations of the Deformed Lie AlgebraIn order to find the energy spectrum of the partner SUSYHamiltonians

2119867minus(119909 120594 (119894120597

120601)) = 119861

minus(119909 120594 (119894120597

120601+ 119896))

times 119861+(119909 120594 (119894120597

120601+ 119896))

2119867aeligminus (aelig 120594aelig (119894120597120601aelig)) = 119861aeligminus (aelig 120594aelig (119894120597120601aelig + 119896aelig))

times 119861aelig+ (aelig 120594aelig (119894120597120601aelig + 119896aelig))

(84)

one must construct the unitary representations of deformedLie algebra defined by (83) [16 18] Using the standardtechnique one defines up to additive constants the functions119892(119869

3) and 119892aelig(119869aelig3) as follows

120585 (1198693) = 119892 (119869

3) minus 119892 (119869

3minus 1)

120585aelig (119869aelig3) = 119892aelig (119869aelig3) minus 119892aelig (119869aelig3 minus 1) (85)

The Casimirs of this algebra can be written as 1198622= 119869

minus119869++

119892(1198693) and accordingly 119862aelig2 = 119869aeligminus119869aelig+ + 119892aelig(119869aelig3) In a basis

in which 1198693and 119862

2are diagonal 119869

minusand 119869

+are lowering

and raising operators (the same holds for aelig-representations)Operating on an arbitrary state |ℎgt they yield

1198693 | ℎ⟩ = ℎ |ℎ⟩ 119869

minus | ℎ⟩ = 119886 (ℎ) |ℎ minus 1⟩

119869+ | ℎ⟩ = 119886

lowast(ℎ + 1) |ℎ + 1⟩

(86)

where

|119886 (ℎ)|2minus |119886 (ℎ + 1)|

2= 119892 (ℎ) minus 119892 (ℎ minus 1) (87)

Similar arguments can be used for the operators 119869(119901119902)3

119862aelig2 and 119869aeligplusmn which yield similar relations for the statesℎaelig The profile of 119892(ℎ) (and thus of 119892aelig(ℎaelig)) determinesthe dimension of the unitary representation Having therepresentation of the algebra associated with a characteristicmodel consequently we obtain the complete spectrum of thesystem For example without ever referring to underlyingdifferential equation we may obtain analytic expressions forthe entire energy spectrum of extended Hamiltonian withself-similar potential A scaling change of parameters is givenas 119886

1= 119876119886

0 119886aelig1 = 119876aelig119886aelig0 at the simple choice119877(119886

0) = minus119903

11198860

where 1199031is a constant That is

120585 (1198693) equiv minus119903

1exp (minus119896119869

3) = 120585

119902(1198691199023) minus 120585

119901(1198691199013)

= minus1199031199021exp (minus119896

1199021198691199023) + 119903

1199011exp (minus119896

1199011198691199013)

(88)

which yields

[119869+ 119869minus] = 120585 (119869

3) = minus119903

1exp (minus119896119869

3) [119869

3 119869plusmn] = plusmn119869

plusmn (89)

This is a deformation of the standard SO(21) Lie algebra andtherefore one gets

119892 (ℎ) =1199031

119890119896 minus 1119890minus119896ℎ

= minus1199031

1 minus 119876119876minusℎ 119876 = 119890

119896

119892aelig (ℎaelig) =119903aelig1

119890119896aelig minus 1119890minus119896aeligℎaelig = minus

119903aelig11 minus 119876aelig

119876minusℎaeligaelig

119876aelig = 119890119896aelig

(90)

For scaling problems [16] one has 0 lt 119902 lt 1 whichleads to 119896 lt 0 The unitary representation of this algebrafor monotonically decreasing profile of the function 119892(ℎ) isinfinite dimensional Let the lowest weight state of the 119869

3be

ℎmin and then 119886(ℎmin) = 0 One can choose the coefficients119886(ℎ) to be real From (87) for an arbitrary ℎ = ℎmin + 119899119899 = 0 1 2 we obtain

119886(ℎ)2= 119892 (ℎ minus 119899 minus 1) minus 119892 (ℎ minus 1) = 1199031

119876119899minus 1

119876 minus 11198761minusℎ (91)

The spectrum of the extended Hamiltonian119867minus(119909 119886

1) reads

119867minus | ℎ⟩ = 119886(ℎ)

2| ℎ⟩ = 1199031

119876119899minus 1

119876 minus 11198761minusℎ| ℎ⟩ (92)


with the eigenenergies

119864119899 (ℎ) = 1199031120572 (ℎ)

119876119899minus 1

119876 minus 1 120572 (ℎ) equiv 119876

1minusℎ (93)

Similar expressions can be obtained for the119867aeligminus and eigenen-ergies 119864

1199021198991and 119864

1199011198992(119899 equiv (119899

1 1198992) 119899

12= 0 1 2 ) as

119867119902minus

10038161003816100381610038161003816ℎ119902⟩ = 119886

119902(ℎ

119902)2 10038161003816100381610038161003816ℎ119902⟩ = 119903

1199021

1198761198991119902minus 1

119876119902minus 1

1198761minusℎ119902

119902

10038161003816100381610038161003816ℎ119902⟩

119867119901minus

10038161003816100381610038161003816ℎ119901⟩ = 119886

119901(ℎ

119901)2 10038161003816100381610038161003816ℎ119901⟩ = 119903

1199011

1198761198992119901minus 1

119876119901minus 1

1198761minusℎ119901

119901

10038161003816100381610038161003816ℎ119901⟩

(94)

and that

1198641199021198991(ℎ

119902) = 119903

1199021120572119902(ℎ

119902)

1198761198991119902minus 1

119876119902minus 1

120572119902(ℎ

119902) equiv 119876

1minusℎ119902

119902

1198641199011198992(ℎ

119901) = 119903

1199011120572119901(ℎ

119901)

1198761198992119901minus 1

119876119901minus 1

120572119901(ℎ

119901) equiv 119876

1minusℎ119901

119901

(95)

Hence

119864119899 (ℎ) = 1198641199021198991

(ℎ119902) minus 119864

1199011198992(ℎ

119901) (96)

7 Conclusions

Weaddressed the classical analog of the extended phase spacequantummechanics of particle which have both bosonic andfermionic degrees of freedom that is the particle with odddegrees of freedom which gives rise to (119873 = 2)-realizationof the supersymmetry algebra We obtain the integrals ofmotion We use the iterative scheme to find the approximateground state solutions to the extended Schrodinger-likeequation and calculate the parameters which measure thebreaking of extended SUSY such as the ground state energyThe approximation which went into the derivation of solu-tions of (44) meets our interest that the ground state energy120576 is supposedly smallThis gives direct evidence for the SUSYbreaking However we calculate a more practical measure forthe SUSY breaking in particular in field theories which is theexpectation value of an auxiliary field We analyze in detailthe nonperturbative mechanism for extended phase spaceSUSY breaking in the instanton picture and show that thishas resulted from tunneling between the classical vacua ofthe theory Finally we present an analysis on the independentgroup theoretical methods with nonlinear extensions of Liealgebras from the extended phase space SUSY quantummechanics Using the factorization procedure we explorethe algebraic property of shape invariance and spectrumgenerating algebra Most of these Hamiltonians possess thisfeature and hence are solvable by an independent grouptheoretical methodWe construct the unitary representationsof the deformed Lie algebra

Acknowledgments

The author would like to thank Y Sobouti for drawinghis attention to the extended phase space formulation ofquantum mechanics The knowledgable comments from theanonymous referees are much appreciated

References

[1] Y Sobouti and S Nasiri ldquoA phase space formulation of quantumstate functionsrdquo International Journal of Modern Physics B vol7 no 18 pp 3255ndash3272 1993

[2] S Nasiri Y Sobouti and F Taati ldquoPhase space quantummechanicsmdashdirectrdquo Journal ofMathematical Physics vol 47 no9 Article ID 092106 15 pages 2006

[3] G T Ter-Kazarian and Y Sobouti ldquoAn extended phase-spacestochastic quantization of constrained Hamiltonian systemsrdquoJournal of Physics A vol 41 no 31 Article ID 315303 8 pages2008

[4] G Ter-Kazarian ldquoAn extended phase-space SUSY quantummechanicsrdquo Journal of Physics A vol 42 no 5 Article ID055302 12 pages 2009

[5] H Nicolai ldquoSupersymmetry and spin systemsrdquo Journal ofPhysics A vol 9 no 9 pp 1497ndash1506 1976

[6] E Witten ldquoDynamical breaking of supersymmetryrdquo NuclearPhysics B vol 188 no 3 pp 513ndash554 1981

[7] E Witten ldquoConstraints on supersymmetry breakingrdquo NuclearPhysics B vol 202 no 2 pp 253ndash316 1982

[8] A Inomata and G Junker ldquoQuasiclassical path-integralapproach to supersymmetric quantum mechanicsrdquo PhysicalReview A vol 50 no 5 pp 3638ndash3649 1994

[9] A Inomata H Kuratsuji and C C Gerry Path Integrals andCoherent States of SU(2) and SU(11) Singapore 1992

[10] J Gamboa andMPlyushchay ldquoClassical anomalies for spinningparticlesrdquoNuclear Physics B vol 512 no 1-2 pp 485ndash504 1998

[11] G Grignani M Plyushchay and P Sodano ldquoA pseudoclassicalmodel for 119875 119879-invariant planar fermionsrdquo Nuclear Physics Bvol 464 no 1-2 pp 189ndash212 1996

[12] J L Cortes M S Plyushchay and L Velazquez ldquoA pseudo-classical model for the massive Dirac particle in 119889 dimensionsrdquoPhysics Letters B vol 306 no 1-2 pp 34ndash40 1993

[13] P Salomonson and J W van Holten ldquoFermionic coordinatesand supersymmetry in quantummechanicsrdquoNuclear Physics Bvol 196 no 3 pp 509ndash531 1982

[14] A M Polyakov ldquoQuark confinement and topology of gaugetheoriesrdquo Nuclear Physics B vol 120 no 3 pp 429ndash458 1977

[15] L Gendenshtein ldquoDerivation of exact spectra of the Schrdingerequation by means of supersymmetryrdquo Pisma Zhurnal Eksperi-mental noi i Teoreticheskoi Fiziki vol 38 pp 299ndash302 1983

[16] A Gangopadhyaya J Mallow and U P Sukhatme ldquoShapeinvariance and its connection to potential algebrardquo in Supersym-metry and IntegrableModels vol 502 of Lecture Notes in Physicspp 341ndash350 Springer 1998

[17] A B Balantekin ldquoAlgebraic approach to shape invariancerdquoPhysical Review A vol 57 no 6 pp 4188ndash4191 1998

[18] F Cooper A Khare and U Sukhatme ldquoSupersymmetry andquantummechanicsrdquo Physics Reports vol 251 no 5-6 pp 267ndash385 1995

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

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Stochastic AnalysisInternational Journal of


2 The Integrals of Motion

For the benefit of those not familiar with the framework ofextended phase space quantization enough details are givenin the following tomake the rest of the paper understandableThe interested reader is invited to consult the original papers[1 2] for further details In the framework of the proposedformalism the extended Lagrangian can be written as

Lext (119901 119902 119902) = minus 119902119894119901119894minus 119902

119894119894+L

119902+L

119901 (1)

where a dynamical system with 119873 degrees of freedomdescribed by the 2119873 independent coordinates 119902= (119902

1 119902

119873)

and momenta 119901 = (1199011 119901

119873) which are not in gen-

eral canonical pairs A Lagrangian L119902(119902 119902) is given in

119902-representation and the corresponding L119901(119901 ) in 119901-

representation Adotwill indicate differentiationwith respectto 119905 The independent nature of 119901 and 119902 gives the freedom ofintroducing a second set of canonical momenta for both 119901and 119902 through the extended Lagrangian

120587119902119894=120597Lext120597 119902

119894

=120597L119902

120597 119902119894

minus 119901119894

120587119901119894=120597Lext120597

119894

=120597L119901

120597119894

minus 119902119894

(2)

One may now define an extended Hamiltonian

119867ext (119901 119902 120587119901 120587119902) = 120587119902119894 119902119894 + 120587119901119894 119894 minusLext (119901 119902 119902)

= 119867 (119901 + 120587119902 119902) minus 119867 (119901 119902 + 120587

119901)

(3)

where 119867(119901 119902) = 119901119894119902119894minusL119902

= 119902119894119894minusL119901 is the conventional

Hamiltonian of the system In particular vanishing of 120587119902or

120587119901is the condition for 119901 and 119902 to constitute a canonical

pair In the language of statistical quantum mechanics thischoice picks up a pure state (actual path) Otherwise one isdealing with a mixed state (virtual path) One may howeverenvisage that the full machinery of the conventional quantummechanical dynamics is extendible to the extended dynamicsas alluded to the above Here 119901 and 119902 will be considered asindependent 119888-number operators on the integrable complexfunction 120594(119902 119901) One of the key assumptions of extendedphase space quantization [1 2] is the differential operatorsand commutation brackets for 120587

119901and 120587

119902borrowed from the

conventional quantum mechanics as follows

120587119902119894= minus119894

120597

120597119902119894

[120587119902119894 119902119895] = minus119894120575

119894119895

120587119901119894= minus119894

120597

120597119901119894

[120587119901119894 119901

119895] = minus119894120575

119894119895

(4)

Note also the following

[119901119894 119902119895] = [119901

119894 119901

119895] = [119902

119894 119902119895] = [120587

119901119894 120587

119902119895]

= [120587119901119894 120587

119901119895] = [120587

119902119894 120587

119902119895] = 0

(5)

By the virtue of (4) and (5) 119867ext is now an operator on 120594Along the trajectories in (119901 119902)-space however it producesthe state functions 120594(119901 119902 119905) via the following Schrodinger-like equation

119894120597

120597119905120594 = 119867ext 120594 (6)

Solutions of (6) are

120594 (119902 119901 119905) = 120594119903(119902 119901 119905) 119890

minus119894119901119902

= 119886120572120573120595120572(119902 119905) 120601

lowast

120573(119901 119905) 119890

minus119894119901119902

(7)

where 119886 = 119886dagger positive definite tr 119886 = 1 and 120595

120572and

120601lowast

120572are solutions of the conventional Schrodinger equation

in 119902- and 119901-representations respectively They are mutuallyFourier transforms Note that the 120572 and 120573 are not in generaleigenindices The normalizable 120594 (int120594119889119901119889119902 = tr(119886) = 1)is a physically acceptable solution The exponential factor isa consequence of the total time derivative minus119889(119902119901)119889119905 in (1)which can be eliminated Actually it is easily verified that

(119901 + 120587119902) 120594

119903(119902 119901 119905) = (120587

119902120594119903(119902 119901 119905)) 119890

minus119894119901119902 (8)

and so on Substitution of (8) in (6) gives

119894120597

120597119905120594 (119902 119901 119905) = 119867ext 120594119903 (119902 119901 119905) (9)

provided by the reducedHamiltonian119867ext From now onwereplace119867ext by119867ext and 120594119903(119902 119901 119905) by 120594119903(119902 119901 119905) respectivelyand retain former notational conventions

It is certainly desirable to derive the classical analog of theextended phase space quantummechanics of the particlewithodd degrees of freedom directly from what may be taken asthe first principleTherefore following [10ndash12] let us considera nonrelativistic particle of unit mass with two (120572 = 1 2)odd (Grassmann) degrees of freedomThe classical extendedLagrangian equation (1) can be written

Lext (119901 119902 119902) = minus 119902119901 minus 119902 +1

21199022minus 119865 (119902) +

1

22

minus 119866 (119901) minus 119877 (119902 119901)119873 +1

2120595120572120572

(10)

provided by 119873 = 12059511205952= minus119894120595

+120595minus Here 119865(119902) R rarr

R 119866(119901) R rarr R and 119877(119902 119901) R rarr R are arbitrarypiecewise continuously differentiable functions given overthe 1-dimensional Euclidean space R The 120595

120572are two odd

(Grassmann) degrees of freedom The nontrivial Poisson-Dirac brackets of the system (10) are

119902 120587119902 = 1 119901 120587

119901 = 1 120595

120572 120595

120573 = 120575

120572120573

120595+ 120595

minus = 1 120595

2

plusmn= 0 120595

plusmn=1

radic2(120595

1plusmn 119894120595

2)

(11)

The extended Hamiltonian119867ext (3) reads

119867ext (119901 119902 120587119901 120587119902) =1

2(119901 + 120587

119902)2

+ 1198652(119902) minus

1

2(119902 + 120587

119901)2

minus 1198662(119901) + 119877 (119902 119901)119873

(12)



119867ext (119901 119902 120587119901 120587119902) =1

21205872

119902+ 119865

2(119902) minus

1

21205872

119901

minus 1198662(119901) + 119877 (119902 119901)119873

(13)


119902 = 120587119902 = 120587

119901

119902= minus119865

1015840

119902(119902) minus 119877

1015840

119902(119902 119901)119873

119901= minus119866

1015840

119901(119901) + 119877

1015840

119901(119902 119901)119873

plusmn= plusmn119894119877 (119902 119901) 120595

plusmn

(14)



119905

1199050

119877 (119902 (120591) 119901 (120591)) 119889120591] (15)


120579plusmn= 120579

plusmn (119905) exp [∓119894 int119905

1199050

119877 (119902 (120591) 119901 (120591)) 119889120591] (16)



plusmn= 120579



(119861119902119901+

)lowast= 119861

119902119901minus


119876119902119901plusmn

= 119861119902119901∓

120595plusmn (17)



119861119902∓L

2(R) 997888rarrL

2(R) 119861

119902∓= [119901 + 120587

119902plusmn 119894119882 (119902)]

119861119901∓L

2(R) 997888rarrL

2(R) 119861

119901∓= [119902 + 120587

119901plusmn 119894119881 (119901)]

(18)


119902(119902)minus119877

119901(119901)


we obtain

119902∓= [minus (119865

1015840

119902+ 119877

1015840

119902119873) plusmn 119894119882

1015840

119902(119902) (119901 + 120587

119902)]

119901∓= [minus (119866

1015840

119901+ 119877

1015840

119901119873) plusmn 119894119881

1015840

119901(119901) (119902 + 120587

119901)]

(19)

Consequently

119902plusmn= plusmn119894 [(119882

1015840

119902minus 119877

1015840

119902plusmn) plusmn 119894 (119865

1015840

119902minus119882119882

1015840

119902) 120595

∓]

119901plusmn= plusmn119894 [(119881

1015840

119901minus 119877

1015840

119901plusmn) plusmn 119894 (119866

1015840

119901minus 119881119881

1015840

119901) 120595

∓]

(20)


119901plusmn= 0 when 1198821015840

119902(119902) =

1198771015840

119902plusmn(119902) and 119865

1015840

119902= (12)(119882

2)1015840

119902or 1198811015840

119901(119901) = 119877

1015840

119901plusmn(119901) and

1198661015840

119901= (12)(119881

2)1015840


119877119902119901


1198771015840

119902plusmn(119902) = 119882

1015840

119902(119902) 119865

119902=1

2(119882

2) + 119862

119902

1198771015840

119901plusmn(119901) = 119881

1015840

119901(119901) 119865

119901=1

2(119881

2) + 119862

119901

(21)

where 119862119902119901



minus 119867119901

where

119867119902=1

21205872

119902+ 119865

2(119902) + 119877

119902119873

119867119901=1

21205872

119901+ 119866

2(119901) + 119877

119901119873

(22)

Then 119876119902119901plusmn


119901form the


119876119902119901+

119876119902119901minus

= minus119894 (119867119902119901minus 119862

119902119901)

119867119902119901 119876

119902119901plusmn = 119876

119902119901plusmn 119876

119902119901plusmn = 0

119873119876119902119901plusmn

= plusmn119894119876119902119901plusmn

119873119867119902119901 = 0

(23)





Lext (119901 119902 119902) =1

21205872

119902minus1

2119882

2(119902) +

1

21205872

119901minus1

21198812(119901)

+ 12059511205952(119882

1015840

119902+ 119881

1015840

119901) +

1

2120595120572120572

(24)








plusmn= radic

1

2(

1plusmn 119894

2)

120572

120573 = 120575

120572120573

+

minus = 1

2

plusmn= 0

(25)



1plusmn120590





119891 =

(12)[+



120594 (119902 119901) = (120594+12

(119902 119901)

120594minus12

(119902 119901)) = (

1205941(119902 119901)

1205942(119902 119901)

) = (

1205951(119902) 120601

1(119901)

1205952(119902) 120601

2(119901))

(26)

where the states 12059512(119902) 120601



0otimes C2

= [L2(R) otimes



119867ext = (119867+

0

0 119867minus

) =1

2(

2

119902+119882

2(119902) + 119894119882

1015840

119902(119902) [

1

2])

minus1

2(

2

119901+ 119881

2(119901) + 119894119881

1015840

119902(119901) [

1

2])

(27)


Lext (119901 119902 119902) = minus 119902119901 minus 119902 +1

2[(119889119902

119889119905)

2

minus1198822(119902)]

+ 1198911198821015840

119902(119902) +

1

2[(119889119901

119889119905)

2

minus 1198812(119901)]

+ 1198911198811015840

119901(119901)

(28)




minusas follows



|00+⟩ = + |00minus⟩ = 0 (30)

then +|00+⟩ = 0


that daggerplusmn=

∓ we get

⟨∓00| plusmn = 0 ⟨∓00| ∓ = ⟨plusmn00| (31)



120572plusmn= minus




|00minus⟩ (33)

and thus

1199021003816100381610038161003816119902120572minus⟩ = 119902

1003816100381610038161003816119902120572minus⟩ minus

1003816100381610038161003816119902120572minus⟩ = 1205721003816100381610038161003816119902120572minus⟩ (34)

Then the 119902and



1003816100381610038161003816 119902120573+⟩ = minusint119889120572119890120572120573 1003816100381610038161003816 119902120572minus⟩

10038161003816100381610038161003816120587119902120572plusmn⟩ = minusint119889119902119890

119894119902120587119902 1003816100381610038161003816 119902120572plusmn⟩

1003816100381610038161003816119901120573+⟩ = minusint119889120572119890120572120573 1003816100381610038161003816119901120572minus⟩

10038161003816100381610038161003816120587119901120572plusmn⟩ = minusint119889119901119890

119894119901120587119901 1003816100381610038161003816119901120572plusmn⟩

(35)

which gives

119902

10038161003816100381610038161003816120587119902120572plusmn⟩ = 120587

119902

10038161003816100381610038161003816120587119902120572plusmn⟩

+

10038161003816100381610038161003816(119902 120587

119902) 120573+⟩ = 120573

10038161003816100381610038161003816(119902 120587

119902) 120573+⟩

119901

10038161003816100381610038161003816120587119901120572plusmn⟩ = 120587

119901

10038161003816100381610038161003816120587119901120572plusmn⟩

+

10038161003816100381610038161003816(119901 120587

119901) 120573+⟩ = 120573

10038161003816100381610038161003816(119901 120587

119901) 120573+⟩

(36)


minus int1198891205721198891199021003816100381610038161003816 119902120572plusmn⟩⟨∓120572

lowast1199021003816100381610038161003816 = 1

minus int119889120572

119889120587119902

2120587

10038161003816100381610038161003816120587119902120572plusmn⟩⟨∓120572

lowast120587119902

10038161003816100381610038161003816= 1

minus int1198891205721198891199011003816100381610038161003816119901120572plusmn⟩⟨∓120572

lowast1199011003816100381610038161003816 = 1

minus int119889120572

119889120587119901

2120587

10038161003816100381610038161003816120587119901120572plusmn⟩⟨∓120572

lowast120587119901

10038161003816100381610038161003816= 1

(37)


120594minus(119902120572119901120573119905) = minusint119889120572

1015840119889119902

1015840119889120573

1015840119889119901

1015840119870(119902120572119901120573119905 | 119902

10158401205721015840119901101584012057310158401199051015840)

(38)

The kernel reads

119870(119902120572119901120573119905 | 11990210158401205721015840119901101584012057310158401199051015840)

= ⟨+119902120572lowast119901120573


1015840) 100381610038161003816100381610038161199021015840120572101584011990110158401205731015840⟩

(39)


K1198911198911015840 (119902119901119905 | 119902

101584011990110158401199051015840) = ⟨119902119901119891

1003816100381610038161003816 119890minus119894119867ext(119905minus119905

1015840) 10038161003816100381610038161003816119902101584011990110158401198911015840⟩ (40)




K1198911198911015840 (119902119901119905 | 119902

101584011990110158401199051015840)

=K1198911198911015840 (119902119905 | 119902

10158401199051015840)K

1198911198911015840 (119901119905 | 119901

10158401199051015840)

= 1205751198911198911015840 int

119902

1199021015840

D119902int119901

1199011015840

D119901 exp(119894int119905

1199051015840

Lext (119901 119902 119902) 119889119905)

(41)


119891(1199021015840 119901

1015840 1199051015840) to obtain all


120594119891(119902 119901 119905) = sum

1198911015840

int1198891199021015840119889119901

1015840K

1198911198911015840 (119902119901119905 | 119902

101584011990110158401199051015840) 120594

1198911015840 (119902

1015840 119901

1015840 1199051015840)

(42)

with 120594plusmn12

(119902 119901 119905) = 12059412(119902 119901 119905) (26) In terms of anti-


radic12 (120578+120577


K (119902120572119901120573119905 | 11990210158401205721015840119901101584012057310158401199051015840) = int

119902120572119901120573

1199021015840120572101584011990110158401205731015840

D119902D119901D120577D120578


1199051015840

Lext (119901 119902 119902) 119889119905)

(43)




119867ext120594 (119902 119901) = (119867119902minus 119867

119901) 120594 (119902 119901) = 120576120594 (119902 119901) (44)


120594uarr(119902 119901) = (

1205941(119902 119901)

0) or 120594

darr(119902 119901) = (

0

1205942(119902 119901)

)

(45)


12(119902)120601

12(119901)




[(120597

120597119902+119882

119902(119902)) minus (

120597

120597119901+ 119881

119901(119901))]120595

1(119902) 120601

1(119901)


(46)

[(minus120597

120597119902+119882

119902(119902)) minus (minus

120597

120597119901+ 119881

119901(119901))]120595

2(119902) 120601

2(119901)


(47)

where 120576 = 120576119902minus120576

119901and 120576

119902and 120576


119902and








119902and 119881

119901are odd then

1205951(minus119902) = 120595

2(119902) 120601

1(minus119901) = 120601

2(119901) (48)


[(120597

120597119902+119882

119902(119902)) minus (

120597

120597119901+ 119881

119901(119901))]120595

1(119902) 120601

1(119901)


(49)


radic2120576119902 = 119882(0) +1205951015840

1(0)

1205951 (0)

radic2120576119901 = 119881 (0) +1206011015840

1(0)

1206011 (0)

(50)


119901(119901) have a maxi-



+and 119901

+ respectively


120595(0)

12(119902) = 120575 (119902 minus 119902

plusmn) 120601

(0)

12(119901) = 120575 (119901 minus 119901

plusmn) (51)



120594(1)

1(119902 119901) =

1

119873119902119873119901


minus)


01198891199021015840119882(1199021015840)+int119901

01198891199011015840119881(1199011015840)

120594(1)

2(119902 119901) =

1

119873119902119873119901

120579 (119902+minus 119902) 120579 (119901

+minus 119901)

times 119890int119902

01198891199021015840119882(1199021015840)minusint119901

01198891199011015840119881(1199011015840)

(52)

where 119873119902and 119873



120594(2)

1(119902 119901) =

1

1198731015840119890minusint119902

01198891199021015840119882(1199021015840)+int119901

01198891199011015840119881(1199011015840)

times int

infin


011988911990210158401015840119882(11990210158401015840)119889119902

1015840

times int

infin


011988911990110158401015840119881(11990110158401015840)119889119901

1015840

120594(2)

2(119902 119901) =

1

1198731015840119890int119902

01198891199021015840119882(1199021015840)minusint119901

01198891199011015840119881(1199011015840)

times int

min(minus119902119902+)

infin

1198902 int1199021015840

011988911990210158401015840119882(11990210158401015840)119889119902

1015840

times int

min(minus119901119901+)

infin

119890minus2int1199011015840

011988911990110158401015840119881(11990110158401015840)119889119901

1015840

(53)


minusthe


0119882(11990210158401015840)11988911990210158401015840



1015840) similarly

1198902 int1199021015840

0119882(11990210158401015840)11988911990210158401015840


1015840)



1198731015840= (int

infin

119902minus

119889119902119890minus2int119902

0119882(1199021015840)1198891199021015840

int

infin

119901minus

119889119901119890minus2 int1199011015840

0119881(1199011015840)1198891199011015840

)

32

(54)


120576 = (1205941 119867ext1205941) (55)



0



+and 119901



120576 =ℎ119882

1015840(119902+)

2120587119890minus2Δ119882ℎ

minusℎ119881

1015840(119901

+)

2120587119890minus2Δ119881ℎ

(56)


Δ119882 = int

119902minus

119902+

119882(119902) 119889119902 Δ119881 = int

119901minus

119901+

119881 (119901) 119889119901 (57)


⟨119865⟩ = (120594uarr 119894 119876+ 120590minus120594uarr) (58)


= 0 with 119876+




120576 = ⟨119867ext⟩ = (120594uarr 119867ext120594uarr) =1

2(120594

uarr 119876

+120594darr) (120594

darr 119876

minus120594uarr)

= 120576119902minus 120576

119901= ⟨119867

119902⟩ minus ⟨119867

119901⟩ = (120595

uarr 119867

119902120595uarr) minus (120601

uarr 119867

119901120601uarr)

=1

2(120595

uarr 119876

119902+120595darr) (120595

darr 119876


minus1

2(120601

uarr 119876

119901+120601darr) (120601

darr 119876


(60)



uarr= (

1205951

0) 120595

darr= (

0

1205952) 120601

uarr= (

1206011

0) and

120601darr= (

0


1

radic120576119876minus120594uarr= 120594

darr=

minus120594uarr

1

radic120576119902

119876119902minus120595uarr= 120595

darr=

minus120595uarr

1

radic120576119901

119876119901minus120601uarr= 120601

darr=

minus120601uarr

(61)


radic120576 (120594uarr 119876

+120594darr) (120594

darr

minus120594uarr)



(62)



119901+ and

minusand the


(120595darr


1198821015840(119902+)

120587119890minusΔ119882

Δ119902

(120601darr


1198811015840(119901

+)

120587119890minusΔ119881

Δ119901

(120595uarr 119876

119902+120595darr) = 119894radic

1198821015840(119902minus)

120587119890minusΔ119882

(120601uarr 119876

119901+120601darr) = 119894radic

1198811015840(119901

minus)

120587119890minusΔ119881

(63)

where Δ119902 = 119902+minus 119902

minusand Δ119901 = 119901

+minus 119901

minus Hence

(120594uarr 119876

+120594darr) (120594

darr

minus120594uarr)

= 2119894radic120576[

[

(

120576119902

120576)radic120576

119902radic119882

1015840(119902minus)

120587Δ119902

minus (

120576119901

120576)radic120576

119901radic1198811015840(119901

minus)

120587Δ119901]

]

(64)

⟨119865⟩ = minus2radic120576 [

[

(

120576119902

120576)radic

1198821015840(119902+)

120587119890minus2Δ119882

Δ119902

minus (

120576119901

120576)radic

1198811015840(119901)

120587119890minus2Δ119881

Δ119901]

]

(65)




plusmn 119876

119902plusmn and 119876



119888and

119888= minus119881

119888


plusmnas follows

radic120576 [⟨+0119902+119901+

1003816100381610038161003816 119890minus119894119867ext(119879minus119905)119876


1003816100381610038161003816



= radic120576119902 [⟨+0119902+1003816100381610038161003816 119890


1003816100381610038161003816


minus

10038161003816100381610038161003816119890minus119894119867119902(119879+119905)119902

+0+⟩]



119901+119890minus119894119867119901(119879+119905)


1003816100381610038161003816 minus

10038161003816100381610038161003816119901+0+⟩ ]


(66)


radic120576 ⟨+0119902+119901+


1003816100381610038161003816 minus1003816100381610038161003816 119902+119901+0+⟩

= radic120576119902 ⟨+0119902+1003816100381610038161003816119876119902+


1003816100381610038161003816 119902+0+⟩

minus radic120576119901 ⟨+0119901+1003816100381610038161003816119876119901+


1003816100381610038161003816119901+0+⟩

(67)





0replacing the




int

infin

0

119882(1199021015840) 119889119902


0

int

infin

0

119882(1199021015840) 119889119902


0

(68)


119902]

119882(1199021015840)119889119902

1015840≫ 1 int119902120583+1

119901120583

119881(1199011015840)119889119901

1015840≫ 1 Around




⟨+0119902+

1003816100381610038161003816119876119902+1003816100381610038161003816 119902minus0minus⟩ = 119894

radic119882

1015840

119888(119902+)

120587119890minusΔ119882119888


1003816100381610038161003816 minus1003816100381610038161003816 119902+0+⟩ =

radic119882

1015840

119888(119902+)

120587119890minusΔ119882119888Δ119902

119888

(69)



⟨+0119902+119901+


1003816100381610038161003816 minus1003816100381610038161003816 119902+119901+0+⟩

= 2119894radic120576[

[

(

120576119902

120576)radic120576119902

radic119882

1015840

119888(119902+)

120587Δ119902

119888

minus (

120576119901

120576)radic120576119901

radic1198811015840

119888(119901

+)

120587Δ119901

119888]

]

(70)




119867plusmn=1

2[120587

2

119902minus 120587

2

119901+ 119880

plusmn(119902 119901)] (71)


119880plusmn(119902 119901) = 119880

119902plusmn(119902) minus 119880

119901plusmn(119901)

119880119902plusmn(119902) = 119882

2(119902 119886

119902) ∓119882

1015840

119902(119902 119886

119902)

119880119901plusmn(119901) = 119881

2(119901 119886

119901) ∓ 119881

1015840

119901(119901 119886

119901)

(72)


119880+(1198860 119902 119901) = 119880

minus(1198861 119902 119901) + 119877 (119886

0) 119886

1= 119891 (119886

0)

(73)

where 1198860and 119886





119867119873=1

2[120587

2

119902minus 120587

2

119901+ 119880

minus(119886119873 119902 119901) +

119873

sum

119896=1

119877 (119886119896)] (74)

where 119886119873= 119891



1198822

119899+1(119902) + 119882

1015840

119902(119899+1)(119902) = 119882

2

119899(119902) minus 119882

1015840

119902119899(119902) minus 120583

119899

1198812

119898+1(119901) + 119881

1015840

119901(119898+1)(119901) = 119881

2

119898(119901) minus 119881

1015840

119901119898(119901) minus ]

119898

(75)


119902119899) 119881

119898(119901) equiv 119881(119901 119886

119901119898)

120583119899equiv 119877

119902(119886119902119899) and ]

119898equiv 119877



respectively are

119864(minus)

1199020= 0 119864

(minus)

119902119899=

119899minus1

sum

119894=0

120583119894 119864

(minus)

1199010= 0 119864

(minus)

119901119898=

119898minus1

sum

119895=0

]119895

(76)


119861+(119909 119886

0) 119861

minus(119909 119886

0) minus 119861

minus(119909 119886

1) 119861

+(119909 119886

1)

= 119877 (1198860) = 120583

0minus ]

0


= (1205830or ]

0)

(77)



operators

119869+= 119890

119894119896120601119861+(119909 120594 (119894120597

120601)) 119869

minus= 119861

minus(119909 120594 (119894120597

120601)) 119890

minus119894119896120601

(78)


119902 120594119901)




(79)






119890119894119896120601119861+(119909 120594 (119894120597

120601)) = 119861

+(119909 120594 (119894120597

120601+ 119896)) 119890

119894119896120601

119861minus(119909 120594 (119894120597

120601)) 119890

minus119894119896120601= 119890

minus119894119896120601119861minus(119909 120594 (119894120597

120601+ 119896))

(80)

and that


119894119896aelig120601aelig




120601+ 119896) =


0rarr 120594(119894120597

120601) and

1198861= 119891(119886

0) rarr 119891[120594(119894120597

120601)] = 120594(119894120597




119861+(119909 120594 (119894120597

120601)) 119861

minus(119909 120594 (119894120597

120601))

minus 119861minus(119909 120594 (119894120597

120601+ 119896)) 119861

+(119909 120594 (119894120597

120601+ 119896)) = 119877 [120594 (119894120597

120601)]




(82)




[119869+ 119869minus] = [119869

119902+ 119869119902minus] minus [119869

119901+ 119869119901minus]

= 120585 (1198693) = 120585

119902(1198691199023) minus 120585

119901(1198691199013)

[1198693 119869plusmn] = plusmn119869



(83)

where 120585(1198693) equiv 119877[120594(119894120597








1198961198860) correspond to 120594(119911) =



2119867minus(119909 120594 (119894120597

120601)) = 119861

minus(119909 120594 (119894120597

120601+ 119896))

times 119861+(119909 120594 (119894120597

120601+ 119896))



(84)



120585 (1198693) = 119892 (119869

3) minus 119892 (119869

3minus 1)



minus119869++


in which 1198693and 119862


minusand 119869

+are lowering


1198693 | ℎ⟩ = ℎ |ℎ⟩ 119869

minus | ℎ⟩ = 119886 (ℎ) |ℎ minus 1⟩

119869+ | ℎ⟩ = 119886

lowast(ℎ + 1) |ℎ + 1⟩

(86)

where

|119886 (ℎ)|2minus |119886 (ℎ + 1)|

2= 119892 (ℎ) minus 119892 (ℎ minus 1) (87)



1= 119876119886


0) = minus119903

11198860


120585 (1198693) equiv minus119903

1exp (minus119896119869

3) = 120585

119902(1198691199023) minus 120585

119901(1198691199013)

= minus1199031199021exp (minus119896

1199021198691199023) + 119903

1199011exp (minus119896

1199011198691199013)

(88)

which yields

[119869+ 119869minus] = 120585 (119869

3) = minus119903

1exp (minus119896119869

3) [119869


plusmn (89)


119892 (ℎ) =1199031

119890119896 minus 1119890minus119896ℎ

= minus1199031

1 minus 119876119876minusℎ 119876 = 119890

119896





119876aelig = 119890119896aelig

(90)


3be



119876119899minus 1

119876 minus 11198761minusℎ (91)


1) reads

119867minus | ℎ⟩ = 119886(ℎ)

2| ℎ⟩ = 1199031

119876119899minus 1

119876 minus 11198761minusℎ| ℎ⟩ (92)



119864119899 (ℎ) = 1199031120572 (ℎ)

119876119899minus 1

119876 minus 1 120572 (ℎ) equiv 119876

1minusℎ (93)


1199021198991and 119864

1199011198992(119899 equiv (119899

1 1198992) 119899

12= 0 1 2 ) as

119867119902minus

10038161003816100381610038161003816ℎ119902⟩ = 119886

119902(ℎ

119902)2 10038161003816100381610038161003816ℎ119902⟩ = 119903

1199021

1198761198991119902minus 1

119876119902minus 1

1198761minusℎ119902

119902

10038161003816100381610038161003816ℎ119902⟩

119867119901minus

10038161003816100381610038161003816ℎ119901⟩ = 119886

119901(ℎ

119901)2 10038161003816100381610038161003816ℎ119901⟩ = 119903

1199011

1198761198992119901minus 1

119876119901minus 1

1198761minusℎ119901

119901

10038161003816100381610038161003816ℎ119901⟩

(94)

and that

1198641199021198991(ℎ

119902) = 119903

1199021120572119902(ℎ

119902)

1198761198991119902minus 1

119876119902minus 1

120572119902(ℎ

119902) equiv 119876

1minusℎ119902

119902

1198641199011198992(ℎ

119901) = 119903

1199011120572119901(ℎ

119901)

1198761198992119901minus 1

119876119901minus 1

120572119901(ℎ

119901) equiv 119876

1minusℎ119901

119901

(95)

Hence

119864119899 (ℎ) = 1198641199021198991

(ℎ119902) minus 119864

1199011198992(ℎ

119901) (96)

7 Conclusions


Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119867ext (119901 119902 120587119901 120587119902) =1

21205872

119902+ 119865

2(119902) minus

1

21205872

119901

minus 1198662(119901) + 119877 (119902 119901)119873

(13)


119902 = 120587119902 = 120587

119901

119902= minus119865

1015840

119902(119902) minus 119877

1015840

119902(119902 119901)119873

119901= minus119866

1015840

119901(119901) + 119877

1015840

119901(119902 119901)119873

plusmn= plusmn119894119877 (119902 119901) 120595

plusmn

(14)



119905

1199050

119877 (119902 (120591) 119901 (120591)) 119889120591] (15)


120579plusmn= 120579

plusmn (119905) exp [∓119894 int119905

1199050

119877 (119902 (120591) 119901 (120591)) 119889120591] (16)



plusmn= 120579



(119861119902119901+

)lowast= 119861

119902119901minus


119876119902119901plusmn

= 119861119902119901∓

120595plusmn (17)



119861119902∓L

2(R) 997888rarrL

2(R) 119861

119902∓= [119901 + 120587

119902plusmn 119894119882 (119902)]

119861119901∓L

2(R) 997888rarrL

2(R) 119861

119901∓= [119902 + 120587

119901plusmn 119894119881 (119901)]

(18)


119902(119902)minus119877

119901(119901)


we obtain

119902∓= [minus (119865

1015840

119902+ 119877

1015840

119902119873) plusmn 119894119882

1015840

119902(119902) (119901 + 120587

119902)]

119901∓= [minus (119866

1015840

119901+ 119877

1015840

119901119873) plusmn 119894119881

1015840

119901(119901) (119902 + 120587

119901)]

(19)

Consequently

119902plusmn= plusmn119894 [(119882

1015840

119902minus 119877

1015840

119902plusmn) plusmn 119894 (119865

1015840

119902minus119882119882

1015840

119902) 120595

∓]

119901plusmn= plusmn119894 [(119881

1015840

119901minus 119877

1015840

119901plusmn) plusmn 119894 (119866

1015840

119901minus 119881119881

1015840

119901) 120595

∓]

(20)


119901plusmn= 0 when 1198821015840

119902(119902) =

1198771015840

119902plusmn(119902) and 119865

1015840

119902= (12)(119882

2)1015840

119902or 1198811015840

119901(119901) = 119877

1015840

119901plusmn(119901) and

1198661015840

119901= (12)(119881

2)1015840


119877119902119901


1198771015840

119902plusmn(119902) = 119882

1015840

119902(119902) 119865

119902=1

2(119882

2) + 119862

119902

1198771015840

119901plusmn(119901) = 119881

1015840

119901(119901) 119865

119901=1

2(119881

2) + 119862

119901

(21)

where 119862119902119901



minus 119867119901

where

119867119902=1

21205872

119902+ 119865

2(119902) + 119877

119902119873

119867119901=1

21205872

119901+ 119866

2(119901) + 119877

119901119873

(22)

Then 119876119902119901plusmn


119901form the


119876119902119901+

119876119902119901minus

= minus119894 (119867119902119901minus 119862

119902119901)

119867119902119901 119876

119902119901plusmn = 119876

119902119901plusmn 119876

119902119901plusmn = 0

119873119876119902119901plusmn

= plusmn119894119876119902119901plusmn

119873119867119902119901 = 0

(23)





Lext (119901 119902 119902) =1

21205872

119902minus1

2119882

2(119902) +

1

21205872

119901minus1

21198812(119901)

+ 12059511205952(119882

1015840

119902+ 119881

1015840

119901) +

1

2120595120572120572

(24)








plusmn= radic

1

2(

1plusmn 119894

2)

120572

120573 = 120575

120572120573

+

minus = 1

2

plusmn= 0

(25)



1plusmn120590





119891 =

(12)[+



120594 (119902 119901) = (120594+12

(119902 119901)

120594minus12

(119902 119901)) = (

1205941(119902 119901)

1205942(119902 119901)

) = (

1205951(119902) 120601

1(119901)

1205952(119902) 120601

2(119901))

(26)

where the states 12059512(119902) 120601



0otimes C2

= [L2(R) otimes



119867ext = (119867+

0

0 119867minus

) =1

2(

2

119902+119882

2(119902) + 119894119882

1015840

119902(119902) [

1

2])

minus1

2(

2

119901+ 119881

2(119901) + 119894119881

1015840

119902(119901) [

1

2])

(27)


Lext (119901 119902 119902) = minus 119902119901 minus 119902 +1

2[(119889119902

119889119905)

2

minus1198822(119902)]

+ 1198911198821015840

119902(119902) +

1

2[(119889119901

119889119905)

2

minus 1198812(119901)]

+ 1198911198811015840

119901(119901)

(28)




minusas follows



|00+⟩ = + |00minus⟩ = 0 (30)

then +|00+⟩ = 0


that daggerplusmn=

∓ we get

⟨∓00| plusmn = 0 ⟨∓00| ∓ = ⟨plusmn00| (31)



120572plusmn= minus




|00minus⟩ (33)

and thus

1199021003816100381610038161003816119902120572minus⟩ = 119902

1003816100381610038161003816119902120572minus⟩ minus

1003816100381610038161003816119902120572minus⟩ = 1205721003816100381610038161003816119902120572minus⟩ (34)

Then the 119902and



1003816100381610038161003816 119902120573+⟩ = minusint119889120572119890120572120573 1003816100381610038161003816 119902120572minus⟩

10038161003816100381610038161003816120587119902120572plusmn⟩ = minusint119889119902119890

119894119902120587119902 1003816100381610038161003816 119902120572plusmn⟩

1003816100381610038161003816119901120573+⟩ = minusint119889120572119890120572120573 1003816100381610038161003816119901120572minus⟩

10038161003816100381610038161003816120587119901120572plusmn⟩ = minusint119889119901119890

119894119901120587119901 1003816100381610038161003816119901120572plusmn⟩

(35)

which gives

119902

10038161003816100381610038161003816120587119902120572plusmn⟩ = 120587

119902

10038161003816100381610038161003816120587119902120572plusmn⟩

+

10038161003816100381610038161003816(119902 120587

119902) 120573+⟩ = 120573

10038161003816100381610038161003816(119902 120587

119902) 120573+⟩

119901

10038161003816100381610038161003816120587119901120572plusmn⟩ = 120587

119901

10038161003816100381610038161003816120587119901120572plusmn⟩

+

10038161003816100381610038161003816(119901 120587

119901) 120573+⟩ = 120573

10038161003816100381610038161003816(119901 120587

119901) 120573+⟩

(36)


minus int1198891205721198891199021003816100381610038161003816 119902120572plusmn⟩⟨∓120572

lowast1199021003816100381610038161003816 = 1

minus int119889120572

119889120587119902

2120587

10038161003816100381610038161003816120587119902120572plusmn⟩⟨∓120572

lowast120587119902

10038161003816100381610038161003816= 1

minus int1198891205721198891199011003816100381610038161003816119901120572plusmn⟩⟨∓120572

lowast1199011003816100381610038161003816 = 1

minus int119889120572

119889120587119901

2120587

10038161003816100381610038161003816120587119901120572plusmn⟩⟨∓120572

lowast120587119901

10038161003816100381610038161003816= 1

(37)


120594minus(119902120572119901120573119905) = minusint119889120572

1015840119889119902

1015840119889120573

1015840119889119901

1015840119870(119902120572119901120573119905 | 119902

10158401205721015840119901101584012057310158401199051015840)

(38)

The kernel reads

119870(119902120572119901120573119905 | 11990210158401205721015840119901101584012057310158401199051015840)

= ⟨+119902120572lowast119901120573


1015840) 100381610038161003816100381610038161199021015840120572101584011990110158401205731015840⟩

(39)


K1198911198911015840 (119902119901119905 | 119902

101584011990110158401199051015840) = ⟨119902119901119891

1003816100381610038161003816 119890minus119894119867ext(119905minus119905

1015840) 10038161003816100381610038161003816119902101584011990110158401198911015840⟩ (40)




K1198911198911015840 (119902119901119905 | 119902

101584011990110158401199051015840)

=K1198911198911015840 (119902119905 | 119902

10158401199051015840)K

1198911198911015840 (119901119905 | 119901

10158401199051015840)

= 1205751198911198911015840 int

119902

1199021015840

D119902int119901

1199011015840

D119901 exp(119894int119905

1199051015840

Lext (119901 119902 119902) 119889119905)

(41)


119891(1199021015840 119901

1015840 1199051015840) to obtain all


120594119891(119902 119901 119905) = sum

1198911015840

int1198891199021015840119889119901

1015840K

1198911198911015840 (119902119901119905 | 119902

101584011990110158401199051015840) 120594

1198911015840 (119902

1015840 119901

1015840 1199051015840)

(42)

with 120594plusmn12

(119902 119901 119905) = 12059412(119902 119901 119905) (26) In terms of anti-


radic12 (120578+120577


K (119902120572119901120573119905 | 11990210158401205721015840119901101584012057310158401199051015840) = int

119902120572119901120573

1199021015840120572101584011990110158401205731015840

D119902D119901D120577D120578


1199051015840

Lext (119901 119902 119902) 119889119905)

(43)




119867ext120594 (119902 119901) = (119867119902minus 119867

119901) 120594 (119902 119901) = 120576120594 (119902 119901) (44)


120594uarr(119902 119901) = (

1205941(119902 119901)

0) or 120594

darr(119902 119901) = (

0

1205942(119902 119901)

)

(45)


12(119902)120601

12(119901)




[(120597

120597119902+119882

119902(119902)) minus (

120597

120597119901+ 119881

119901(119901))]120595

1(119902) 120601

1(119901)


(46)

[(minus120597

120597119902+119882

119902(119902)) minus (minus

120597

120597119901+ 119881

119901(119901))]120595

2(119902) 120601

2(119901)


(47)

where 120576 = 120576119902minus120576

119901and 120576

119902and 120576


119902and








119902and 119881

119901are odd then

1205951(minus119902) = 120595

2(119902) 120601

1(minus119901) = 120601

2(119901) (48)


[(120597

120597119902+119882

119902(119902)) minus (

120597

120597119901+ 119881

119901(119901))]120595

1(119902) 120601

1(119901)


(49)


radic2120576119902 = 119882(0) +1205951015840

1(0)

1205951 (0)

radic2120576119901 = 119881 (0) +1206011015840

1(0)

1206011 (0)

(50)


119901(119901) have a maxi-



+and 119901

+ respectively


120595(0)

12(119902) = 120575 (119902 minus 119902

plusmn) 120601

(0)

12(119901) = 120575 (119901 minus 119901

plusmn) (51)



120594(1)

1(119902 119901) =

1

119873119902119873119901


minus)


01198891199021015840119882(1199021015840)+int119901

01198891199011015840119881(1199011015840)

120594(1)

2(119902 119901) =

1

119873119902119873119901

120579 (119902+minus 119902) 120579 (119901

+minus 119901)

times 119890int119902

01198891199021015840119882(1199021015840)minusint119901

01198891199011015840119881(1199011015840)

(52)

where 119873119902and 119873



120594(2)

1(119902 119901) =

1

1198731015840119890minusint119902

01198891199021015840119882(1199021015840)+int119901

01198891199011015840119881(1199011015840)

times int

infin


011988911990210158401015840119882(11990210158401015840)119889119902

1015840

times int

infin


011988911990110158401015840119881(11990110158401015840)119889119901

1015840

120594(2)

2(119902 119901) =

1

1198731015840119890int119902

01198891199021015840119882(1199021015840)minusint119901

01198891199011015840119881(1199011015840)

times int

min(minus119902119902+)

infin

1198902 int1199021015840

011988911990210158401015840119882(11990210158401015840)119889119902

1015840

times int

min(minus119901119901+)

infin

119890minus2int1199011015840

011988911990110158401015840119881(11990110158401015840)119889119901

1015840

(53)


minusthe


0119882(11990210158401015840)11988911990210158401015840



1015840) similarly

1198902 int1199021015840

0119882(11990210158401015840)11988911990210158401015840


1015840)



1198731015840= (int

infin

119902minus

119889119902119890minus2int119902

0119882(1199021015840)1198891199021015840

int

infin

119901minus

119889119901119890minus2 int1199011015840

0119881(1199011015840)1198891199011015840

)

32

(54)


120576 = (1205941 119867ext1205941) (55)



0



+and 119901



120576 =ℎ119882

1015840(119902+)

2120587119890minus2Δ119882ℎ

minusℎ119881

1015840(119901

+)

2120587119890minus2Δ119881ℎ

(56)


Δ119882 = int

119902minus

119902+

119882(119902) 119889119902 Δ119881 = int

119901minus

119901+

119881 (119901) 119889119901 (57)


⟨119865⟩ = (120594uarr 119894 119876+ 120590minus120594uarr) (58)


= 0 with 119876+




120576 = ⟨119867ext⟩ = (120594uarr 119867ext120594uarr) =1

2(120594

uarr 119876

+120594darr) (120594

darr 119876

minus120594uarr)

= 120576119902minus 120576

119901= ⟨119867

119902⟩ minus ⟨119867

119901⟩ = (120595

uarr 119867

119902120595uarr) minus (120601

uarr 119867

119901120601uarr)

=1

2(120595

uarr 119876

119902+120595darr) (120595

darr 119876


minus1

2(120601

uarr 119876

119901+120601darr) (120601

darr 119876


(60)



uarr= (

1205951

0) 120595

darr= (

0

1205952) 120601

uarr= (

1206011

0) and

120601darr= (

0


1

radic120576119876minus120594uarr= 120594

darr=

minus120594uarr

1

radic120576119902

119876119902minus120595uarr= 120595

darr=

minus120595uarr

1

radic120576119901

119876119901minus120601uarr= 120601

darr=

minus120601uarr

(61)


radic120576 (120594uarr 119876

+120594darr) (120594

darr

minus120594uarr)



(62)



119901+ and

minusand the


(120595darr


1198821015840(119902+)

120587119890minusΔ119882

Δ119902

(120601darr


1198811015840(119901

+)

120587119890minusΔ119881

Δ119901

(120595uarr 119876

119902+120595darr) = 119894radic

1198821015840(119902minus)

120587119890minusΔ119882

(120601uarr 119876

119901+120601darr) = 119894radic

1198811015840(119901

minus)

120587119890minusΔ119881

(63)

where Δ119902 = 119902+minus 119902

minusand Δ119901 = 119901

+minus 119901

minus Hence

(120594uarr 119876

+120594darr) (120594

darr

minus120594uarr)

= 2119894radic120576[

[

(

120576119902

120576)radic120576

119902radic119882

1015840(119902minus)

120587Δ119902

minus (

120576119901

120576)radic120576

119901radic1198811015840(119901

minus)

120587Δ119901]

]

(64)

⟨119865⟩ = minus2radic120576 [

[

(

120576119902

120576)radic

1198821015840(119902+)

120587119890minus2Δ119882

Δ119902

minus (

120576119901

120576)radic

1198811015840(119901)

120587119890minus2Δ119881

Δ119901]

]

(65)




plusmn 119876

119902plusmn and 119876



119888and

119888= minus119881

119888


plusmnas follows

radic120576 [⟨+0119902+119901+

1003816100381610038161003816 119890minus119894119867ext(119879minus119905)119876


1003816100381610038161003816



= radic120576119902 [⟨+0119902+1003816100381610038161003816 119890


1003816100381610038161003816


minus

10038161003816100381610038161003816119890minus119894119867119902(119879+119905)119902

+0+⟩]



119901+119890minus119894119867119901(119879+119905)


1003816100381610038161003816 minus

10038161003816100381610038161003816119901+0+⟩ ]


(66)


radic120576 ⟨+0119902+119901+


1003816100381610038161003816 minus1003816100381610038161003816 119902+119901+0+⟩

= radic120576119902 ⟨+0119902+1003816100381610038161003816119876119902+


1003816100381610038161003816 119902+0+⟩

minus radic120576119901 ⟨+0119901+1003816100381610038161003816119876119901+


1003816100381610038161003816119901+0+⟩

(67)





0replacing the




int

infin

0

119882(1199021015840) 119889119902


0

int

infin

0

119882(1199021015840) 119889119902


0

(68)


119902]

119882(1199021015840)119889119902

1015840≫ 1 int119902120583+1

119901120583

119881(1199011015840)119889119901

1015840≫ 1 Around




⟨+0119902+

1003816100381610038161003816119876119902+1003816100381610038161003816 119902minus0minus⟩ = 119894

radic119882

1015840

119888(119902+)

120587119890minusΔ119882119888


1003816100381610038161003816 minus1003816100381610038161003816 119902+0+⟩ =

radic119882

1015840

119888(119902+)

120587119890minusΔ119882119888Δ119902

119888

(69)



⟨+0119902+119901+


1003816100381610038161003816 minus1003816100381610038161003816 119902+119901+0+⟩

= 2119894radic120576[

[

(

120576119902

120576)radic120576119902

radic119882

1015840

119888(119902+)

120587Δ119902

119888

minus (

120576119901

120576)radic120576119901

radic1198811015840

119888(119901

+)

120587Δ119901

119888]

]

(70)




119867plusmn=1

2[120587

2

119902minus 120587

2

119901+ 119880

plusmn(119902 119901)] (71)


119880plusmn(119902 119901) = 119880

119902plusmn(119902) minus 119880

119901plusmn(119901)

119880119902plusmn(119902) = 119882

2(119902 119886

119902) ∓119882

1015840

119902(119902 119886

119902)

119880119901plusmn(119901) = 119881

2(119901 119886

119901) ∓ 119881

1015840

119901(119901 119886

119901)

(72)


119880+(1198860 119902 119901) = 119880

minus(1198861 119902 119901) + 119877 (119886

0) 119886

1= 119891 (119886

0)

(73)

where 1198860and 119886





119867119873=1

2[120587

2

119902minus 120587

2

119901+ 119880

minus(119886119873 119902 119901) +

119873

sum

119896=1

119877 (119886119896)] (74)

where 119886119873= 119891



1198822

119899+1(119902) + 119882

1015840

119902(119899+1)(119902) = 119882

2

119899(119902) minus 119882

1015840

119902119899(119902) minus 120583

119899

1198812

119898+1(119901) + 119881

1015840

119901(119898+1)(119901) = 119881

2

119898(119901) minus 119881

1015840

119901119898(119901) minus ]

119898

(75)


119902119899) 119881

119898(119901) equiv 119881(119901 119886

119901119898)

120583119899equiv 119877

119902(119886119902119899) and ]

119898equiv 119877



respectively are

119864(minus)

1199020= 0 119864

(minus)

119902119899=

119899minus1

sum

119894=0

120583119894 119864

(minus)

1199010= 0 119864

(minus)

119901119898=

119898minus1

sum

119895=0

]119895

(76)


119861+(119909 119886

0) 119861

minus(119909 119886

0) minus 119861

minus(119909 119886

1) 119861

+(119909 119886

1)

= 119877 (1198860) = 120583

0minus ]

0


= (1205830or ]

0)

(77)



operators

119869+= 119890

119894119896120601119861+(119909 120594 (119894120597

120601)) 119869

minus= 119861

minus(119909 120594 (119894120597

120601)) 119890

minus119894119896120601

(78)


119902 120594119901)




(79)






119890119894119896120601119861+(119909 120594 (119894120597

120601)) = 119861

+(119909 120594 (119894120597

120601+ 119896)) 119890

119894119896120601

119861minus(119909 120594 (119894120597

120601)) 119890

minus119894119896120601= 119890

minus119894119896120601119861minus(119909 120594 (119894120597

120601+ 119896))

(80)

and that


119894119896aelig120601aelig




120601+ 119896) =


0rarr 120594(119894120597

120601) and

1198861= 119891(119886

0) rarr 119891[120594(119894120597

120601)] = 120594(119894120597




119861+(119909 120594 (119894120597

120601)) 119861

minus(119909 120594 (119894120597

120601))

minus 119861minus(119909 120594 (119894120597

120601+ 119896)) 119861

+(119909 120594 (119894120597

120601+ 119896)) = 119877 [120594 (119894120597

120601)]




(82)




[119869+ 119869minus] = [119869

119902+ 119869119902minus] minus [119869

119901+ 119869119901minus]

= 120585 (1198693) = 120585

119902(1198691199023) minus 120585

119901(1198691199013)

[1198693 119869plusmn] = plusmn119869



(83)

where 120585(1198693) equiv 119877[120594(119894120597








1198961198860) correspond to 120594(119911) =



2119867minus(119909 120594 (119894120597

120601)) = 119861

minus(119909 120594 (119894120597

120601+ 119896))

times 119861+(119909 120594 (119894120597

120601+ 119896))



(84)



120585 (1198693) = 119892 (119869

3) minus 119892 (119869

3minus 1)



minus119869++


in which 1198693and 119862


minusand 119869

+are lowering


1198693 | ℎ⟩ = ℎ |ℎ⟩ 119869

minus | ℎ⟩ = 119886 (ℎ) |ℎ minus 1⟩

119869+ | ℎ⟩ = 119886

lowast(ℎ + 1) |ℎ + 1⟩

(86)

where

|119886 (ℎ)|2minus |119886 (ℎ + 1)|

2= 119892 (ℎ) minus 119892 (ℎ minus 1) (87)



1= 119876119886


0) = minus119903

11198860


120585 (1198693) equiv minus119903

1exp (minus119896119869

3) = 120585

119902(1198691199023) minus 120585

119901(1198691199013)

= minus1199031199021exp (minus119896

1199021198691199023) + 119903

1199011exp (minus119896

1199011198691199013)

(88)

which yields

[119869+ 119869minus] = 120585 (119869

3) = minus119903

1exp (minus119896119869

3) [119869


plusmn (89)


119892 (ℎ) =1199031

119890119896 minus 1119890minus119896ℎ

= minus1199031

1 minus 119876119876minusℎ 119876 = 119890

119896





119876aelig = 119890119896aelig

(90)


3be



119876119899minus 1

119876 minus 11198761minusℎ (91)


1) reads

119867minus | ℎ⟩ = 119886(ℎ)

2| ℎ⟩ = 1199031

119876119899minus 1

119876 minus 11198761minusℎ| ℎ⟩ (92)



119864119899 (ℎ) = 1199031120572 (ℎ)

119876119899minus 1

119876 minus 1 120572 (ℎ) equiv 119876

1minusℎ (93)


1199021198991and 119864

1199011198992(119899 equiv (119899

1 1198992) 119899

12= 0 1 2 ) as

119867119902minus

10038161003816100381610038161003816ℎ119902⟩ = 119886

119902(ℎ

119902)2 10038161003816100381610038161003816ℎ119902⟩ = 119903

1199021

1198761198991119902minus 1

119876119902minus 1

1198761minusℎ119902

119902

10038161003816100381610038161003816ℎ119902⟩

119867119901minus

10038161003816100381610038161003816ℎ119901⟩ = 119886

119901(ℎ

119901)2 10038161003816100381610038161003816ℎ119901⟩ = 119903

1199011

1198761198992119901minus 1

119876119901minus 1

1198761minusℎ119901

119901

10038161003816100381610038161003816ℎ119901⟩

(94)

and that

1198641199021198991(ℎ

119902) = 119903

1199021120572119902(ℎ

119902)

1198761198991119902minus 1

119876119902minus 1

120572119902(ℎ

119902) equiv 119876

1minusℎ119902

119902

1198641199011198992(ℎ

119901) = 119903

1199011120572119901(ℎ

119901)

1198761198992119901minus 1

119876119901minus 1

120572119901(ℎ

119901) equiv 119876

1minusℎ119901

119901

(95)

Hence

119864119899 (ℎ) = 1198641199021198991

(ℎ119902) minus 119864

1199011198992(ℎ

119901) (96)

7 Conclusions


Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119891 =

(12)[+



120594 (119902 119901) = (120594+12

(119902 119901)

120594minus12

(119902 119901)) = (

1205941(119902 119901)

1205942(119902 119901)

) = (

1205951(119902) 120601

1(119901)

1205952(119902) 120601

2(119901))

(26)

where the states 12059512(119902) 120601



0otimes C2

= [L2(R) otimes



119867ext = (119867+

0

0 119867minus

) =1

2(

2

119902+119882

2(119902) + 119894119882

1015840

119902(119902) [

1

2])

minus1

2(

2

119901+ 119881

2(119901) + 119894119881

1015840

119902(119901) [

1

2])

(27)


Lext (119901 119902 119902) = minus 119902119901 minus 119902 +1

2[(119889119902

119889119905)

2

minus1198822(119902)]

+ 1198911198821015840

119902(119902) +

1

2[(119889119901

119889119905)

2

minus 1198812(119901)]

+ 1198911198811015840

119901(119901)

(28)




minusas follows



|00+⟩ = + |00minus⟩ = 0 (30)

then +|00+⟩ = 0


that daggerplusmn=

∓ we get

⟨∓00| plusmn = 0 ⟨∓00| ∓ = ⟨plusmn00| (31)



120572plusmn= minus




|00minus⟩ (33)

and thus

1199021003816100381610038161003816119902120572minus⟩ = 119902

1003816100381610038161003816119902120572minus⟩ minus

1003816100381610038161003816119902120572minus⟩ = 1205721003816100381610038161003816119902120572minus⟩ (34)

Then the 119902and



1003816100381610038161003816 119902120573+⟩ = minusint119889120572119890120572120573 1003816100381610038161003816 119902120572minus⟩

10038161003816100381610038161003816120587119902120572plusmn⟩ = minusint119889119902119890

119894119902120587119902 1003816100381610038161003816 119902120572plusmn⟩

1003816100381610038161003816119901120573+⟩ = minusint119889120572119890120572120573 1003816100381610038161003816119901120572minus⟩

10038161003816100381610038161003816120587119901120572plusmn⟩ = minusint119889119901119890

119894119901120587119901 1003816100381610038161003816119901120572plusmn⟩

(35)

which gives

119902

10038161003816100381610038161003816120587119902120572plusmn⟩ = 120587

119902

10038161003816100381610038161003816120587119902120572plusmn⟩

+

10038161003816100381610038161003816(119902 120587

119902) 120573+⟩ = 120573

10038161003816100381610038161003816(119902 120587

119902) 120573+⟩

119901

10038161003816100381610038161003816120587119901120572plusmn⟩ = 120587

119901

10038161003816100381610038161003816120587119901120572plusmn⟩

+

10038161003816100381610038161003816(119901 120587

119901) 120573+⟩ = 120573

10038161003816100381610038161003816(119901 120587

119901) 120573+⟩

(36)


minus int1198891205721198891199021003816100381610038161003816 119902120572plusmn⟩⟨∓120572

lowast1199021003816100381610038161003816 = 1

minus int119889120572

119889120587119902

2120587

10038161003816100381610038161003816120587119902120572plusmn⟩⟨∓120572

lowast120587119902

10038161003816100381610038161003816= 1

minus int1198891205721198891199011003816100381610038161003816119901120572plusmn⟩⟨∓120572

lowast1199011003816100381610038161003816 = 1

minus int119889120572

119889120587119901

2120587

10038161003816100381610038161003816120587119901120572plusmn⟩⟨∓120572

lowast120587119901

10038161003816100381610038161003816= 1

(37)


120594minus(119902120572119901120573119905) = minusint119889120572

1015840119889119902

1015840119889120573

1015840119889119901

1015840119870(119902120572119901120573119905 | 119902

10158401205721015840119901101584012057310158401199051015840)

(38)

The kernel reads

119870(119902120572119901120573119905 | 11990210158401205721015840119901101584012057310158401199051015840)

= ⟨+119902120572lowast119901120573


1015840) 100381610038161003816100381610038161199021015840120572101584011990110158401205731015840⟩

(39)


K1198911198911015840 (119902119901119905 | 119902

101584011990110158401199051015840) = ⟨119902119901119891

1003816100381610038161003816 119890minus119894119867ext(119905minus119905

1015840) 10038161003816100381610038161003816119902101584011990110158401198911015840⟩ (40)




K1198911198911015840 (119902119901119905 | 119902

101584011990110158401199051015840)

=K1198911198911015840 (119902119905 | 119902

10158401199051015840)K

1198911198911015840 (119901119905 | 119901

10158401199051015840)

= 1205751198911198911015840 int

119902

1199021015840

D119902int119901

1199011015840

D119901 exp(119894int119905

1199051015840

Lext (119901 119902 119902) 119889119905)

(41)


119891(1199021015840 119901

1015840 1199051015840) to obtain all


120594119891(119902 119901 119905) = sum

1198911015840

int1198891199021015840119889119901

1015840K

1198911198911015840 (119902119901119905 | 119902

101584011990110158401199051015840) 120594

1198911015840 (119902

1015840 119901

1015840 1199051015840)

(42)

with 120594plusmn12

(119902 119901 119905) = 12059412(119902 119901 119905) (26) In terms of anti-


radic12 (120578+120577


K (119902120572119901120573119905 | 11990210158401205721015840119901101584012057310158401199051015840) = int

119902120572119901120573

1199021015840120572101584011990110158401205731015840

D119902D119901D120577D120578


1199051015840

Lext (119901 119902 119902) 119889119905)

(43)




119867ext120594 (119902 119901) = (119867119902minus 119867

119901) 120594 (119902 119901) = 120576120594 (119902 119901) (44)


120594uarr(119902 119901) = (

1205941(119902 119901)

0) or 120594

darr(119902 119901) = (

0

1205942(119902 119901)

)

(45)


12(119902)120601

12(119901)




[(120597

120597119902+119882

119902(119902)) minus (

120597

120597119901+ 119881

119901(119901))]120595

1(119902) 120601

1(119901)


(46)

[(minus120597

120597119902+119882

119902(119902)) minus (minus

120597

120597119901+ 119881

119901(119901))]120595

2(119902) 120601

2(119901)


(47)

where 120576 = 120576119902minus120576

119901and 120576

119902and 120576


119902and








119902and 119881

119901are odd then

1205951(minus119902) = 120595

2(119902) 120601

1(minus119901) = 120601

2(119901) (48)


[(120597

120597119902+119882

119902(119902)) minus (

120597

120597119901+ 119881

119901(119901))]120595

1(119902) 120601

1(119901)


(49)


radic2120576119902 = 119882(0) +1205951015840

1(0)

1205951 (0)

radic2120576119901 = 119881 (0) +1206011015840

1(0)

1206011 (0)

(50)


119901(119901) have a maxi-



+and 119901

+ respectively


120595(0)

12(119902) = 120575 (119902 minus 119902

plusmn) 120601

(0)

12(119901) = 120575 (119901 minus 119901

plusmn) (51)



120594(1)

1(119902 119901) =

1

119873119902119873119901


minus)


01198891199021015840119882(1199021015840)+int119901

01198891199011015840119881(1199011015840)

120594(1)

2(119902 119901) =

1

119873119902119873119901

120579 (119902+minus 119902) 120579 (119901

+minus 119901)

times 119890int119902

01198891199021015840119882(1199021015840)minusint119901

01198891199011015840119881(1199011015840)

(52)

where 119873119902and 119873



120594(2)

1(119902 119901) =

1

1198731015840119890minusint119902

01198891199021015840119882(1199021015840)+int119901

01198891199011015840119881(1199011015840)

times int

infin


011988911990210158401015840119882(11990210158401015840)119889119902

1015840

times int

infin


011988911990110158401015840119881(11990110158401015840)119889119901

1015840

120594(2)

2(119902 119901) =

1

1198731015840119890int119902

01198891199021015840119882(1199021015840)minusint119901

01198891199011015840119881(1199011015840)

times int

min(minus119902119902+)

infin

1198902 int1199021015840

011988911990210158401015840119882(11990210158401015840)119889119902

1015840

times int

min(minus119901119901+)

infin

119890minus2int1199011015840

011988911990110158401015840119881(11990110158401015840)119889119901

1015840

(53)


minusthe


0119882(11990210158401015840)11988911990210158401015840



1015840) similarly

1198902 int1199021015840

0119882(11990210158401015840)11988911990210158401015840


1015840)



1198731015840= (int

infin

119902minus

119889119902119890minus2int119902

0119882(1199021015840)1198891199021015840

int

infin

119901minus

119889119901119890minus2 int1199011015840

0119881(1199011015840)1198891199011015840

)

32

(54)


120576 = (1205941 119867ext1205941) (55)



0



+and 119901



120576 =ℎ119882

1015840(119902+)

2120587119890minus2Δ119882ℎ

minusℎ119881

1015840(119901

+)

2120587119890minus2Δ119881ℎ

(56)


Δ119882 = int

119902minus

119902+

119882(119902) 119889119902 Δ119881 = int

119901minus

119901+

119881 (119901) 119889119901 (57)


⟨119865⟩ = (120594uarr 119894 119876+ 120590minus120594uarr) (58)


= 0 with 119876+




120576 = ⟨119867ext⟩ = (120594uarr 119867ext120594uarr) =1

2(120594

uarr 119876

+120594darr) (120594

darr 119876

minus120594uarr)

= 120576119902minus 120576

119901= ⟨119867

119902⟩ minus ⟨119867

119901⟩ = (120595

uarr 119867

119902120595uarr) minus (120601

uarr 119867

119901120601uarr)

=1

2(120595

uarr 119876

119902+120595darr) (120595

darr 119876


minus1

2(120601

uarr 119876

119901+120601darr) (120601

darr 119876


(60)



uarr= (

1205951

0) 120595

darr= (

0

1205952) 120601

uarr= (

1206011

0) and

120601darr= (

0


1

radic120576119876minus120594uarr= 120594

darr=

minus120594uarr

1

radic120576119902

119876119902minus120595uarr= 120595

darr=

minus120595uarr

1

radic120576119901

119876119901minus120601uarr= 120601

darr=

minus120601uarr

(61)


radic120576 (120594uarr 119876

+120594darr) (120594

darr

minus120594uarr)



(62)



119901+ and

minusand the


(120595darr


1198821015840(119902+)

120587119890minusΔ119882

Δ119902

(120601darr


1198811015840(119901

+)

120587119890minusΔ119881

Δ119901

(120595uarr 119876

119902+120595darr) = 119894radic

1198821015840(119902minus)

120587119890minusΔ119882

(120601uarr 119876

119901+120601darr) = 119894radic

1198811015840(119901

minus)

120587119890minusΔ119881

(63)

where Δ119902 = 119902+minus 119902

minusand Δ119901 = 119901

+minus 119901

minus Hence

(120594uarr 119876

+120594darr) (120594

darr

minus120594uarr)

= 2119894radic120576[

[

(

120576119902

120576)radic120576

119902radic119882

1015840(119902minus)

120587Δ119902

minus (

120576119901

120576)radic120576

119901radic1198811015840(119901

minus)

120587Δ119901]

]

(64)

⟨119865⟩ = minus2radic120576 [

[

(

120576119902

120576)radic

1198821015840(119902+)

120587119890minus2Δ119882

Δ119902

minus (

120576119901

120576)radic

1198811015840(119901)

120587119890minus2Δ119881

Δ119901]

]

(65)




plusmn 119876

119902plusmn and 119876



119888and

119888= minus119881

119888


plusmnas follows

radic120576 [⟨+0119902+119901+

1003816100381610038161003816 119890minus119894119867ext(119879minus119905)119876


1003816100381610038161003816



= radic120576119902 [⟨+0119902+1003816100381610038161003816 119890


1003816100381610038161003816


minus

10038161003816100381610038161003816119890minus119894119867119902(119879+119905)119902

+0+⟩]



119901+119890minus119894119867119901(119879+119905)


1003816100381610038161003816 minus

10038161003816100381610038161003816119901+0+⟩ ]


(66)


radic120576 ⟨+0119902+119901+


1003816100381610038161003816 minus1003816100381610038161003816 119902+119901+0+⟩

= radic120576119902 ⟨+0119902+1003816100381610038161003816119876119902+


1003816100381610038161003816 119902+0+⟩

minus radic120576119901 ⟨+0119901+1003816100381610038161003816119876119901+


1003816100381610038161003816119901+0+⟩

(67)





0replacing the




int

infin

0

119882(1199021015840) 119889119902


0

int

infin

0

119882(1199021015840) 119889119902


0

(68)


119902]

119882(1199021015840)119889119902

1015840≫ 1 int119902120583+1

119901120583

119881(1199011015840)119889119901

1015840≫ 1 Around




⟨+0119902+

1003816100381610038161003816119876119902+1003816100381610038161003816 119902minus0minus⟩ = 119894

radic119882

1015840

119888(119902+)

120587119890minusΔ119882119888


1003816100381610038161003816 minus1003816100381610038161003816 119902+0+⟩ =

radic119882

1015840

119888(119902+)

120587119890minusΔ119882119888Δ119902

119888

(69)



⟨+0119902+119901+


1003816100381610038161003816 minus1003816100381610038161003816 119902+119901+0+⟩

= 2119894radic120576[

[

(

120576119902

120576)radic120576119902

radic119882

1015840

119888(119902+)

120587Δ119902

119888

minus (

120576119901

120576)radic120576119901

radic1198811015840

119888(119901

+)

120587Δ119901

119888]

]

(70)




119867plusmn=1

2[120587

2

119902minus 120587

2

119901+ 119880

plusmn(119902 119901)] (71)


119880plusmn(119902 119901) = 119880

119902plusmn(119902) minus 119880

119901plusmn(119901)

119880119902plusmn(119902) = 119882

2(119902 119886

119902) ∓119882

1015840

119902(119902 119886

119902)

119880119901plusmn(119901) = 119881

2(119901 119886

119901) ∓ 119881

1015840

119901(119901 119886

119901)

(72)


119880+(1198860 119902 119901) = 119880

minus(1198861 119902 119901) + 119877 (119886

0) 119886

1= 119891 (119886

0)

(73)

where 1198860and 119886





119867119873=1

2[120587

2

119902minus 120587

2

119901+ 119880

minus(119886119873 119902 119901) +

119873

sum

119896=1

119877 (119886119896)] (74)

where 119886119873= 119891



1198822

119899+1(119902) + 119882

1015840

119902(119899+1)(119902) = 119882

2

119899(119902) minus 119882

1015840

119902119899(119902) minus 120583

119899

1198812

119898+1(119901) + 119881

1015840

119901(119898+1)(119901) = 119881

2

119898(119901) minus 119881

1015840

119901119898(119901) minus ]

119898

(75)


119902119899) 119881

119898(119901) equiv 119881(119901 119886

119901119898)

120583119899equiv 119877

119902(119886119902119899) and ]

119898equiv 119877



respectively are

119864(minus)

1199020= 0 119864

(minus)

119902119899=

119899minus1

sum

119894=0

120583119894 119864

(minus)

1199010= 0 119864

(minus)

119901119898=

119898minus1

sum

119895=0

]119895

(76)


119861+(119909 119886

0) 119861

minus(119909 119886

0) minus 119861

minus(119909 119886

1) 119861

+(119909 119886

1)

= 119877 (1198860) = 120583

0minus ]

0


= (1205830or ]

0)

(77)



operators

119869+= 119890

119894119896120601119861+(119909 120594 (119894120597

120601)) 119869

minus= 119861

minus(119909 120594 (119894120597

120601)) 119890

minus119894119896120601

(78)


119902 120594119901)




(79)






119890119894119896120601119861+(119909 120594 (119894120597

120601)) = 119861

+(119909 120594 (119894120597

120601+ 119896)) 119890

119894119896120601

119861minus(119909 120594 (119894120597

120601)) 119890

minus119894119896120601= 119890

minus119894119896120601119861minus(119909 120594 (119894120597

120601+ 119896))

(80)

and that


119894119896aelig120601aelig




120601+ 119896) =


0rarr 120594(119894120597

120601) and

1198861= 119891(119886

0) rarr 119891[120594(119894120597

120601)] = 120594(119894120597




119861+(119909 120594 (119894120597

120601)) 119861

minus(119909 120594 (119894120597

120601))

minus 119861minus(119909 120594 (119894120597

120601+ 119896)) 119861

+(119909 120594 (119894120597

120601+ 119896)) = 119877 [120594 (119894120597

120601)]




(82)




[119869+ 119869minus] = [119869

119902+ 119869119902minus] minus [119869

119901+ 119869119901minus]

= 120585 (1198693) = 120585

119902(1198691199023) minus 120585

119901(1198691199013)

[1198693 119869plusmn] = plusmn119869



(83)

where 120585(1198693) equiv 119877[120594(119894120597








1198961198860) correspond to 120594(119911) =



2119867minus(119909 120594 (119894120597

120601)) = 119861

minus(119909 120594 (119894120597

120601+ 119896))

times 119861+(119909 120594 (119894120597

120601+ 119896))



(84)



120585 (1198693) = 119892 (119869

3) minus 119892 (119869

3minus 1)



minus119869++


in which 1198693and 119862


minusand 119869

+are lowering


1198693 | ℎ⟩ = ℎ |ℎ⟩ 119869

minus | ℎ⟩ = 119886 (ℎ) |ℎ minus 1⟩

119869+ | ℎ⟩ = 119886

lowast(ℎ + 1) |ℎ + 1⟩

(86)

where

|119886 (ℎ)|2minus |119886 (ℎ + 1)|

2= 119892 (ℎ) minus 119892 (ℎ minus 1) (87)



1= 119876119886


0) = minus119903

11198860


120585 (1198693) equiv minus119903

1exp (minus119896119869

3) = 120585

119902(1198691199023) minus 120585

119901(1198691199013)

= minus1199031199021exp (minus119896

1199021198691199023) + 119903

1199011exp (minus119896

1199011198691199013)

(88)

which yields

[119869+ 119869minus] = 120585 (119869

3) = minus119903

1exp (minus119896119869

3) [119869


plusmn (89)


119892 (ℎ) =1199031

119890119896 minus 1119890minus119896ℎ

= minus1199031

1 minus 119876119876minusℎ 119876 = 119890

119896





119876aelig = 119890119896aelig

(90)


3be



119876119899minus 1

119876 minus 11198761minusℎ (91)


1) reads

119867minus | ℎ⟩ = 119886(ℎ)

2| ℎ⟩ = 1199031

119876119899minus 1

119876 minus 11198761minusℎ| ℎ⟩ (92)



119864119899 (ℎ) = 1199031120572 (ℎ)

119876119899minus 1

119876 minus 1 120572 (ℎ) equiv 119876

1minusℎ (93)


1199021198991and 119864

1199011198992(119899 equiv (119899

1 1198992) 119899

12= 0 1 2 ) as

119867119902minus

10038161003816100381610038161003816ℎ119902⟩ = 119886

119902(ℎ

119902)2 10038161003816100381610038161003816ℎ119902⟩ = 119903

1199021

1198761198991119902minus 1

119876119902minus 1

1198761minusℎ119902

119902

10038161003816100381610038161003816ℎ119902⟩

119867119901minus

10038161003816100381610038161003816ℎ119901⟩ = 119886

119901(ℎ

119901)2 10038161003816100381610038161003816ℎ119901⟩ = 119903

1199011

1198761198992119901minus 1

119876119901minus 1

1198761minusℎ119901

119901

10038161003816100381610038161003816ℎ119901⟩

(94)

and that

1198641199021198991(ℎ

119902) = 119903

1199021120572119902(ℎ

119902)

1198761198991119902minus 1

119876119902minus 1

120572119902(ℎ

119902) equiv 119876

1minusℎ119902

119902

1198641199011198992(ℎ

119901) = 119903

1199011120572119901(ℎ

119901)

1198761198992119901minus 1

119876119901minus 1

120572119901(ℎ

119901) equiv 119876

1minusℎ119901

119901

(95)

Hence

119864119899 (ℎ) = 1198641199021198991

(ℎ119902) minus 119864

1199011198992(ℎ

119901) (96)

7 Conclusions


Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











K1198911198911015840 (119902119901119905 | 119902

101584011990110158401199051015840)

=K1198911198911015840 (119902119905 | 119902

10158401199051015840)K

1198911198911015840 (119901119905 | 119901

10158401199051015840)

= 1205751198911198911015840 int

119902

1199021015840

D119902int119901

1199011015840

D119901 exp(119894int119905

1199051015840

Lext (119901 119902 119902) 119889119905)

(41)


119891(1199021015840 119901

1015840 1199051015840) to obtain all


120594119891(119902 119901 119905) = sum

1198911015840

int1198891199021015840119889119901

1015840K

1198911198911015840 (119902119901119905 | 119902

101584011990110158401199051015840) 120594

1198911015840 (119902

1015840 119901

1015840 1199051015840)

(42)

with 120594plusmn12

(119902 119901 119905) = 12059412(119902 119901 119905) (26) In terms of anti-


radic12 (120578+120577


K (119902120572119901120573119905 | 11990210158401205721015840119901101584012057310158401199051015840) = int

119902120572119901120573

1199021015840120572101584011990110158401205731015840

D119902D119901D120577D120578


1199051015840

Lext (119901 119902 119902) 119889119905)

(43)




119867ext120594 (119902 119901) = (119867119902minus 119867

119901) 120594 (119902 119901) = 120576120594 (119902 119901) (44)


120594uarr(119902 119901) = (

1205941(119902 119901)

0) or 120594

darr(119902 119901) = (

0

1205942(119902 119901)

)

(45)


12(119902)120601

12(119901)




[(120597

120597119902+119882

119902(119902)) minus (

120597

120597119901+ 119881

119901(119901))]120595

1(119902) 120601

1(119901)


(46)

[(minus120597

120597119902+119882

119902(119902)) minus (minus

120597

120597119901+ 119881

119901(119901))]120595

2(119902) 120601

2(119901)


(47)

where 120576 = 120576119902minus120576

119901and 120576

119902and 120576


119902and








119902and 119881

119901are odd then

1205951(minus119902) = 120595

2(119902) 120601

1(minus119901) = 120601

2(119901) (48)


[(120597

120597119902+119882

119902(119902)) minus (

120597

120597119901+ 119881

119901(119901))]120595

1(119902) 120601

1(119901)


(49)


radic2120576119902 = 119882(0) +1205951015840

1(0)

1205951 (0)

radic2120576119901 = 119881 (0) +1206011015840

1(0)

1206011 (0)

(50)


119901(119901) have a maxi-



+and 119901

+ respectively


120595(0)

12(119902) = 120575 (119902 minus 119902

plusmn) 120601

(0)

12(119901) = 120575 (119901 minus 119901

plusmn) (51)



120594(1)

1(119902 119901) =

1

119873119902119873119901


minus)


01198891199021015840119882(1199021015840)+int119901

01198891199011015840119881(1199011015840)

120594(1)

2(119902 119901) =

1

119873119902119873119901

120579 (119902+minus 119902) 120579 (119901

+minus 119901)

times 119890int119902

01198891199021015840119882(1199021015840)minusint119901

01198891199011015840119881(1199011015840)

(52)

where 119873119902and 119873



120594(2)

1(119902 119901) =

1

1198731015840119890minusint119902

01198891199021015840119882(1199021015840)+int119901

01198891199011015840119881(1199011015840)

times int

infin


011988911990210158401015840119882(11990210158401015840)119889119902

1015840

times int

infin


011988911990110158401015840119881(11990110158401015840)119889119901

1015840

120594(2)

2(119902 119901) =

1

1198731015840119890int119902

01198891199021015840119882(1199021015840)minusint119901

01198891199011015840119881(1199011015840)

times int

min(minus119902119902+)

infin

1198902 int1199021015840

011988911990210158401015840119882(11990210158401015840)119889119902

1015840

times int

min(minus119901119901+)

infin

119890minus2int1199011015840

011988911990110158401015840119881(11990110158401015840)119889119901

1015840

(53)


minusthe


0119882(11990210158401015840)11988911990210158401015840



1015840) similarly

1198902 int1199021015840

0119882(11990210158401015840)11988911990210158401015840


1015840)



1198731015840= (int

infin

119902minus

119889119902119890minus2int119902

0119882(1199021015840)1198891199021015840

int

infin

119901minus

119889119901119890minus2 int1199011015840

0119881(1199011015840)1198891199011015840

)

32

(54)


120576 = (1205941 119867ext1205941) (55)



0



+and 119901



120576 =ℎ119882

1015840(119902+)

2120587119890minus2Δ119882ℎ

minusℎ119881

1015840(119901

+)

2120587119890minus2Δ119881ℎ

(56)


Δ119882 = int

119902minus

119902+

119882(119902) 119889119902 Δ119881 = int

119901minus

119901+

119881 (119901) 119889119901 (57)


⟨119865⟩ = (120594uarr 119894 119876+ 120590minus120594uarr) (58)


= 0 with 119876+




120576 = ⟨119867ext⟩ = (120594uarr 119867ext120594uarr) =1

2(120594

uarr 119876

+120594darr) (120594

darr 119876

minus120594uarr)

= 120576119902minus 120576

119901= ⟨119867

119902⟩ minus ⟨119867

119901⟩ = (120595

uarr 119867

119902120595uarr) minus (120601

uarr 119867

119901120601uarr)

=1

2(120595

uarr 119876

119902+120595darr) (120595

darr 119876


minus1

2(120601

uarr 119876

119901+120601darr) (120601

darr 119876


(60)



uarr= (

1205951

0) 120595

darr= (

0

1205952) 120601

uarr= (

1206011

0) and

120601darr= (

0


1

radic120576119876minus120594uarr= 120594

darr=

minus120594uarr

1

radic120576119902

119876119902minus120595uarr= 120595

darr=

minus120595uarr

1

radic120576119901

119876119901minus120601uarr= 120601

darr=

minus120601uarr

(61)


radic120576 (120594uarr 119876

+120594darr) (120594

darr

minus120594uarr)



(62)



119901+ and

minusand the


(120595darr


1198821015840(119902+)

120587119890minusΔ119882

Δ119902

(120601darr


1198811015840(119901

+)

120587119890minusΔ119881

Δ119901

(120595uarr 119876

119902+120595darr) = 119894radic

1198821015840(119902minus)

120587119890minusΔ119882

(120601uarr 119876

119901+120601darr) = 119894radic

1198811015840(119901

minus)

120587119890minusΔ119881

(63)

where Δ119902 = 119902+minus 119902

minusand Δ119901 = 119901

+minus 119901

minus Hence

(120594uarr 119876

+120594darr) (120594

darr

minus120594uarr)

= 2119894radic120576[

[

(

120576119902

120576)radic120576

119902radic119882

1015840(119902minus)

120587Δ119902

minus (

120576119901

120576)radic120576

119901radic1198811015840(119901

minus)

120587Δ119901]

]

(64)

⟨119865⟩ = minus2radic120576 [

[

(

120576119902

120576)radic

1198821015840(119902+)

120587119890minus2Δ119882

Δ119902

minus (

120576119901

120576)radic

1198811015840(119901)

120587119890minus2Δ119881

Δ119901]

]

(65)




plusmn 119876

119902plusmn and 119876



119888and

119888= minus119881

119888


plusmnas follows

radic120576 [⟨+0119902+119901+

1003816100381610038161003816 119890minus119894119867ext(119879minus119905)119876


1003816100381610038161003816



= radic120576119902 [⟨+0119902+1003816100381610038161003816 119890


1003816100381610038161003816


minus

10038161003816100381610038161003816119890minus119894119867119902(119879+119905)119902

+0+⟩]



119901+119890minus119894119867119901(119879+119905)


1003816100381610038161003816 minus

10038161003816100381610038161003816119901+0+⟩ ]


(66)


radic120576 ⟨+0119902+119901+


1003816100381610038161003816 minus1003816100381610038161003816 119902+119901+0+⟩

= radic120576119902 ⟨+0119902+1003816100381610038161003816119876119902+


1003816100381610038161003816 119902+0+⟩

minus radic120576119901 ⟨+0119901+1003816100381610038161003816119876119901+


1003816100381610038161003816119901+0+⟩

(67)





0replacing the




int

infin

0

119882(1199021015840) 119889119902


0

int

infin

0

119882(1199021015840) 119889119902


0

(68)


119902]

119882(1199021015840)119889119902

1015840≫ 1 int119902120583+1

119901120583

119881(1199011015840)119889119901

1015840≫ 1 Around




⟨+0119902+

1003816100381610038161003816119876119902+1003816100381610038161003816 119902minus0minus⟩ = 119894

radic119882

1015840

119888(119902+)

120587119890minusΔ119882119888


1003816100381610038161003816 minus1003816100381610038161003816 119902+0+⟩ =

radic119882

1015840

119888(119902+)

120587119890minusΔ119882119888Δ119902

119888

(69)



⟨+0119902+119901+


1003816100381610038161003816 minus1003816100381610038161003816 119902+119901+0+⟩

= 2119894radic120576[

[

(

120576119902

120576)radic120576119902

radic119882

1015840

119888(119902+)

120587Δ119902

119888

minus (

120576119901

120576)radic120576119901

radic1198811015840

119888(119901

+)

120587Δ119901

119888]

]

(70)




119867plusmn=1

2[120587

2

119902minus 120587

2

119901+ 119880

plusmn(119902 119901)] (71)


119880plusmn(119902 119901) = 119880

119902plusmn(119902) minus 119880

119901plusmn(119901)

119880119902plusmn(119902) = 119882

2(119902 119886

119902) ∓119882

1015840

119902(119902 119886

119902)

119880119901plusmn(119901) = 119881

2(119901 119886

119901) ∓ 119881

1015840

119901(119901 119886

119901)

(72)


119880+(1198860 119902 119901) = 119880

minus(1198861 119902 119901) + 119877 (119886

0) 119886

1= 119891 (119886

0)

(73)

where 1198860and 119886





119867119873=1

2[120587

2

119902minus 120587

2

119901+ 119880

minus(119886119873 119902 119901) +

119873

sum

119896=1

119877 (119886119896)] (74)

where 119886119873= 119891



1198822

119899+1(119902) + 119882

1015840

119902(119899+1)(119902) = 119882

2

119899(119902) minus 119882

1015840

119902119899(119902) minus 120583

119899

1198812

119898+1(119901) + 119881

1015840

119901(119898+1)(119901) = 119881

2

119898(119901) minus 119881

1015840

119901119898(119901) minus ]

119898

(75)


119902119899) 119881

119898(119901) equiv 119881(119901 119886

119901119898)

120583119899equiv 119877

119902(119886119902119899) and ]

119898equiv 119877



respectively are

119864(minus)

1199020= 0 119864

(minus)

119902119899=

119899minus1

sum

119894=0

120583119894 119864

(minus)

1199010= 0 119864

(minus)

119901119898=

119898minus1

sum

119895=0

]119895

(76)


119861+(119909 119886

0) 119861

minus(119909 119886

0) minus 119861

minus(119909 119886

1) 119861

+(119909 119886

1)

= 119877 (1198860) = 120583

0minus ]

0


= (1205830or ]

0)

(77)



operators

119869+= 119890

119894119896120601119861+(119909 120594 (119894120597

120601)) 119869

minus= 119861

minus(119909 120594 (119894120597

120601)) 119890

minus119894119896120601

(78)


119902 120594119901)




(79)






119890119894119896120601119861+(119909 120594 (119894120597

120601)) = 119861

+(119909 120594 (119894120597

120601+ 119896)) 119890

119894119896120601

119861minus(119909 120594 (119894120597

120601)) 119890

minus119894119896120601= 119890

minus119894119896120601119861minus(119909 120594 (119894120597

120601+ 119896))

(80)

and that


119894119896aelig120601aelig




120601+ 119896) =


0rarr 120594(119894120597

120601) and

1198861= 119891(119886

0) rarr 119891[120594(119894120597

120601)] = 120594(119894120597




119861+(119909 120594 (119894120597

120601)) 119861

minus(119909 120594 (119894120597

120601))

minus 119861minus(119909 120594 (119894120597

120601+ 119896)) 119861

+(119909 120594 (119894120597

120601+ 119896)) = 119877 [120594 (119894120597

120601)]




(82)




[119869+ 119869minus] = [119869

119902+ 119869119902minus] minus [119869

119901+ 119869119901minus]

= 120585 (1198693) = 120585

119902(1198691199023) minus 120585

119901(1198691199013)

[1198693 119869plusmn] = plusmn119869



(83)

where 120585(1198693) equiv 119877[120594(119894120597








1198961198860) correspond to 120594(119911) =



2119867minus(119909 120594 (119894120597

120601)) = 119861

minus(119909 120594 (119894120597

120601+ 119896))

times 119861+(119909 120594 (119894120597

120601+ 119896))



(84)



120585 (1198693) = 119892 (119869

3) minus 119892 (119869

3minus 1)



minus119869++


in which 1198693and 119862


minusand 119869

+are lowering


1198693 | ℎ⟩ = ℎ |ℎ⟩ 119869

minus | ℎ⟩ = 119886 (ℎ) |ℎ minus 1⟩

119869+ | ℎ⟩ = 119886

lowast(ℎ + 1) |ℎ + 1⟩

(86)

where

|119886 (ℎ)|2minus |119886 (ℎ + 1)|

2= 119892 (ℎ) minus 119892 (ℎ minus 1) (87)



1= 119876119886


0) = minus119903

11198860


120585 (1198693) equiv minus119903

1exp (minus119896119869

3) = 120585

119902(1198691199023) minus 120585

119901(1198691199013)

= minus1199031199021exp (minus119896

1199021198691199023) + 119903

1199011exp (minus119896

1199011198691199013)

(88)

which yields

[119869+ 119869minus] = 120585 (119869

3) = minus119903

1exp (minus119896119869

3) [119869


plusmn (89)


119892 (ℎ) =1199031

119890119896 minus 1119890minus119896ℎ

= minus1199031

1 minus 119876119876minusℎ 119876 = 119890

119896





119876aelig = 119890119896aelig

(90)


3be



119876119899minus 1

119876 minus 11198761minusℎ (91)


1) reads

119867minus | ℎ⟩ = 119886(ℎ)

2| ℎ⟩ = 1199031

119876119899minus 1

119876 minus 11198761minusℎ| ℎ⟩ (92)



119864119899 (ℎ) = 1199031120572 (ℎ)

119876119899minus 1

119876 minus 1 120572 (ℎ) equiv 119876

1minusℎ (93)


1199021198991and 119864

1199011198992(119899 equiv (119899

1 1198992) 119899

12= 0 1 2 ) as

119867119902minus

10038161003816100381610038161003816ℎ119902⟩ = 119886

119902(ℎ

119902)2 10038161003816100381610038161003816ℎ119902⟩ = 119903

1199021

1198761198991119902minus 1

119876119902minus 1

1198761minusℎ119902

119902

10038161003816100381610038161003816ℎ119902⟩

119867119901minus

10038161003816100381610038161003816ℎ119901⟩ = 119886

119901(ℎ

119901)2 10038161003816100381610038161003816ℎ119901⟩ = 119903

1199011

1198761198992119901minus 1

119876119901minus 1

1198761minusℎ119901

119901

10038161003816100381610038161003816ℎ119901⟩

(94)

and that

1198641199021198991(ℎ

119902) = 119903

1199021120572119902(ℎ

119902)

1198761198991119902minus 1

119876119902minus 1

120572119902(ℎ

119902) equiv 119876

1minusℎ119902

119902

1198641199011198992(ℎ

119901) = 119903

1199011120572119901(ℎ

119901)

1198761198992119901minus 1

119876119901minus 1

120572119901(ℎ

119901) equiv 119876

1minusℎ119901

119901

(95)

Hence

119864119899 (ℎ) = 1198641199021198991

(ℎ119902) minus 119864

1199011198992(ℎ

119901) (96)

7 Conclusions


Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120594(1)

1(119902 119901) =

1

119873119902119873119901


minus)


01198891199021015840119882(1199021015840)+int119901

01198891199011015840119881(1199011015840)

120594(1)

2(119902 119901) =

1

119873119902119873119901

120579 (119902+minus 119902) 120579 (119901

+minus 119901)

times 119890int119902

01198891199021015840119882(1199021015840)minusint119901

01198891199011015840119881(1199011015840)

(52)

where 119873119902and 119873



120594(2)

1(119902 119901) =

1

1198731015840119890minusint119902

01198891199021015840119882(1199021015840)+int119901

01198891199011015840119881(1199011015840)

times int

infin


011988911990210158401015840119882(11990210158401015840)119889119902

1015840

times int

infin


011988911990110158401015840119881(11990110158401015840)119889119901

1015840

120594(2)

2(119902 119901) =

1

1198731015840119890int119902

01198891199021015840119882(1199021015840)minusint119901

01198891199011015840119881(1199011015840)

times int

min(minus119902119902+)

infin

1198902 int1199021015840

011988911990210158401015840119882(11990210158401015840)119889119902

1015840

times int

min(minus119901119901+)

infin

119890minus2int1199011015840

011988911990110158401015840119881(11990110158401015840)119889119901

1015840

(53)


minusthe


0119882(11990210158401015840)11988911990210158401015840



1015840) similarly

1198902 int1199021015840

0119882(11990210158401015840)11988911990210158401015840


1015840)



1198731015840= (int

infin

119902minus

119889119902119890minus2int119902

0119882(1199021015840)1198891199021015840

int

infin

119901minus

119889119901119890minus2 int1199011015840

0119881(1199011015840)1198891199011015840

)

32

(54)


120576 = (1205941 119867ext1205941) (55)



0



+and 119901



120576 =ℎ119882

1015840(119902+)

2120587119890minus2Δ119882ℎ

minusℎ119881

1015840(119901

+)

2120587119890minus2Δ119881ℎ

(56)


Δ119882 = int

119902minus

119902+

119882(119902) 119889119902 Δ119881 = int

119901minus

119901+

119881 (119901) 119889119901 (57)


⟨119865⟩ = (120594uarr 119894 119876+ 120590minus120594uarr) (58)


= 0 with 119876+




120576 = ⟨119867ext⟩ = (120594uarr 119867ext120594uarr) =1

2(120594

uarr 119876

+120594darr) (120594

darr 119876

minus120594uarr)

= 120576119902minus 120576

119901= ⟨119867

119902⟩ minus ⟨119867

119901⟩ = (120595

uarr 119867

119902120595uarr) minus (120601

uarr 119867

119901120601uarr)

=1

2(120595

uarr 119876

119902+120595darr) (120595

darr 119876


minus1

2(120601

uarr 119876

119901+120601darr) (120601

darr 119876


(60)



uarr= (

1205951

0) 120595

darr= (

0

1205952) 120601

uarr= (

1206011

0) and

120601darr= (

0


1

radic120576119876minus120594uarr= 120594

darr=

minus120594uarr

1

radic120576119902

119876119902minus120595uarr= 120595

darr=

minus120595uarr

1

radic120576119901

119876119901minus120601uarr= 120601

darr=

minus120601uarr

(61)


radic120576 (120594uarr 119876

+120594darr) (120594

darr

minus120594uarr)



(62)



119901+ and

minusand the


(120595darr


1198821015840(119902+)

120587119890minusΔ119882

Δ119902

(120601darr


1198811015840(119901

+)

120587119890minusΔ119881

Δ119901

(120595uarr 119876

119902+120595darr) = 119894radic

1198821015840(119902minus)

120587119890minusΔ119882

(120601uarr 119876

119901+120601darr) = 119894radic

1198811015840(119901

minus)

120587119890minusΔ119881

(63)

where Δ119902 = 119902+minus 119902

minusand Δ119901 = 119901

+minus 119901

minus Hence

(120594uarr 119876

+120594darr) (120594

darr

minus120594uarr)

= 2119894radic120576[

[

(

120576119902

120576)radic120576

119902radic119882

1015840(119902minus)

120587Δ119902

minus (

120576119901

120576)radic120576

119901radic1198811015840(119901

minus)

120587Δ119901]

]

(64)

⟨119865⟩ = minus2radic120576 [

[

(

120576119902

120576)radic

1198821015840(119902+)

120587119890minus2Δ119882

Δ119902

minus (

120576119901

120576)radic

1198811015840(119901)

120587119890minus2Δ119881

Δ119901]

]

(65)




plusmn 119876

119902plusmn and 119876



119888and

119888= minus119881

119888


plusmnas follows

radic120576 [⟨+0119902+119901+

1003816100381610038161003816 119890minus119894119867ext(119879minus119905)119876


1003816100381610038161003816



= radic120576119902 [⟨+0119902+1003816100381610038161003816 119890


1003816100381610038161003816


minus

10038161003816100381610038161003816119890minus119894119867119902(119879+119905)119902

+0+⟩]



119901+119890minus119894119867119901(119879+119905)


1003816100381610038161003816 minus

10038161003816100381610038161003816119901+0+⟩ ]


(66)


radic120576 ⟨+0119902+119901+


1003816100381610038161003816 minus1003816100381610038161003816 119902+119901+0+⟩

= radic120576119902 ⟨+0119902+1003816100381610038161003816119876119902+


1003816100381610038161003816 119902+0+⟩

minus radic120576119901 ⟨+0119901+1003816100381610038161003816119876119901+


1003816100381610038161003816119901+0+⟩

(67)





0replacing the




int

infin

0

119882(1199021015840) 119889119902


0

int

infin

0

119882(1199021015840) 119889119902


0

(68)


119902]

119882(1199021015840)119889119902

1015840≫ 1 int119902120583+1

119901120583

119881(1199011015840)119889119901

1015840≫ 1 Around




⟨+0119902+

1003816100381610038161003816119876119902+1003816100381610038161003816 119902minus0minus⟩ = 119894

radic119882

1015840

119888(119902+)

120587119890minusΔ119882119888


1003816100381610038161003816 minus1003816100381610038161003816 119902+0+⟩ =

radic119882

1015840

119888(119902+)

120587119890minusΔ119882119888Δ119902

119888

(69)



⟨+0119902+119901+


1003816100381610038161003816 minus1003816100381610038161003816 119902+119901+0+⟩

= 2119894radic120576[

[

(

120576119902

120576)radic120576119902

radic119882

1015840

119888(119902+)

120587Δ119902

119888

minus (

120576119901

120576)radic120576119901

radic1198811015840

119888(119901

+)

120587Δ119901

119888]

]

(70)




119867plusmn=1

2[120587

2

119902minus 120587

2

119901+ 119880

plusmn(119902 119901)] (71)


119880plusmn(119902 119901) = 119880

119902plusmn(119902) minus 119880

119901plusmn(119901)

119880119902plusmn(119902) = 119882

2(119902 119886

119902) ∓119882

1015840

119902(119902 119886

119902)

119880119901plusmn(119901) = 119881

2(119901 119886

119901) ∓ 119881

1015840

119901(119901 119886

119901)

(72)


119880+(1198860 119902 119901) = 119880

minus(1198861 119902 119901) + 119877 (119886

0) 119886

1= 119891 (119886

0)

(73)

where 1198860and 119886





119867119873=1

2[120587

2

119902minus 120587

2

119901+ 119880

minus(119886119873 119902 119901) +

119873

sum

119896=1

119877 (119886119896)] (74)

where 119886119873= 119891



1198822

119899+1(119902) + 119882

1015840

119902(119899+1)(119902) = 119882

2

119899(119902) minus 119882

1015840

119902119899(119902) minus 120583

119899

1198812

119898+1(119901) + 119881

1015840

119901(119898+1)(119901) = 119881

2

119898(119901) minus 119881

1015840

119901119898(119901) minus ]

119898

(75)


119902119899) 119881

119898(119901) equiv 119881(119901 119886

119901119898)

120583119899equiv 119877

119902(119886119902119899) and ]

119898equiv 119877



respectively are

119864(minus)

1199020= 0 119864

(minus)

119902119899=

119899minus1

sum

119894=0

120583119894 119864

(minus)

1199010= 0 119864

(minus)

119901119898=

119898minus1

sum

119895=0

]119895

(76)


119861+(119909 119886

0) 119861

minus(119909 119886

0) minus 119861

minus(119909 119886

1) 119861

+(119909 119886

1)

= 119877 (1198860) = 120583

0minus ]

0


= (1205830or ]

0)

(77)



operators

119869+= 119890

119894119896120601119861+(119909 120594 (119894120597

120601)) 119869

minus= 119861

minus(119909 120594 (119894120597

120601)) 119890

minus119894119896120601

(78)


119902 120594119901)




(79)






119890119894119896120601119861+(119909 120594 (119894120597

120601)) = 119861

+(119909 120594 (119894120597

120601+ 119896)) 119890

119894119896120601

119861minus(119909 120594 (119894120597

120601)) 119890

minus119894119896120601= 119890

minus119894119896120601119861minus(119909 120594 (119894120597

120601+ 119896))

(80)

and that


119894119896aelig120601aelig




120601+ 119896) =


0rarr 120594(119894120597

120601) and

1198861= 119891(119886

0) rarr 119891[120594(119894120597

120601)] = 120594(119894120597




119861+(119909 120594 (119894120597

120601)) 119861

minus(119909 120594 (119894120597

120601))

minus 119861minus(119909 120594 (119894120597

120601+ 119896)) 119861

+(119909 120594 (119894120597

120601+ 119896)) = 119877 [120594 (119894120597

120601)]




(82)




[119869+ 119869minus] = [119869

119902+ 119869119902minus] minus [119869

119901+ 119869119901minus]

= 120585 (1198693) = 120585

119902(1198691199023) minus 120585

119901(1198691199013)

[1198693 119869plusmn] = plusmn119869



(83)

where 120585(1198693) equiv 119877[120594(119894120597








1198961198860) correspond to 120594(119911) =



2119867minus(119909 120594 (119894120597

120601)) = 119861

minus(119909 120594 (119894120597

120601+ 119896))

times 119861+(119909 120594 (119894120597

120601+ 119896))



(84)



120585 (1198693) = 119892 (119869

3) minus 119892 (119869

3minus 1)



minus119869++


in which 1198693and 119862


minusand 119869

+are lowering


1198693 | ℎ⟩ = ℎ |ℎ⟩ 119869

minus | ℎ⟩ = 119886 (ℎ) |ℎ minus 1⟩

119869+ | ℎ⟩ = 119886

lowast(ℎ + 1) |ℎ + 1⟩

(86)

where

|119886 (ℎ)|2minus |119886 (ℎ + 1)|

2= 119892 (ℎ) minus 119892 (ℎ minus 1) (87)



1= 119876119886


0) = minus119903

11198860


120585 (1198693) equiv minus119903

1exp (minus119896119869

3) = 120585

119902(1198691199023) minus 120585

119901(1198691199013)

= minus1199031199021exp (minus119896

1199021198691199023) + 119903

1199011exp (minus119896

1199011198691199013)

(88)

which yields

[119869+ 119869minus] = 120585 (119869

3) = minus119903

1exp (minus119896119869

3) [119869


plusmn (89)


119892 (ℎ) =1199031

119890119896 minus 1119890minus119896ℎ

= minus1199031

1 minus 119876119876minusℎ 119876 = 119890

119896





119876aelig = 119890119896aelig

(90)


3be



119876119899minus 1

119876 minus 11198761minusℎ (91)


1) reads

119867minus | ℎ⟩ = 119886(ℎ)

2| ℎ⟩ = 1199031

119876119899minus 1

119876 minus 11198761minusℎ| ℎ⟩ (92)



119864119899 (ℎ) = 1199031120572 (ℎ)

119876119899minus 1

119876 minus 1 120572 (ℎ) equiv 119876

1minusℎ (93)


1199021198991and 119864

1199011198992(119899 equiv (119899

1 1198992) 119899

12= 0 1 2 ) as

119867119902minus

10038161003816100381610038161003816ℎ119902⟩ = 119886

119902(ℎ

119902)2 10038161003816100381610038161003816ℎ119902⟩ = 119903

1199021

1198761198991119902minus 1

119876119902minus 1

1198761minusℎ119902

119902

10038161003816100381610038161003816ℎ119902⟩

119867119901minus

10038161003816100381610038161003816ℎ119901⟩ = 119886

119901(ℎ

119901)2 10038161003816100381610038161003816ℎ119901⟩ = 119903

1199011

1198761198992119901minus 1

119876119901minus 1

1198761minusℎ119901

119901

10038161003816100381610038161003816ℎ119901⟩

(94)

and that

1198641199021198991(ℎ

119902) = 119903

1199021120572119902(ℎ

119902)

1198761198991119902minus 1

119876119902minus 1

120572119902(ℎ

119902) equiv 119876

1minusℎ119902

119902

1198641199011198992(ℎ

119901) = 119903

1199011120572119901(ℎ

119901)

1198761198992119901minus 1

119876119901minus 1

120572119901(ℎ

119901) equiv 119876

1minusℎ119901

119901

(95)

Hence

119864119899 (ℎ) = 1198641199021198991

(ℎ119902) minus 119864

1199011198992(ℎ

119901) (96)

7 Conclusions


Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119901+ and

minusand the


(120595darr


1198821015840(119902+)

120587119890minusΔ119882

Δ119902

(120601darr


1198811015840(119901

+)

120587119890minusΔ119881

Δ119901

(120595uarr 119876

119902+120595darr) = 119894radic

1198821015840(119902minus)

120587119890minusΔ119882

(120601uarr 119876

119901+120601darr) = 119894radic

1198811015840(119901

minus)

120587119890minusΔ119881

(63)

where Δ119902 = 119902+minus 119902

minusand Δ119901 = 119901

+minus 119901

minus Hence

(120594uarr 119876

+120594darr) (120594

darr

minus120594uarr)

= 2119894radic120576[

[

(

120576119902

120576)radic120576

119902radic119882

1015840(119902minus)

120587Δ119902

minus (

120576119901

120576)radic120576

119901radic1198811015840(119901

minus)

120587Δ119901]

]

(64)

⟨119865⟩ = minus2radic120576 [

[

(

120576119902

120576)radic

1198821015840(119902+)

120587119890minus2Δ119882

Δ119902

minus (

120576119901

120576)radic

1198811015840(119901)

120587119890minus2Δ119881

Δ119901]

]

(65)




plusmn 119876

119902plusmn and 119876



119888and

119888= minus119881

119888


plusmnas follows

radic120576 [⟨+0119902+119901+

1003816100381610038161003816 119890minus119894119867ext(119879minus119905)119876


1003816100381610038161003816



= radic120576119902 [⟨+0119902+1003816100381610038161003816 119890


1003816100381610038161003816


minus

10038161003816100381610038161003816119890minus119894119867119902(119879+119905)119902

+0+⟩]



119901+119890minus119894119867119901(119879+119905)


1003816100381610038161003816 minus

10038161003816100381610038161003816119901+0+⟩ ]


(66)


radic120576 ⟨+0119902+119901+


1003816100381610038161003816 minus1003816100381610038161003816 119902+119901+0+⟩

= radic120576119902 ⟨+0119902+1003816100381610038161003816119876119902+


1003816100381610038161003816 119902+0+⟩

minus radic120576119901 ⟨+0119901+1003816100381610038161003816119876119901+


1003816100381610038161003816119901+0+⟩

(67)





0replacing the




int

infin

0

119882(1199021015840) 119889119902


0

int

infin

0

119882(1199021015840) 119889119902


0

(68)


119902]

119882(1199021015840)119889119902

1015840≫ 1 int119902120583+1

119901120583

119881(1199011015840)119889119901

1015840≫ 1 Around




⟨+0119902+

1003816100381610038161003816119876119902+1003816100381610038161003816 119902minus0minus⟩ = 119894

radic119882

1015840

119888(119902+)

120587119890minusΔ119882119888


1003816100381610038161003816 minus1003816100381610038161003816 119902+0+⟩ =

radic119882

1015840

119888(119902+)

120587119890minusΔ119882119888Δ119902

119888

(69)



⟨+0119902+119901+


1003816100381610038161003816 minus1003816100381610038161003816 119902+119901+0+⟩

= 2119894radic120576[

[

(

120576119902

120576)radic120576119902

radic119882

1015840

119888(119902+)

120587Δ119902

119888

minus (

120576119901

120576)radic120576119901

radic1198811015840

119888(119901

+)

120587Δ119901

119888]

]

(70)




119867plusmn=1

2[120587

2

119902minus 120587

2

119901+ 119880

plusmn(119902 119901)] (71)


119880plusmn(119902 119901) = 119880

119902plusmn(119902) minus 119880

119901plusmn(119901)

119880119902plusmn(119902) = 119882

2(119902 119886

119902) ∓119882

1015840

119902(119902 119886

119902)

119880119901plusmn(119901) = 119881

2(119901 119886

119901) ∓ 119881

1015840

119901(119901 119886

119901)

(72)


119880+(1198860 119902 119901) = 119880

minus(1198861 119902 119901) + 119877 (119886

0) 119886

1= 119891 (119886

0)

(73)

where 1198860and 119886





119867119873=1

2[120587

2

119902minus 120587

2

119901+ 119880

minus(119886119873 119902 119901) +

119873

sum

119896=1

119877 (119886119896)] (74)

where 119886119873= 119891



1198822

119899+1(119902) + 119882

1015840

119902(119899+1)(119902) = 119882

2

119899(119902) minus 119882

1015840

119902119899(119902) minus 120583

119899

1198812

119898+1(119901) + 119881

1015840

119901(119898+1)(119901) = 119881

2

119898(119901) minus 119881

1015840

119901119898(119901) minus ]

119898

(75)


119902119899) 119881

119898(119901) equiv 119881(119901 119886

119901119898)

120583119899equiv 119877

119902(119886119902119899) and ]

119898equiv 119877



respectively are

119864(minus)

1199020= 0 119864

(minus)

119902119899=

119899minus1

sum

119894=0

120583119894 119864

(minus)

1199010= 0 119864

(minus)

119901119898=

119898minus1

sum

119895=0

]119895

(76)


119861+(119909 119886

0) 119861

minus(119909 119886

0) minus 119861

minus(119909 119886

1) 119861

+(119909 119886

1)

= 119877 (1198860) = 120583

0minus ]

0


= (1205830or ]

0)

(77)



operators

119869+= 119890

119894119896120601119861+(119909 120594 (119894120597

120601)) 119869

minus= 119861

minus(119909 120594 (119894120597

120601)) 119890

minus119894119896120601

(78)


119902 120594119901)




(79)






119890119894119896120601119861+(119909 120594 (119894120597

120601)) = 119861

+(119909 120594 (119894120597

120601+ 119896)) 119890

119894119896120601

119861minus(119909 120594 (119894120597

120601)) 119890

minus119894119896120601= 119890

minus119894119896120601119861minus(119909 120594 (119894120597

120601+ 119896))

(80)

and that


119894119896aelig120601aelig




120601+ 119896) =


0rarr 120594(119894120597

120601) and

1198861= 119891(119886

0) rarr 119891[120594(119894120597

120601)] = 120594(119894120597




119861+(119909 120594 (119894120597

120601)) 119861

minus(119909 120594 (119894120597

120601))

minus 119861minus(119909 120594 (119894120597

120601+ 119896)) 119861

+(119909 120594 (119894120597

120601+ 119896)) = 119877 [120594 (119894120597

120601)]




(82)




[119869+ 119869minus] = [119869

119902+ 119869119902minus] minus [119869

119901+ 119869119901minus]

= 120585 (1198693) = 120585

119902(1198691199023) minus 120585

119901(1198691199013)

[1198693 119869plusmn] = plusmn119869



(83)

where 120585(1198693) equiv 119877[120594(119894120597








1198961198860) correspond to 120594(119911) =



2119867minus(119909 120594 (119894120597

120601)) = 119861

minus(119909 120594 (119894120597

120601+ 119896))

times 119861+(119909 120594 (119894120597

120601+ 119896))



(84)



120585 (1198693) = 119892 (119869

3) minus 119892 (119869

3minus 1)



minus119869++


in which 1198693and 119862


minusand 119869

+are lowering


1198693 | ℎ⟩ = ℎ |ℎ⟩ 119869

minus | ℎ⟩ = 119886 (ℎ) |ℎ minus 1⟩

119869+ | ℎ⟩ = 119886

lowast(ℎ + 1) |ℎ + 1⟩

(86)

where

|119886 (ℎ)|2minus |119886 (ℎ + 1)|

2= 119892 (ℎ) minus 119892 (ℎ minus 1) (87)



1= 119876119886


0) = minus119903

11198860


120585 (1198693) equiv minus119903

1exp (minus119896119869

3) = 120585

119902(1198691199023) minus 120585

119901(1198691199013)

= minus1199031199021exp (minus119896

1199021198691199023) + 119903

1199011exp (minus119896

1199011198691199013)

(88)

which yields

[119869+ 119869minus] = 120585 (119869

3) = minus119903

1exp (minus119896119869

3) [119869


plusmn (89)


119892 (ℎ) =1199031

119890119896 minus 1119890minus119896ℎ

= minus1199031

1 minus 119876119876minusℎ 119876 = 119890

119896





119876aelig = 119890119896aelig

(90)


3be



119876119899minus 1

119876 minus 11198761minusℎ (91)


1) reads

119867minus | ℎ⟩ = 119886(ℎ)

2| ℎ⟩ = 1199031

119876119899minus 1

119876 minus 11198761minusℎ| ℎ⟩ (92)



119864119899 (ℎ) = 1199031120572 (ℎ)

119876119899minus 1

119876 minus 1 120572 (ℎ) equiv 119876

1minusℎ (93)


1199021198991and 119864

1199011198992(119899 equiv (119899

1 1198992) 119899

12= 0 1 2 ) as

119867119902minus

10038161003816100381610038161003816ℎ119902⟩ = 119886

119902(ℎ

119902)2 10038161003816100381610038161003816ℎ119902⟩ = 119903

1199021

1198761198991119902minus 1

119876119902minus 1

1198761minusℎ119902

119902

10038161003816100381610038161003816ℎ119902⟩

119867119901minus

10038161003816100381610038161003816ℎ119901⟩ = 119886

119901(ℎ

119901)2 10038161003816100381610038161003816ℎ119901⟩ = 119903

1199011

1198761198992119901minus 1

119876119901minus 1

1198761minusℎ119901

119901

10038161003816100381610038161003816ℎ119901⟩

(94)

and that

1198641199021198991(ℎ

119902) = 119903

1199021120572119902(ℎ

119902)

1198761198991119902minus 1

119876119902minus 1

120572119902(ℎ

119902) equiv 119876

1minusℎ119902

119902

1198641199011198992(ℎ

119901) = 119903

1199011120572119901(ℎ

119901)

1198761198992119901minus 1

119876119901minus 1

120572119901(ℎ

119901) equiv 119876

1minusℎ119901

119901

(95)

Hence

119864119899 (ℎ) = 1198641199021198991

(ℎ119902) minus 119864

1199011198992(ℎ

119901) (96)

7 Conclusions


Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











⟨+0119902+119901+


1003816100381610038161003816 minus1003816100381610038161003816 119902+119901+0+⟩

= 2119894radic120576[

[

(

120576119902

120576)radic120576119902

radic119882

1015840

119888(119902+)

120587Δ119902

119888

minus (

120576119901

120576)radic120576119901

radic1198811015840

119888(119901

+)

120587Δ119901

119888]

]

(70)




119867plusmn=1

2[120587

2

119902minus 120587

2

119901+ 119880

plusmn(119902 119901)] (71)


119880plusmn(119902 119901) = 119880

119902plusmn(119902) minus 119880

119901plusmn(119901)

119880119902plusmn(119902) = 119882

2(119902 119886

119902) ∓119882

1015840

119902(119902 119886

119902)

119880119901plusmn(119901) = 119881

2(119901 119886

119901) ∓ 119881

1015840

119901(119901 119886

119901)

(72)


119880+(1198860 119902 119901) = 119880

minus(1198861 119902 119901) + 119877 (119886

0) 119886

1= 119891 (119886

0)

(73)

where 1198860and 119886





119867119873=1

2[120587

2

119902minus 120587

2

119901+ 119880

minus(119886119873 119902 119901) +

119873

sum

119896=1

119877 (119886119896)] (74)

where 119886119873= 119891



1198822

119899+1(119902) + 119882

1015840

119902(119899+1)(119902) = 119882

2

119899(119902) minus 119882

1015840

119902119899(119902) minus 120583

119899

1198812

119898+1(119901) + 119881

1015840

119901(119898+1)(119901) = 119881

2

119898(119901) minus 119881

1015840

119901119898(119901) minus ]

119898

(75)


119902119899) 119881

119898(119901) equiv 119881(119901 119886

119901119898)

120583119899equiv 119877

119902(119886119902119899) and ]

119898equiv 119877



respectively are

119864(minus)

1199020= 0 119864

(minus)

119902119899=

119899minus1

sum

119894=0

120583119894 119864

(minus)

1199010= 0 119864

(minus)

119901119898=

119898minus1

sum

119895=0

]119895

(76)


119861+(119909 119886

0) 119861

minus(119909 119886

0) minus 119861

minus(119909 119886

1) 119861

+(119909 119886

1)

= 119877 (1198860) = 120583

0minus ]

0


= (1205830or ]

0)

(77)



operators

119869+= 119890

119894119896120601119861+(119909 120594 (119894120597

120601)) 119869

minus= 119861

minus(119909 120594 (119894120597

120601)) 119890

minus119894119896120601

(78)


119902 120594119901)




(79)






119890119894119896120601119861+(119909 120594 (119894120597

120601)) = 119861

+(119909 120594 (119894120597

120601+ 119896)) 119890

119894119896120601

119861minus(119909 120594 (119894120597

120601)) 119890

minus119894119896120601= 119890

minus119894119896120601119861minus(119909 120594 (119894120597

120601+ 119896))

(80)

and that


119894119896aelig120601aelig




120601+ 119896) =


0rarr 120594(119894120597

120601) and

1198861= 119891(119886

0) rarr 119891[120594(119894120597

120601)] = 120594(119894120597




119861+(119909 120594 (119894120597

120601)) 119861

minus(119909 120594 (119894120597

120601))

minus 119861minus(119909 120594 (119894120597

120601+ 119896)) 119861

+(119909 120594 (119894120597

120601+ 119896)) = 119877 [120594 (119894120597

120601)]




(82)




[119869+ 119869minus] = [119869

119902+ 119869119902minus] minus [119869

119901+ 119869119901minus]

= 120585 (1198693) = 120585

119902(1198691199023) minus 120585

119901(1198691199013)

[1198693 119869plusmn] = plusmn119869



(83)

where 120585(1198693) equiv 119877[120594(119894120597








1198961198860) correspond to 120594(119911) =



2119867minus(119909 120594 (119894120597

120601)) = 119861

minus(119909 120594 (119894120597

120601+ 119896))

times 119861+(119909 120594 (119894120597

120601+ 119896))



(84)



120585 (1198693) = 119892 (119869

3) minus 119892 (119869

3minus 1)



minus119869++


in which 1198693and 119862


minusand 119869

+are lowering


1198693 | ℎ⟩ = ℎ |ℎ⟩ 119869

minus | ℎ⟩ = 119886 (ℎ) |ℎ minus 1⟩

119869+ | ℎ⟩ = 119886

lowast(ℎ + 1) |ℎ + 1⟩

(86)

where

|119886 (ℎ)|2minus |119886 (ℎ + 1)|

2= 119892 (ℎ) minus 119892 (ℎ minus 1) (87)



1= 119876119886


0) = minus119903

11198860


120585 (1198693) equiv minus119903

1exp (minus119896119869

3) = 120585

119902(1198691199023) minus 120585

119901(1198691199013)

= minus1199031199021exp (minus119896

1199021198691199023) + 119903

1199011exp (minus119896

1199011198691199013)

(88)

which yields

[119869+ 119869minus] = 120585 (119869

3) = minus119903

1exp (minus119896119869

3) [119869


plusmn (89)


119892 (ℎ) =1199031

119890119896 minus 1119890minus119896ℎ

= minus1199031

1 minus 119876119876minusℎ 119876 = 119890

119896





119876aelig = 119890119896aelig

(90)


3be



119876119899minus 1

119876 minus 11198761minusℎ (91)


1) reads

119867minus | ℎ⟩ = 119886(ℎ)

2| ℎ⟩ = 1199031

119876119899minus 1

119876 minus 11198761minusℎ| ℎ⟩ (92)



119864119899 (ℎ) = 1199031120572 (ℎ)

119876119899minus 1

119876 minus 1 120572 (ℎ) equiv 119876

1minusℎ (93)


1199021198991and 119864

1199011198992(119899 equiv (119899

1 1198992) 119899

12= 0 1 2 ) as

119867119902minus

10038161003816100381610038161003816ℎ119902⟩ = 119886

119902(ℎ

119902)2 10038161003816100381610038161003816ℎ119902⟩ = 119903

1199021

1198761198991119902minus 1

119876119902minus 1

1198761minusℎ119902

119902

10038161003816100381610038161003816ℎ119902⟩

119867119901minus

10038161003816100381610038161003816ℎ119901⟩ = 119886

119901(ℎ

119901)2 10038161003816100381610038161003816ℎ119901⟩ = 119903

1199011

1198761198992119901minus 1

119876119901minus 1

1198761minusℎ119901

119901

10038161003816100381610038161003816ℎ119901⟩

(94)

and that

1198641199021198991(ℎ

119902) = 119903

1199021120572119902(ℎ

119902)

1198761198991119902minus 1

119876119902minus 1

120572119902(ℎ

119902) equiv 119876

1minusℎ119902

119902

1198641199011198992(ℎ

119901) = 119903

1199011120572119901(ℎ

119901)

1198761198992119901minus 1

119876119901minus 1

120572119901(ℎ

119901) equiv 119876

1minusℎ119901

119901

(95)

Hence

119864119899 (ℎ) = 1198641199021198991

(ℎ119902) minus 119864

1199011198992(ℎ

119901) (96)

7 Conclusions


Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119861+(119909 120594 (119894120597

120601)) 119861

minus(119909 120594 (119894120597

120601))

minus 119861minus(119909 120594 (119894120597

120601+ 119896)) 119861

+(119909 120594 (119894120597

120601+ 119896)) = 119877 [120594 (119894120597

120601)]




(82)




[119869+ 119869minus] = [119869

119902+ 119869119902minus] minus [119869

119901+ 119869119901minus]

= 120585 (1198693) = 120585

119902(1198691199023) minus 120585

119901(1198691199013)

[1198693 119869plusmn] = plusmn119869



(83)

where 120585(1198693) equiv 119877[120594(119894120597








1198961198860) correspond to 120594(119911) =



2119867minus(119909 120594 (119894120597

120601)) = 119861

minus(119909 120594 (119894120597

120601+ 119896))

times 119861+(119909 120594 (119894120597

120601+ 119896))



(84)



120585 (1198693) = 119892 (119869

3) minus 119892 (119869

3minus 1)



minus119869++


in which 1198693and 119862


minusand 119869

+are lowering


1198693 | ℎ⟩ = ℎ |ℎ⟩ 119869

minus | ℎ⟩ = 119886 (ℎ) |ℎ minus 1⟩

119869+ | ℎ⟩ = 119886

lowast(ℎ + 1) |ℎ + 1⟩

(86)

where

|119886 (ℎ)|2minus |119886 (ℎ + 1)|

2= 119892 (ℎ) minus 119892 (ℎ minus 1) (87)



1= 119876119886


0) = minus119903

11198860


120585 (1198693) equiv minus119903

1exp (minus119896119869

3) = 120585

119902(1198691199023) minus 120585

119901(1198691199013)

= minus1199031199021exp (minus119896

1199021198691199023) + 119903

1199011exp (minus119896

1199011198691199013)

(88)

which yields

[119869+ 119869minus] = 120585 (119869

3) = minus119903

1exp (minus119896119869

3) [119869


plusmn (89)


119892 (ℎ) =1199031

119890119896 minus 1119890minus119896ℎ

= minus1199031

1 minus 119876119876minusℎ 119876 = 119890

119896





119876aelig = 119890119896aelig

(90)


3be



119876119899minus 1

119876 minus 11198761minusℎ (91)


1) reads

119867minus | ℎ⟩ = 119886(ℎ)

2| ℎ⟩ = 1199031

119876119899minus 1

119876 minus 11198761minusℎ| ℎ⟩ (92)



119864119899 (ℎ) = 1199031120572 (ℎ)

119876119899minus 1

119876 minus 1 120572 (ℎ) equiv 119876

1minusℎ (93)


1199021198991and 119864

1199011198992(119899 equiv (119899

1 1198992) 119899

12= 0 1 2 ) as

119867119902minus

10038161003816100381610038161003816ℎ119902⟩ = 119886

119902(ℎ

119902)2 10038161003816100381610038161003816ℎ119902⟩ = 119903

1199021

1198761198991119902minus 1

119876119902minus 1

1198761minusℎ119902

119902

10038161003816100381610038161003816ℎ119902⟩

119867119901minus

10038161003816100381610038161003816ℎ119901⟩ = 119886

119901(ℎ

119901)2 10038161003816100381610038161003816ℎ119901⟩ = 119903

1199011

1198761198992119901minus 1

119876119901minus 1

1198761minusℎ119901

119901

10038161003816100381610038161003816ℎ119901⟩

(94)

and that

1198641199021198991(ℎ

119902) = 119903

1199021120572119902(ℎ

119902)

1198761198991119902minus 1

119876119902minus 1

120572119902(ℎ

119902) equiv 119876

1minusℎ119902

119902

1198641199011198992(ℎ

119901) = 119903

1199011120572119901(ℎ

119901)

1198761198992119901minus 1

119876119901minus 1

120572119901(ℎ

119901) equiv 119876

1minusℎ119901

119901

(95)

Hence

119864119899 (ℎ) = 1198641199021198991

(ℎ119902) minus 119864

1199011198992(ℎ

119901) (96)

7 Conclusions


Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119864119899 (ℎ) = 1199031120572 (ℎ)

119876119899minus 1

119876 minus 1 120572 (ℎ) equiv 119876

1minusℎ (93)


1199021198991and 119864

1199011198992(119899 equiv (119899

1 1198992) 119899

12= 0 1 2 ) as

119867119902minus

10038161003816100381610038161003816ℎ119902⟩ = 119886

119902(ℎ

119902)2 10038161003816100381610038161003816ℎ119902⟩ = 119903

1199021

1198761198991119902minus 1

119876119902minus 1

1198761minusℎ119902

119902

10038161003816100381610038161003816ℎ119902⟩

119867119901minus

10038161003816100381610038161003816ℎ119901⟩ = 119886

119901(ℎ

119901)2 10038161003816100381610038161003816ℎ119901⟩ = 119903

1199011

1198761198992119901minus 1

119876119901minus 1

1198761minusℎ119901

119901

10038161003816100381610038161003816ℎ119901⟩

(94)

and that

1198641199021198991(ℎ

119902) = 119903

1199021120572119902(ℎ

119902)

1198761198991119902minus 1

119876119902minus 1

120572119902(ℎ

119902) equiv 119876

1minusℎ119902

119902

1198641199011198992(ℎ

119901) = 119903

1199011120572119901(ℎ

119901)

1198761198992119901minus 1

119876119901minus 1

120572119901(ℎ

119901) equiv 119876

1minusℎ119901

119901

(95)

Hence

119864119899 (ℎ) = 1198641199021198991

(ℎ119902) minus 119864

1199011198992(ℎ

119901) (96)

7 Conclusions


Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Classical Analog of Extended Phase Space SUSY...

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