Coherent state maps related to the bounded positive operatorsAnatol Odzijewicz, Tomasz Goliński, and Agnieszka Tereszkiewicz Citation: J. Math. Phys. 48, 123514 (2007); doi: 10.1063/1.2821615 View online: http://dx.doi.org/10.1063/1.2821615 View Table of Contents: http://jmp.aip.org/resource/1/JMAPAQ/v48/i12 Published by the American Institute of Physics. Related ArticlesA note on the Liouvillian integrability and the qualitative properties of the mass rate equation for black holes J. Math. Phys. 53, 062701 (2012) Hamiltonian magnetohydrodynamics: Helically symmetric formulation, Casimir invariants, and equilibriumvariational principles Phys. Plasmas 19, 052102 (2012) Super-Poincarè algebras, space-times, and supergravities. II J. Math. Phys. 53, 032505 (2012) Whittaker pairs for the Virasoro algebra and the Gaiotto-Bonelli-Maruyoshi-Tanzini states J. Math. Phys. 53, 033504 (2012) Neural networks and chaos: Construction, evaluation of chaotic networks, and prediction of chaos with multilayerfeedforward networks Chaos 22, 013122 (2012) Additional information on J. Math. Phys.Journal Homepage: http://jmp.aip.org/ Journal Information: http://jmp.aip.org/about/about_the_journal Top downloads: http://jmp.aip.org/features/most_downloaded Information for Authors: http://jmp.aip.org/authors

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Coherent state maps related to the bounded positiveoperators

Anatol Odzijewicz,a� Tomasz Goliński,b� and Agnieszka Tereszkiewiczc�

Institute of Mathematics, University in Białystok, Akademicka 2, 15-267 Białystok, Poland

�Received 11 September 2007; accepted 13 November 2007;published online 18 December 2007�

We show that for any bounded positive operator H with the simple spectrum, onecan canonically define two coherent state maps. The algebras generated by annihi-lation operators defined by these coherent state maps are studied. We describe alsohow the Toda isospectral deformation of H deforms the corresponding coherentstate maps and the related operator algebras. © 2007 American Institute ofPhysics. �DOI: 10.1063/1.2821615�

I. INTRODUCTION

The coherent state method appears prominently in many domains of mathematical physicsincluding quantization of physical systems as well as quantum optics.4,9,13 For these reasons, it issignificant to construct the coherent state map for the system under the investigation.8

In this paper, we present a construction method of the coherent state map of the disk D�Cinto the Hilbert space H using the spectral resolution of the positive bounded operatorH�L��H� with simple spectrum. We explain the relation of the reproducing measure of thiscoherent state map with the spectral measure of H. Also, necessary and sufficient conditions on themoments �5� of H are given for which the algebra AA of annihilation A and creation A* operatorsis the Toeplitz algebra T.

In Sec. III, we study in which manner the Toda isospectral deformation of H �see Ref. 10�defines the deformation equations �46� for the canonical relation �24� as well as the deformationequations for the creation and annihilation operators �47� and �48�. Finally, we show that theproperty AA=T is preserved by the Toda isospectral deformation.

The results obtained in Sec. III lead to the dynamics of the infinite system of points on thepositive axis which form the spectrum of the operator A*A. As it follows from the given con-struction, this system is equivalent to the seminfinite Toda lattice10 and is hence integrable. In Sec.IV, we present a few examples illustrating the investigated model.

II. COHERENT STATE MAPS AND CORRESPONDING CANONICAL RELATIONS

Let us consider a bounded positive operator H with simple spectrum acting on the Hilbertspace H. We recall that the simplicity of the spectrum means that there exists a cyclic vector�0��H, i.e., a vector �0� such that the set �E����0����B�R� is linearly dense in H, whereE :B�R�→L��H� is the spectral measure of H, B�R� is the �-ring of Borel subsets of R, andL��H� is a C*-algebra of bounded operators. We use Dirac notation and we also assume that0 �0�=1. Since H is bounded, this condition is equivalent to the condition that the set�Hn�0��n�N��0� is linearly dense in H. Thus, the map

a�Electronic mail: aodzijew@uwb.edu.plb�Electronic mail: tomaszg@alpha.uwb.edu.plc�Electronic mail: tereszk@uwb.edu.pl

JOURNAL OF MATHEMATICAL PHYSICS 48, 123514 �2007�

48, 123514-10022-2488/2007/48�12�/123514/14/$23.00 © 2007 American Institute of Physics

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I:L2�R,d�� � f � R

f���E�d���0� � H

is an isomorphism of the Hilbert spaces L2�R ,d�� and H, where the measure � is defined by

���� = 0�E���0� . �1�

Moreover, the operator I* �H � I acts on L2�R ,d�� as multiplication by the argument �. One hasthat I��k�=Hk�0� and from the cyclicity of �0�, it follows that the set of all polynomials is dense inL2�R ,d��. Therefore, by the Gram-Schmidt orthonormalization, one obtains the family of poly-nomials Pn orthonormal with respect to the measure � which form a basis of L2�R ,d��. Weobserve that the coefficients �k of the polynomial Pn���=�n�n+ ¯ +�0 are real. Let us define theorthonormal basis in H by

�n� ª Pn�H��0� = I�Pn� , �2�

for n�N� �0�. From the three-term recurrence relation for the orthonormal polynomials, it fol-lows that

H�n� = bn−1�n − 1� + an�n� + bn�n + 1�

and thus the matrix Jª �n �Hm��n,m�N��0� of H in the basis �2� is a Jacobi matrix, i.e., it is athree-diagonal symmetric matrix

J =�a0 b0 0 ¯

b0 a1 b1

0 b1 a2 �

] � �

� , �3�

with bn�0, where b−1=0 and an ,bn�R are given by

an = R

�Pn2�����d��, bn =

R�Pn���Pn+1�����d�� .

The fact that there exists a basis for self-adjoint operators with simple spectrum such that itsmatrix is Jacobi is true also for unbounded operators �see Refs. 1 and 12�

The moments

�k ª R

�k��d�� � 0 �4�

of the measure � can be expressed as

�k = 0�Hk0�, k � N � �0� . �5�

Let us remark that from the positivity of the scalar product, it follows that the Hankel matrix��k+l�k,l�N��0� is positive definite.

One can express the orthonormal polynomials Pn and the coefficients of the Jacobi matrix J interms of moments by the following formulas �see Ref. 1�:

123514-2 Odzijewicz, Goliński, and Tereszkiewicz J. Math. Phys. 48, 123514 �2007�

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Pn��� =1

Dn−1Dn��0 �1 �2 . . . �n

�1 �2 �3 . . . �n+1

] ] ] � ]

�n−1 �n �n+1 . . . �2n−1

1 � �2 . . . �n� for n � N , �6�

and P0����1,

an =�n

Dn−

�n−1

Dn−1, bn =

Dn−1Dn+1

Dn, �7�

where

Dn = ��0 �1 . . . �n

�1 �2 . . . �n+1

] ] � ]

�n �n+1 . . . �2n

�and

�n = ��0 �1 . . . �n−1 �n+1

�1 �2 . . . �n �n+2

] ] � ] ]

�n �n+1 . . . �2n−1 �2n+1

� .

The vacuum expectation value of the resolvent R�ª �H−�1�−1 of H can be expressed as

0�R�0� = R

��dx�x − �

= − �k=0

��k

�k+1 , �8�

where the second equality in �8� is valid for ���� �H�. Recall that the norm of a bounded self-adjoint operator is equal to its spectral radius.

In this paper, by a coherent state map, we will mean a complex analytic map, K :D→H fromthe disk D�C to the Hilbert space H whose image is linearly dense in H. We will assume that�K�v�� is rotationally invariant, i.e., �K�v��= �K�ei�v�� for v�D and 0�2�. Then, K can beexpressed �up to composition with automorphism of H� by the series

K�z� = �n=0

�

cnzn�n� ,

where 0cn�R, for z belonging to the disk of radius R= �lim supn→� n cn�−1. One defines the

annihilation operator A related to K by

AK�z� = zK�z� . �9�

Direct calculations show that

A�n� ªcn−1

cn�n − 1�, n � N �10�

and

A�0� = 0.

The creation operator is defined by

123514-3 Coherent state maps related to the bounded operators J. Math. Phys. 48, 123514 �2007�

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A*�n� ªcn

cn+1�n + 1� �11�

in order to have

An�m� = n�A*m� .

The complex analytic function E defined by

E�v̄w� ª K�v��K�w�� ,

for �v̄w�R2, can be considered as a generalized exponential function. Such functions were in-vestigated in Ref. 8.

We will consider below two coherent state maps defined canonically by the operator H,namely,

KA�z� ª �n=0

�

�nzn�n� �12�

for z�DAª �z�C � �z� �H�−1/2� and

KB�z� ª �n=0

�1

�n

zn�n� �13�

for z�DBª �z�C � �z� �H�1/2�.Let us note that the exponential function EA defined for KA is related to the resolvent of H by

EA�z� = −1

z0�R1/z0� = �

n=0

�

�nzn. �14�

The coherent state map KB defines the resolution of unity,

DB

�KB�z��KB�z���B�dz� = 1 , �15�

with respect to the measure

�B�dz� ª1

2�d�f*��dr� ,

where z=rei� and the radial part of �B is the pullback f*� of the spectral measure �1� by the mapf�x�ªx2. Formula �15� can be expressed in the form

EB�v̄w� = DB

EB�v̄z�EB�z̄w��B�dz� , �16�

where the exponential function

EB�v̄w� = KB�v��KB�w�� = �n=0

�1

�n�v̄w�n �17�

is the reproducing kernel in the Hilbert space L2O�DB ,�B� of holomorphic functions square inte-grable with respect to �B on the disk DB.

Let us denote by T the C*-algebra generated by the unilateral shift operator S defined by

S�n� = �n − 1�, n � N ,

123514-4 Odzijewicz, Goliński, and Tereszkiewicz J. Math. Phys. 48, 123514 �2007�

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S�0� = 0.

The C*-algebra T �called Toeplitz algebra� plays a crucial role in various problems of mathematics�see, e.g., Ref. 3�. Let us observe that T contains the projectors

�n�n� = �S*�nSn − �S*�n+1Sn+1 � T

and

�n + k�n� = �S*�k�n�n� ,

where according to the Dirac notation, one defines ��n�m���v�ª m �v��n� for �v��H. Thus, Tcontains all operators of finite rank and as a consequence of that, it also contains

K � T ,

the C*-ideal K of all compact operators.Proposition 1:

(i) Let us assume that the annihilation operator A defined by

A�n� = �n−1

�n�n − 1� for n � N ,

A�0� = 0 �18�

is bounded. Then, the C*-algebra AA generated by A is the Toeplitz algebra, i.e., AA=T ifand only if the sequence ��n−1 /�n�n�N is convergent.

(ii) Let us assume that the annihilation operator B defined by

B�n� = �n

�n−1�n − 1� for n � N ,

B�0� = 0 �19�

is bounded. Then, the C*-algebra AB generated by B is the Toeplitz algebra, i.e., AB=T ifand only if the sequence ��n /�n−1�n�N is convergent.

(iii) If both A and B are bounded, then AA=AB.

Proof:

�i� Let us assume that the limit limn→� �n−1 /�n exists. From the inequalities

limn→�

�n

�n−1 lim inf

n→�

n �n lim supn→�

n �n limn→�

�n

�n−1,

it follows that the limit limn→� n �n=limn→� �n /�n−1 also exists. Moreover,

0 �H� = limn→�

n �n �

and

infn�N

�n−1

�n

= limn→�

�n−1

�n= �H�−1 � 0. �20�

Due to �20�, the operator AA* is invertible and we can express the shift operator S by

123514-5 Coherent state maps related to the bounded operators J. Math. Phys. 48, 123514 �2007�

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S = �AA*�−1/2A , �21�

which gives T�AA. On the other hand, the operator CªAA*− �H�−11 is compact and 1=SS*�T. So, by �21� and AA*= �H�−11+C, we see that A= �AA*�1/2S�T. Since AA isgenerated by A, one has AA�T.Assuming T=AA, we obtain that the diagonal operators A*A and AA*,

A*A�n� =�n−1

�n�n�, AA*�n� =

�n

�n+1�n� ,

belong to the Toeplitz algebra T. Since the subalgebra of diagonal operators in T is gener-ated by the unity 1 and the diagonal compact operators, we obtain that the sequence�n−1 /�n converges.

�ii� The proof of �ii� is analogous to the proof of �i� of Proposition 1.�iii� Since

AA* = �BB*�−1,

one has that if A is bounded, then BB* is bounded from below and thus invertible. Let usobserve that

A = �BB*�−1B and B = �AA*�−1A .

Thus, the C*-algebra generated by A coincides with the C*-algebra generated by B. �

Note that A ,B�K for any choice of bounded operator H. Everywhere below we will restrictour considerations to the case when AA=AB=T.

The operator Q defined by

Q ª �n=0

�

qn�n�n� , �22�

for 0q1, is compact; thus, it belongs to T. Moreover, the C*-algebra generated by Q and 1coincides with a subalgebra of diagonal operators in T. So, there exists a continuous functionR : spec Q→spec A*A defined by

R�qn� ª�n−1

�nfor n � N � �0� ,

R�0� ª limn→�

�n−1

�n= �H�−1, �23�

for 0q1, which maps continuously the spectra of the operators Q and A*A. We assume asusual that �−1ª0; thus, R�1�ª0. Using R, we obtain the canonical relations

A*A = R�Q� ,

AA* = R�qQ� ,

qQA = AQ ,

qA*Q = QA* �24�

for the operators A* ,A ,Q=Q*�T �see Ref. 8�. If we denote by N the occupation numberoperator

123514-6 Odzijewicz, Goliński, and Tereszkiewicz J. Math. Phys. 48, 123514 �2007�

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N�n� ª n�n� ,

then Q=qN and we can write the relations �24� in the form

A*A = R�qN�, �A,N� = A ,

AA* = R�qN+1�, �A*,N� = − A*. �25�

Let us note that from �24� one has

A*kAk = R�Q� . . . R�q−�k−1�Q� ,

AkA*k = R�qQ� . . . R�qkQ� , �26�

�k =1

R�q� . . . R�qk�=

1

0�AkA*k0�, �27�

and

EA�z� ª 1 + �k=1

�zk

R�q� . . . R�qk�, �28�

EB�z� ª 1 + �k=1

�

R�q� . . . R�qk�zk. �29�

Let us briefly summarize the scheme described above. We start with a bounded operator Hwith fixed cyclic vector �0�. By the spectral theorem, we obtain a probability measure � on R andorthonormal polynomials which allow us to construct a basis in which matrix of H is threediagonal. The moments �k of this measure were expressed as vacuum expectation values ofpowers of H. Under these assumptions, the moment problem has a unique solution; thus, we canreconstruct from the moments the measure � and the matrix of H �see Ref. 1�. So, the momentsdefine the operator H up to unitary equivalence.

Using the moments �k, one defines the coherent state maps KA and KB and the related operatoralgebras AA and AB of the annihilation and creation operators. These algebras are defined by thecanonical relation �24� or, equivalently, by relations �25� which are known in the physical litera-ture as relations corresponding to the deformed harmonic oscillator, e.g., Refs. 2 and 7. Theoperator algebras of this type have applications in quantum optics4 as well as in the theory ofintegrable systems.8,11

In the framework of this paper, one can consider the algebra AA �or AB� as a symmetryalgebra of the physical system whose dynamics is described by the Hamiltonian H.

In this paper, we do not investigate the cases when the algebra AA is different from theToeplitz algebra T. They are the subject for another paper. However, some interesting examples, inour opinion, will be presented in Sec. IV.

III. TODA ISOSPECTRAL DEFORMATION

Now, let us discuss the dependence of the Jacobi matrix J on the choice of the cyclic vector.Let �0�� be another cyclic vector for H. We can generate from Hn�0�� an orthonormal basis ��n���by the procedure described in the preceding section. In this way, one obtains another three-diagonal matrix J� and orthonormal polynomials Pn�. We can define a unitary map U by�n��=U�n�. Let us assume that we have a one-parameter subgroup

123514-7 Coherent state maps related to the bounded operators J. Math. Phys. 48, 123514 �2007�

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R � t � Ut � Aut H

of the group of all unitary operators obtained in this way, i.e., such that the matrix Jt defined by therelation

H = �n,m=0

�

�Jt�nm�m�ttn� �30�

is three diagonal, where

�n�t ª Ut�n� . �31�

By direct calculation, one gets that

d

dt�n�t = Bt

*�n�t,

where Bt is an anti-Hermitian operator defined by

Bt ª � d

dtUt�Ut

*.

Instead of the evolution of the basis �31�, one can fix it and consider the evolution

Ht ª Ut*HUt = �

n,m=0

�

�Jt�nm�m�n� �32�

of H which satisfies the following equation:

d

dtHt = �Ht,Bt� , �33�

with the condition that �0� is cyclic for Ht and Jt is three diagonal for all t�R. From �30� and �32�,it follows that the matrix of Ht in the basis ��n�� coincides with the matrix of H in the basis ��n�t�,i.e.,

n�Htm� = tn�Hm�t.

Following Ref. 10, let us consider now the case when the operator Ht depends on infinitenumber of “times” which we will denote by the same symbol t= �t1 , t2 , . . . �. We will assume thatHt satisfies the Toda lattice equations

�

�tkHt = �Ht,Bkt� , �34�

with respect to all times tk, k�N, i.e., the operators Bkt in �34� are defined as follows:

Bkt ª Htk − P0�Ht

k� − 2P+�Htk� ,

where P0�Htk� is the diagonal operator defined by

P0�Htk� ª �

n=0

�

�Jtk�nn�n�n� �35�

and P+�Htk� is the upper-triangular operator defined by

123514-8 Odzijewicz, Goliński, and Tereszkiewicz J. Math. Phys. 48, 123514 �2007�

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P+�Htk� ª �

n=1

�

�m=0

n−1

�Jtk�mn�m�n� . �36�

Let us stress that the basis in �35� and �36� does not depend on t. From

P0�Htk��0� = 0�Ht

k0��0�

and

P+�Htk��0� = 0,

we find that

Bkt�0� = �Htk − �k�t���0� . �37�

It follows from �34� that the moments �k�t�= 0 �Htk0� satisfy the system of nonlinear difference-

differential equations

�

�tl�k�t� = 2��k+l�t� − �k�t��l�t�� �38�

for k , l�NFrom �38� one has �� /�tk��l�t�= �� /�tl��k�t�. Thus, there exists a function �t�= �t1 , t2 , . . . �

such that

�k�t� =1

2

�

�tklog �t� . �39�

Substituting �39� into �38�, we obtain the system of linear differential equations

�

�tk

�

�tl �t� = 2

�

�tk+l �t� �40�

on the function .Let us now find the system of equations on the measure �t���ª 0 �Et���0�, where

Et :B�R�→L��H� is the spectral measure for Ht. From

�

�tkEt��� = �Et���,Bkt� , �41�

by direct application of �37�, we obtain

�

�tk�t�d�� = ��k −

R�k�t�d����t�d�� . �42�

Summing up, one has the evolution equations for the following objects: the operator Ht, themeasure �t, the moments �k�t�, and the function �t�. We can reconstruct any of these objects fromanother, so the solution of evolution equation for one of them solves the problem for all.

The equations �40� for are linear and their solution is given by

�t� = �0�R

e2�l=1� tl�

l�0�d�� �43�

for the initial conditions �k�0�=�k. Let us remark that the series �l=1� tl�

l converges iflim supk�N

k tk �H0�−1. For example, this condition is satisfied for ��ktk�� l�, where ��R+ sat-isfies 0��H0�1.

From �43�, we get that

123514-9 Coherent state maps related to the bounded operators J. Math. Phys. 48, 123514 �2007�

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�t�d�� =e2�l=1

� tl�l

R

e2�l=1� tl�

l�0�d��

�0�d�� �44�

and

�k�t� =1

R

e2�l=1� tl�

l�0�d��

R

�ke2�l=1� tl�

l�0�d�� , �45�

where �0� and �0 are initial data for the corresponding quantities.In order to solve the Toda equation �34�, one needs to apply the formulas �7� following from

the orthogonal polynomial theory, which express the Jacobi matrix Jt by the moments �k�t� �seeRef. 1�.

Since Rt�qk�=�k−1�t� /�k�t�, from �38�, we find the system of evolution equations

�

�tlRt�Q� = 2Rt�Q�� 1

Rt�Q�Rt�qQ� . . . Rt�ql−1Q�−

1

Rt�qQ�Rt�q2Q� . . . Rt�qlQ�� �46�

on the structural function Rt.Using

At = Rt�Q�S and AtlAt

*l = Rt�qQ� . . . Rt�qlQ� , �47�

one obtains the hierarchy of equations

�

�tlAt = ��At

lAt*l�−1,At�,

�

�tlAt

* = − ��AtlAt

*l�−1,At*� �48�

on the creation and annihilation operators At and At*, which is equivalent to the hierarchy of Toda

equations on the operator Ht.Proposition 2:

(i) If there exist limn→� �n−1�0� /�n�0�¬� (which in the considered case is �H0�−1), then

limn→�

�n−1�t��n�t�

= �

for t�R.(ii) If AA�0�=T for t=0, it is also true that AA�t�=T for t�R.

Proof:

�i� One has

�n−1�t��n�t�

=

R

�n−1e2�l=1� tl�

l�0�d��

R

�ne2�l=1� tl�

l�0�d��

=

�l�N��0�

sl�t��l+n−1�0�

�l�N��0�

sl�t��l+n�0�,

where sl�t�=sl�t1 , t2 , . . . , tl� are elementary Schur polynomials defined by e�l=1� tl�

l

= :�l=0� sl�t1 , t2 , . . . , tl��l, e.g., see. Ref. 5. For ��0 and appropriate N�N, we have that for

n�N and tl�0 the inequality holds

123514-10 Odzijewicz, Goliński, and Tereszkiewicz J. Math. Phys. 48, 123514 �2007�

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� − � �l+n−1�0��l+n�0�

� + � .

Thus, we obtain

� − �

�l�N��0�

sl�t��l+n−1�0��l+n�0�

�l+n�0�

�l�N��0�

sl�t��l+n�0� � + � . �49�

The inequality �49� shows that

limn→�

�n−1�t��n�t�

= � �50�

for ti�0. Thus, because of the group property for the t evolution, �50� is valid for any t.�ii� This statement follows from point �i� of the proposition and point �i� of Proposition 1.

�We conclude from Proposition 2 that the isospectral deformation changes the canonical rela-

tions �24� but does not change the algebra AA=T.

IV. EXAMPLES AND APPLICATIONS

In this section, we illustrate by few examples the notions presented in Sec. I.Example 1: Let us take for a structural function an arbitrary rational function

R�x� =�1 − x��1 − b1q−1x� . . . �1 − bj−1q−1x�

�1 − a1q−1x� . . . �1 − ajq−1x�

�1 − ��1��x�� , �51�

where ��1� is the characteristic function of the set �1� and ai ,bi1. This function is continuous onspec Q and R�1�=0. The generalized exponential functions �14� and �17� for R given by �51� arebasic hypergeometric series j� j−1,

EA�z� = j� j−1�� a1 . . . aj

b1 . . . bj−1�q;z� = �

n=0

��a1;q�n . . . �aj;q�n

�q;q�n�b1;q�n . . . �bj−1;q�nzn, �52�

EB�z� = j+1� j��qqb1 . . . bj−1

a1 . . . aj�q;z� , �53�

defined for �z�1. Thus, the moments �n are given by

�n =�a1;q�n . . . �aj;q�n

�q;q�n�b1;q�n . . . �bj−1;q�n, �54�

where

��;q�n ª �1 − ���1 − �q� . . . �1 − �qn−1�

is the q-Pochhammer symbol.Let us present the case j=1 in detail. One has

R�x� =1 − x

1 − ax, �55�

with the restriction that 1aq−1. The relations �24� take the form

123514-11 Coherent state maps related to the bounded operators J. Math. Phys. 48, 123514 �2007�

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1 − Q = �1 − aQ�A*A ,

1 − qQ = �1 − aqQ�AA*. �56�

The moments �k and coefficients an, bn of the Jacobi matrix are

�k =�aq;q�k

�q;q�k,

an =qn�1 − aqn+1��1 − qn��1 − q2n��1 − q2n+1�

+aqn�1 − qn��1 − a−1qn−1�

�1 − q2n−1��1 − q2n�,

bn = aq2n+1�1 − qn+1��1 − aqn+1��1 − a−1qn��1 − qn��1 − q2n��1 − q2n+1�2�1 − q2n+2�

.

In this case, the measure � is discrete,

��d�� =�aq;q���a−1,q��

�q;q���q;q���n=0

��q�;q��an�

�a−1�;q��

��� − qn�d� .

The polynomials orthonormal with respect to this measure are a subclass of little q-Jacobi poly-nomials �see Ref. 6� The exponential functions EA and EB are the following:

EA�z� = 1�0��aq

−�q;z� ,

EB�z� = 2�1��q , q

a�q;z� .

The reproducing property �16� takes the form

EB�v̄w� =1

2�

0

1 �aq;q���a−1,q��

�q;q���q;q���n=0

��qr;q��anr

�a−1r;q��

��r − q2n�dr0

2�

d�EB�v̄rei��EB�re−i�w� .

For j=2 and a1=q, b1a2, 0a21, we get

R�x� =�1 − b1q−1x��1 − a2q−1x�

�1 − ��1��x�� . �57�

The relations �24� take the form

�1 − a2q−1Q�A*A = �1 − b1q−1Q��1 − �0�0�� ,

�1 − a2Q�AA* = 1 − b1Q . �58�

The moments �n and the coefficients of the Jacobi matrix are

�n =�a2;q�n

�b1;q�n,

an =qn�1 − a2qn��1 − b1qn−1��1 − b1q2n−1��1 − b1q2n�

+a2qn−1�1 − qn��1 − qn−1b1/a2�

�1 − b1q2n−2��1 − b1q2n−1�,

123514-12 Odzijewicz, Goliński, and Tereszkiewicz J. Math. Phys. 48, 123514 �2007�

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bn = a2q2n�1 − qn+1��1 − qnb1/a2��1 − a2qn��1 − b1qn−1��1 − b1q2n−1��1 − b1q2n�2�1 − b1q2n+1�

.

The measure is

��d�� =�a2;q���b1/a2;q��

�b1;q���q;q���n=0

��q�;q��a2

n

��b1/a2;q��

��� − qn�d� .

The exponential functions EA and EB are the following

EA�z� = 2�1��a2, q

b1�q;z� ,

EB�z� = 2�1��b1, q

a2�q;z� .

The reproducing property �16� takes the form

EB�v̄w� =1

2�

0

1 �a2;q���b1/a2;q��

�b1;q���q;q���n=0

��rq;q��a2

n

�rb1/a2;q��

��r − q2n�dr0

2�

d�EB�v̄rei��EB�re−i�w� .

The orthogonal polynomials corresponding to this case are the little q-Jacobi polynomials. Forb1=q, we get the previous case j=1. For b1=q2, a2=q, we get the little q-Legendre polynomialsand for b1=0, 0a21, we get the little q-Laguerre/Wall polynomials �see Ref. 6�.

Example 2: This example concerns the case related to the classical Jacobi polynomials. Itmeans that the Jacobi matrix �3� of the operator H is given by

an =�2 − �2

�2n + � + ���2n + � + � + 2�,

bn = 2 �n + 1��n + 1 + ���n + 1 + ���n + 1 + � + ���2n + � + � + 1��2n + � + � + 2�2�2n + � + � + 3�

for � ,��−1. In this case, the measure �1� and the moments �4� assume the following form:

��d�� = �1 − ����� ��� + � + 2���� + 1���� + 1�

��0,1����d� ,

�n =�� + 1�n

�� + � + 2�n.

One easily obtains that the structural function �23� is

R�qx� =� + � + 1 + x

� + x�1 − ��0��x�� , �59�

which leads to the canonical relations

�� + N�A*A = �� + � + 1 + N� ,

�� + 1 + N�AA* = � + � + 2 + N , �60�

involving the occupation number operators N.The exponential functions EA and EB are given by

123514-13 Coherent state maps related to the bounded operators J. Math. Phys. 48, 123514 �2007�

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EA�z� = 2F1�� � + 1, 1

� + � + 2�z� ,

EB�z� = 2F1��� + � + 2, 1

� + 1�z� ,

respectively. The exponential function EB satisfies the reproduction property �16� with the measure�B�dz� given by

�B�dz� =1

2��1 − r2��r2� ��� + � + 2�

��� + 1���� + 1���0,1��r�d�dr .

The case considered in this example contains the following subcases �see Ref. 6�:

�i� for �=�=�− 12 , ��0, we get Gegenbauer/ultraspherical polynomials;

�ii� for �=�=− 12 , we get Chebychev I kind polynomials;

�iii� for �=�= 12 , we get Chebychev II kind polynomials;

�iv� for �=�=0, we get Legendre/spherical polynomials.

ACKNOWLEDGMENTS

We would like to thank T. S. Ratiu for his interest in the subject. The authors also highlyappreciate the referee for remarks and corrections which made the paper more readable. Theauthors wish to acknowledge partial support by the Polish Grant No. 1 PO3A 001 29.

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