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Bivariate raising and lowering differential operators for eigenfunctions of a 2D Fourier

transform

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2015 J. Phys. A: Math. Theor. 48 075201

(http://iopscience.iop.org/1751-8121/48/7/075201)

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Bivariate raising and lowering differentialoperators for eigenfunctions of a 2D Fouriertransform

Iván Area1, Natig Atakishiyev2, Eduardo Godoy3 andKurt Bernardo Wolf4

1Departamento de Matemática Aplicada II, E.E. Telecomunicación, Universidade deVigo, Campus Lagoas-Marcosende, E-36310 Vigo, Spain2 Instituto de Matemáticas, Unidad Cuernavaca, Universidad Nacional Autónoma deMéxico, Av. Universidad s/n, Cuernavaca, Morelos 62210, México3Departamento de Matemática Aplicada II, E.E. Industrial, Universidade de Vigo,Campus Lagoas-Marcosende, E-36310 Vigo, Spain4 Instituto de Ciencias Físicas,Universidad Nacional Autónoma de México, Av.Universidad s/n, Cuernavaca, Morelos 62210, México

E-mail: area@uvigo.es, natig@matcuer.unam.mx, egodoy@dma.uvigo.es andbwolf@fis.unam.mx

Received 6 October 2014, revised 9 December 2014Accepted for publication 23 December 2014Published 28 January 2015

AbstractWe define a two-dimensional (2D) Fourier transform that self-reproduces aone-parameter family of bivariate Hermite functions; these are eigenfunctionsof a Hamiltonian differential operator of second order, whose exponential isthat transform. We find explicit forms of the bivariate raising and loweringpartial differential operators of first degree for the eigenfunctions of this 2DFourier transform.

Keywords: bivariate Fourier transform, bivariate Hermite polynomials, raisingand lowering partial differential operators, eigenfunctions of the bivariateFourier transform, two-dimensional harmonic oscillator

(Some figures may appear in colour only in the online journal)

1. Introduction

The Fourier transform, the quantum harmonic oscillator and the associated Lie algebraicrelations, have been an unusually fruitful field of research, because beyond the classicaldefinitions and theory, several extensions have been developed which continue to display

Journal of Physics A: Mathematical and Theoretical

J. Phys. A: Math. Theor. 48 (2015) 075201 (12pp) doi:10.1088/1751-8113/48/7/075201

1751-8113/15/075201+12$33.00 © 2015 IOP Publishing Ltd Printed in the UK 1

their intimate connections, applying to diverse physical systems. These include signal ana-lysis and image processing in their discrete and finite versions, which are fundamental formost nuclear models, and particularly their raising and lowering operators become realized inphotonic devices that harbor entangled states of light. Deformations of the original mathe-matical frame have also yielded interesting results when the main components retain theirroles: similar integral kernels, Hermite functions, and their raising and lowering operatorsclosing into a Heisenberg–Weyl Lie algebra.

The interest of mathematical physics researchers in multivariate special functions hasincreased in connection with various possible generalizations of the well-known one-variablespecial functions to more than one-dimensions (1D) [1] that are not separable as Cartesianproducts. For instance, the multivariable generalization of Bessel functions [2], originallyintroduced by Appell and Kampé de Feriet [3], which arise in the problem of the ellipticmotion of planets [4]. Correspondingly, the multivariate generalization of classical orthogonalpolynomials has attracted great interest recently [5–7]; the Hermite polynomials of severalvariables appear quite naturally in the description of quantum systems governed by quadraticHamiltonians, in the kinetic theory of gases, of fluctuations, and in optical systems (see e.g.[8–10] and references therein).

The subject of this work is the study of a new one-parameter family of two-dimensional(2D) generalizations of the classical Fourier transform, their Hermite function eigenvectorsbound through raising and lowering operators. The basic and well-known frame is laid out insection 2 to serve as comparison for the expressions that are thereafter subject to general-ization. Section 3 includes the definition of the multivariate Hermite polynomials and func-tions, which goes back to Charles Hermite [11] in 1864, and some of their integral anddifferential properties; however, their systematic use for the study of mathematical physicsproblems of Hamiltonian phase spaces is much more recent. In section 4 we introduce thedefinition of a bivariate Fourier transform that reproduces this particular family of bivariateHermite functions. The bivariate raising and lowering differential operators for this set ofdeformed 2D oscillator eigenstates are given in section 5, and confirm that, here too, thebivariate Fourier transform is the exponential of the Hamiltonian operator. The concludingsection 6 offers a brief discussion of some further research directions that may be of interest.

2. The harmonic oscillator and Fourier transform

The 1D Hermite polynomials and their properties are very well known [12, section 5.5]; herewe mention only those germane to the solution of the harmonic oscillator system in quantummechanics, whose wave functions in the dimensionless variable x are

ψπ

=√

− = …( )xn

H x x n( )1

2 !( ) exp 2 , 0, 1, 2, , (1)n

nn

2

where Hn(x) are the Hermite polynomials [17, (10.13.9)]

∑= − −− − = −

−=

−H x x F n n

xx

x

s n s( ) (2 ) (1 ) 2 2 1

(2 )( 1) (2 )

! ( 2 )!, (2)n

n n

s

n s n s

2 0 20

[ 2] 2⎛⎝⎜

⎞⎠⎟

orthogonal with respect to the weight function −xexp ( )2 ,

∫πδ− =

−∞

∞( )x H x H x x n

1exp ( ) ( )d 2 ! ,n m

nn m

2,

and where δn m, denotes the Kronecker delta.

J. Phys. A: Math. Theor. 48 (2015) 075201 I Area et al

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In quantum mechanics they are eigenfunctions of the linear harmonic oscillator Hamil-tonian

ψ ψ ψ= − = +x xx

x n xH ( )1

2

d

d( )

1

2( ), (3)n n n

22

2⎜ ⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝

⎞⎠

which is a self-adjoint second order differential operator. This is an orthonormal set offunctions in the Hilbert space ( )2 defined by closure with respect to the sesquilinear innerproduct

∫ψ ψ ψ ψ δ= =

x x x( , ) ( ) ( )d ,n m n m n m,

where integration is over the full real line ∈ x . The basis of functions ψ =∞{ }n n 0 is dense in

( )2 : any function ∈ f x( ) ( )2 can be approximated in the norm by the expansionψ= ∑ =

∞f x c x( ) ( )n n n0 and the expansion coefficients =∞c{ }n n 0 are determined by the integrals

ψ=( )c f,n n for =n 0, 1,... .

The Fourier transform of Lebesgue square-integrable functions on the real line is theunitary map of the Hilbert space ( )2 given by

∫π

→ = = ′ ′ ′∼

f x f x f x xx f x xF( ) ( ) ( )( )1

2exp (i ) ( )d . (4)

Among its many well-known properties is that of being the fourth root of the unit operator,=F 14 , and that it transforms the quantum harmonic oscillator wave functions (1) into

themselves, multiplying them by the phase i n,

∫ψπ

ψ ψ= ′ ′ ′ =

( ) x xx x x xF ( )1

2exp (i ) ( )d i ( ). (5)n n

nn

Passing now to 2D of space, x y, , the isotropic 2D harmonic oscillator is characterizedby the Hamiltonian

= − ∂∂

+ ∂∂

+ + ( )x y

x y1

2

1

2.

2

2

2

22 2

⎛⎝⎜

⎞⎠⎟

The associated Schrödinger equation ψ ψ= separates in Cartesian coordinates, yieldingthe 2D oscillator wave functions

ψπ

= − ++

x yn m

x yH x H y( , )

1

2 ! !exp

2( ) ( ), (6)n m n m

n m,

2 2⎛⎝⎜

⎞⎠⎟

with energy eigenvalues = + + n m 1.The Cartesian product of two ordinary Fourier transforms (4) in x and y is the well-

known 2D Fourier transform

∫∫π= = ′ ′ ′ + ′ ′ ′∼

f x y f x y f x y xx yy x y( )( , ) ( , )

1

2( , ) exp(i( ))d d . (7)

2

The 2D harmonic oscillator wavefunctions (6) are naturally its eigenfunctions

ψ ψ= +( ) x y x y( , ) i ( , ). (8)n mn m

n m, ,

The 2D Fourier transform (7) represents a bivariate extension of (4) in separated variables,characterized by the integral kernel

J. Phys. A: Math. Theor. 48 (2015) 075201 I Area et al

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′ ′ = ′ + ′K x y x y xx yy( , ; , ) exp(i( )). (9)

Evidently, similar expressions are valid for the isotropic three- and higher-dimensionalharmonic oscillators [5, 13, 14].

3. Bivariate Hermite polynomials and functions

Other ways of generalizing classical univariate Hermite polynomials [15] to bivariateexpressions were proposed by Hermite himself [11], who introduced a class of polynomials inseveral variables with generating functions that are exponentials of quadratic forms. Webriefly recall here some properties of the bivariate Hermite polynomials that were analyzedlater by Appell and Kampé de Feriet [3].

Consider the positive definite quadratic form defined by

φ Λ= = = + +Λ ( )( )( )x y x y a bb c

xy ax bxy cyx x( , ) 2 , (10)t 2 2

where Λ = ( )a

b

b

c, = x yx ( )t , >a c, 0, and δ Λ= = − >ac bdet( ) 02 . Then, as introduced

by Hermite [3, p 373], and studied by means of a Rodrigues-type formula, the bivariatehypergeometric Hermite polynomials are defined by

Λ

φ φ= − ∂∂ ∂

− =Λ Λ+

+( ) ( )

H x y

x yx y

x y n m

( , ; )

( 1) exp ( , ) exp ( , ) , , 0, 1, 2 ,... . (11)

n m

n mn m

n m

,

⎡⎣ ⎤⎦One particular case of (11), which has turned out to be a useful tool in the study of

quantum-mechanical harmonic oscillator entangled states [16], corresponds to an antidiagonalΛ matrix, with = =a c0 and τ=b 2 . In that case, the incomplete two-variable Hermitepolynomials are defined by

∑

τ ττ τ

τ

ττ

= − ∂∂ ∂

=− −

= −

++

=

− −

− + − − − + −

h x yxy

x y

xy

n mx y

r n r m r

n m x y Lxy

( , ) exp exp

! !! ( ) ! ( )!

min ( , ) ! , (12)

n mn m

n m

n m

r

n m r n r m r

n m m n n m m n n mn m

n m

,

0

min ( , )

( ) ( ) min( , )min ( , )

12

12

⎜ ⎟ ⎜ ⎟

⎜ ⎟

⎛⎝

⎞⎠

⎡⎣⎢

⎛⎝

⎞⎠

⎤⎦⎥

⎛⎝

⎞⎠

where

∑ α= +−

−α

=

L xn

n k

x

k( )

( )

!,n

k

n k

0

⎜ ⎟⎛⎝

⎞⎠

are the associated Laguerre polynomials [10].

Finally, when Λ is the identity matrix =I 1 00 1

⎜ ⎟⎛⎝

⎞⎠ ( = =a c 1 and b = 0), these bivariate

polynomials return to the product of two univariate Hermite polynomials in separated vari-ables

=H x y H x H yI( , ; ) ( ) ( ), (13)n m n m,

where Hn(x) and Hm(y) are defined as in (2).For general values of the parameters a b c, , , the polynomials (11) can be expressed in

terms of Hn(x) by the bilinear expression [3, p 370, equation (21)]

J. Phys. A: Math. Theor. 48 (2015) 075201 I Area et al

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∑

Λ

= − + +

=− −

− −( )( )

H x y

k a b c Hax by

aH

bx cy

c

( , ; )

( 2) ! . (14)

n m

k

n mk m

k

n

kk

n k m k

,

0

min ( , )n k m k

2 2⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

These bivariate Hermite polynomials Λ =H x y u x y( , ; ) ( , )n m, are of total degree +n m, andin particular Λ =H x y( , ; ) 10,0 . They also satisfy the following second-order partialdifferential equation [17, p 288, equation (10)]

δ δ

∂∂

− ∂∂ ∂

+ ∂∂

− ∂∂

+ ∂∂

+ + =

cu x y

xb

u x y

x ya

u x y

y

xu x y

xy

u x y

yn m u x y

( , )2

( , ) ( , )

2( , ) ( , )

2 ( ) ( , ) 0, (15)

2

2

2 2

2

⎛⎝⎜

⎞⎠⎟

where we recall that δ = − >ac b 02 . This is a bivariate second-order linear partialdifferential equation that is admissible, potentially self-adjoint and of hypergeometric type[18]. Finally, a generating function for these bivariate Hermite polynomials has been given in[3, p 370, equation (18)].

The bivariate Hermite polynomials (11) of distinct total degrees are mutually orthogonal[15, 19] with respect to the weight function φ− Λ x yexp ( ( , )),

∫∫ φ Λ Λ Ω δ− =Λ

( )x y H x y H x y x y n j m kexp ( , ) ( , ; ) ( , ; ) d d ( , , , ) , (16)n j m k N M, , ,2

where = +N n j, = +M m k , and

∑Ω π=−

− − − + >+ −

=

+−n j m k

n a b c

ac bk m s

j

s

b

ac( , , , )

2 !( 1) ( ) ( 1) 0,

n j m k j j

s

jj s

j s s

s

20

2⎜ ⎟⎛⎝

⎞⎠

⎛⎝⎜

⎞⎠⎟

where Γ Γ= +u u m u( ) : ( ) ( )m is the Pochhammer symbol. The orthogonality relation (16) isreadily derived from (14) and (2) by using repeatedly the formula [20, equation (3.462.2), p337]. Observe carefully that the orthogonality relation (16) implies that a polynomial

ΛH x y( , ; )n j, of total degree +n j and ≠b 0, is not orthogonal to any other polynomialΛH x y( , ; )m k, of the same total degree + = +m k n j. Only in the ‘Cartesian’ case b = 0

orthogonality holds between any two bivariate Hermite polynomials.We remind the reader that it is well known from the book by Appell and Kampé de Feriet

[3] that the bivariate Hermite polynomials (11) are also biorthogonal. This existence of twotypes of orthogonality relations is intimately related with the fact that the spectrum of thesecond-order linear partial differential equation (16), that governs the bivariate Hermitepolynomials (11), is degenerate.

Properties of the zeros of the bivariate Hermite polynomials (11) have been obtainedrecently [19] by considering the affine transformation

= + = +s

ax by

at

bx cy

c, , (17)

which enables one to transform (14) into

∑Λ = −=

− −− −

H s t km

k

n

ka b c H s H t( , ; ) ( 2) ! ( ) ( ), (18)n m

k

n mk k

n k m k,

0

min ( , )n k m k

2 2⎜ ⎟ ⎜ ⎟⎛⎝

⎞⎠

⎛⎝

⎞⎠

for ⩾n m, 0.Notice in the definition of the quadratic form φΛ x y( , ) in (10), that the parameters a and c

essentially produce only re-scalings in the directions of the x y, coordinates axes; we shall

J. Phys. A: Math. Theor. 48 (2015) 075201 I Area et al

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thus fix = =a c1 , keeping only the parameter b in what follows, writing

∑= −=

− −H s t b km

k

n

kb H s H t( , ; ) ( 2) ! ( ) ( ), (19)n m

k

n mk k

n k m k,

0

min ( , )

⎜ ⎟ ⎜ ⎟⎛⎝

⎞⎠

⎛⎝

⎞⎠

with δ = − >b1 02 for ⩾n m, 0. These polynomials are solutions of the partial differentialequation

∂∂

+ ∂∂ ∂

+ ∂∂

− ∂∂

+ ∂∂

+ + =s

bs t t

ss

tt

n m H s t b2 2 2( ) ( , ; ) 0, (20)n m

2

2

2 2

2 ,⎜ ⎟⎡⎣⎢

⎛⎝

⎞⎠

⎤⎦⎥

satisfy the differential shift relations

∂∂

=∂

∂=

− −

H s t b

sn H s t b

H s t b

tm H s t b

( , ; )2 ( , ; ),

( , ; )2 ( , ; ), (21)

n mn m

n mn m

,1,

,, 1

and exhibit the orthogonality relation

∫∫ Ω δ− −−

=

bst s t

bH s t b H s t b s t n j m kexp

2

1( , ; ) ( , ; )d d ( , , , ) , (22)n j m k N M

2 2

2 , , ,2

⎛⎝⎜

⎞⎠⎟

with = +N n j, = +M m k, and

∑Ω π= − −

× − − + >

+ −

=

+

−

( )n j m k n b b

k m s b

( , , , ) 2 ! 1 ( 1)

( ) ( 1) 0. (23)

n j k j

s

jj s j

s

j s ss

2

0

2

In particular, for m = n, k = j and ∈ −b ( 1, 1), one has the normalization in the form

∫∫π

− −−

=

= − − − >

+

( )

H s t b H s t bbst s t

bs t d b

d b b n j F j n b

( , ; ) ( , ; ) exp2

1d d ( )

( ) 2 1 ! ! , ; 1; 0. (24)

n j n j n j

n jn j

, ,

2 2

2 ,

,2

2 12

2

⎛⎝⎜

⎞⎠⎟

Finally, when b = 0, the case of separated variables described above, is recovered.Following the common quantum mechanical oscillator functions (1), which incorporate

the measure and normalization of the Hermite polynomials, attending (22) it is natural todefine the bivariate Hermite functions associated to the bivariate Hermite polynomialsH s t b( , ; )n m, , as

ψ = − − +−

( )

s t bd b

s bst t

bH s t b( , ; )

1

( )exp

2

2 1( , ; ), (25)n m

n mn m,

,

2 2

2,

⎛

⎝⎜⎜

⎞

⎠⎟⎟

for ⩾n m, 0. In particular, the ground state wave function is

ψπ

=−

− − +−( )

s t bb

s bst t

b( , ; )

1

1exp

2

2 1. (26)0,0

2

2 2

2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

This is a 2D Gaussian function, squeezed with the ratio + −b b(1 ) (1 ) in the direction givenby the angle π bsign( )1

4. In figure 1 we show three ground states corresponding to different

values of b.

J. Phys. A: Math. Theor. 48 (2015) 075201 I Area et al

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From (21) one readily derives the differential shift relations

ψ ψ∂∂

+ −−

= −−s

s bt

bs t b n

d b

d bs t b

1( , ; ) 2

( )

( )( , ; ), (27)n m

n m

n mn m2 ,

1,

,1,

⎛⎝⎜

⎞⎠⎟

ψ ψ∂∂

+ −−

= −−t

t bs

bs t b m

d b

d bs t b

1( , ; ) 2

( )

( )( , ; ). (28)n m

n m

n mn m2 ,

, 1

,, 1

⎛⎝⎜

⎞⎠⎟

As a consequence, the bivariate Hermite functions will shift both indices as

ψ ψ∂∂

+ −−

∂∂

+ −−

= − −− −s

s bt

b t

t bs

bs t b nm

d b

d bs t b

1 1( , ; ) 4

( )

( )( , ; ).n m

n m

n mn m2 2 ,

1, 1

,1, 1

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

The bivariate Hermite functions are solutions of the partial differential equation (20), thatwe can cast in the form of a Hamiltonian second-order partial differential eigenfunctionequation, which entangles the s and t directions of a 2D harmonic oscilator as

ψ ψ= + + s t b n m s t b( , ; ) ( 1) ( , ; ), (29)bn m n m

( ), ,

where the Hamiltonian operator is

= − +−

− ∂∂

− ∂∂ ∂

− ∂∂

s bst t

b sb

s t t:

1

2

2

12 . (30)b( )

2 2

2

2

2

2 2

2

⎡⎣⎢

⎤⎦⎥

Observe that, as in the case of ordinary Cartesian separation (b = 0), the spectrum of thebivariate second-order differential Hamiltonian operator b( ) is finitely degenerate: +N 1functions ψ s t b( , ; )n m, , + =n m N , with indices =n m N( , ) {( , 0); − …N( 1, 1); ;

−N N(1, 1); (0, )} correspond to the same eigenvalue N of the operator b( ).The inner product of two bivariate Hermite functions corresponding to the 1D case (3) is,

from (22)

∫∫ ψ ψ δ Ω=

+ +

s t b s t b s tn j m k

d b d b( , ; ) ( , ; )d d

( , , , )

( ) ( ), (31)n j m k n j m k

n j m k, , ,

, ,2

where Ω n j m k( , , , ) is defined as in (23), which implies that each of these functions is indeednormalized

∫∫ ψ =

( )s t b s t( , ; ) d d 1. (32)n j,

2

2

0.40.20.0-5

0s

5-5

0 t

50.80.60.40.20.0-5

0s

5-5

0 t

5

0.60.40.20.0-5

0s

5 -5

0 t

5

Figure 1. Ground states ψ s t b( , ; )0,0 , for values of =b 1 10, 9 10 and −1 2.

J. Phys. A: Math. Theor. 48 (2015) 075201 I Area et al

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Moreover, if + > +n j m k ( >N M ), then

∫∫ ψ ψ =

s t b s t b s t( , ; ) ( , ; )d d 0. (33)n j m k, ,2

4. The bivariate Fourier transform and its eigenfunctions

We define the bivariate Fourier transform as the following integral transform

∫∫π= =∼

( )f x y f s t b f x y K s t x y b x y( , ) ( , ; )

1

2( , ) ( , ; , ; ) d d , (34)b

2

with the kernel

δ=

−+ − +K s t x y b

bsx ty b sy tx( , ; , ; )

1

1exp

i[ ( )] , (35)

2⎜ ⎟⎛⎝

⎞⎠

and δ = − b1 2; the exponent can be written as

+ − + = −−( )( )sx ty b sy tx s t b

bxy( ) ( , ) 1

1. (36)

When b = 0 this kernel reduces to ordinary Fourier kernel (9).One can verify that the bivariate Fourier transform b reproduces the ground state

ψ x y b( , ; )0,0 in (26)

∫∫

∫∫

∫

ψπ

π

π

π

πψ

=−

− − +−

=−

− − +−

=−

− + + + − + +−

=−

+ − + + −−

=−

− + +−

=

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( ) ( )

x y bb

x bxy y

b

b

x bxy y

bK s t x y b x y

b

y t b s x x bt s x y

bx y

b

b t y bst s y y t

by

b

bst s t

bs t b

( , ; )1

1exp

2

2 1

1

2 1exp

2

2 1( , , , ; ) d d

1

2 1exp

2i ( ( i )) (2i 2i )

2 1d d

1

2 1exp

( i ) 2 ( 2i )

2 1d

1

1exp

2

2 1( , ; ). (37)

b b0,02

2 2

2

3 2 2 3 4

2 2

2

3 2 2 3 4

2

2

2 1 4

2 2 2

2

2 1 4

2 2

2 0,0

2

2

⎛

⎝⎜⎜

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎞

⎠⎟⎟

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎛

⎝⎜⎜

⎞

⎠⎟⎟

Then, for the all other functions ψ s t b( , ; )n m, defined in (25), for any n and m nonnegativeintegers, from (18) and (25) it follows by integration that

ψ ψ= +( ) x y b s t b( , ; ) i ( , ; ). (38)b n mn m

n m, ,

Finally, we note that equation (29) indicates that the integral bivariate Fourier transform(34) can also be written in operator form as

J. Phys. A: Math. Theor. 48 (2015) 075201 I Area et al

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π= − ( )exp i2

1 . (39)bb( )⎜ ⎟

⎛⎝

⎞⎠

This type of exponential relation is well known to hold for the ordinary Fourier transform andthe quantum harmonic oscillator Hamiltonian [21].

5. Bivariate raising and lowering operators

To construct raising and lowering operators for eigenfunctions of the Hamiltonian b( ) in (30)and of the bivariate Fourier transform b, we derive first some differential identities for itskernel (35), noting that one can variously write the exponent as

+ − + = − + − = − + −sx ty b sy tx s bt x t bs y x by s y bx t( ) ( ) ( ) ( ) ( ) ,

so that its partial derivatives easily yield

δ∂∂

= −s

K s t x y b x by K s t x y b( , ; , ; )i

( ) ( , ; , ; ), (40)

δ∂∂

= −t

K s t x y b y bx K s t x y b( , ; , ; )i

( ) ( , ; , ; ), (41)

and

= ∂∂

+ ∂∂

s K s t x y bx

by

K s t x y b( , ; , ; )1

i( , ; , ; ), (42)

⎛⎝⎜

⎞⎠⎟

= ∂∂

+ ∂∂

t K s t x y b bx y

K s t x y b( , ; , ; )1

i( , ; , ; ). (43)

⎛⎝⎜

⎞⎠⎟

With the aid of these identities the following relations for the bivariate Fourier transform(34) are proven

∫∫δ∂∂

= −∼s

f s t b K s t x y b x b y f x y x y( , ; )i

( , ; , ; )( ) ( , ) d d , (44)2

∫∫δ∂∂

= −∼t

f s t b K s t x y b y b x f x y x y( , ; )i

( , ; , ; )( ) ( , ) d d , (45)2

∫∫= ∂∂

+ ∂∂

∼

s f s t b K s t x y bx

by

f x y x y( , ; ) i ( , ; , ; ) ( , ) d d , (46)2

⎛⎝⎜

⎞⎠⎟

∫∫= ∂∂

+ ∂∂

∼

t f s t b K s t x y b bx y

f x y x y( , ; ) i ( , ; , ; ) ( , ) d d . (47)2

⎛⎝⎜

⎞⎠⎟

Out of these relations one deduces that some linear combinations of the variables s and tand their partial derivatives ∂ ∂s and ∂ ∂t are reproduced by the bivariate Fourier transform .b In particular, for the two linear combinations of the form

= + + + ∂∂

+ ∂∂

s t bs t

A :1

2(1 ) ,⎜ ⎟

⎡⎣⎢

⎛⎝

⎞⎠

⎤⎦⎥

= + − + ∂∂

+ ∂∂

s t bs t

A :1

2(1 ) , (48)† ⎜ ⎟

⎡⎣⎢

⎛⎝

⎞⎠

⎤⎦⎥

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one obtains their intertwining relations with this transform

= = − A A A Ai , i . (49)b b b b† †

This means that these two linear combinations (48) actually represent the bivariate loweringand raising differential operators for the eigenfunctions of the bivariate Fourier transformoperator b.

It is important to notice that although the identities (49) exactly reproduce similaridentities between the univariable lowering = +( )xa :

x

1

2

d

dand raising = −( )xa :

x† 1

2

d

doperators, and the standard Fourier transform operator (4) on , the action of the bivariatelowering A and raising A† differential operators on the eigenfunctions (25) of the bivariateFourier transform operator b is distinct from the one-dimensional case. Indeed, by using thedifferential shift relations (27) and (28) it is not hard to evaluate that

ψ ψ

ψ

= +

+

−−

−−

s t b b nd b

d bs t b

md b

d bs t b

A ( , ; ) 2 (1 )( )

( )( , ; )

( )

( )( , ; ) , (50)

n mn m

n mn m

n m

n mn m

,1,

,1,

, 1

,, 1

⎡⎣⎢⎢

⎤⎦⎥⎥

ψ ψ

ψ

=

+

++

++

s t bd b

d bs t b

d b

d bs t b

A ( , ; )1

2

( )

( )( , ; )

( )

( )( , ; ) . (51)

n mn m

n mn m

n m

n mn m

†,

1,

,1,

, 1

,, 1

⎡⎣⎢⎢

⎤⎦⎥⎥

Thus, if a function ψ s t( , )n m, is an eigenfunction of the operator b, associated with the

eigenvalue +in m, then the functions ψ s tA ( , )n m, and ψ s tA ( , )n m†

, are definite linear

combinations of two eigenfunctions of b, associated with its eigenvalues + −in m 1 and+ +in m 1, respectively.

It is also readily verified that A and A† obey, after a simple renormalization of both, thestandard Heisenberg commutation relation:

= − = + bA A AA A A I, 2(1 ) , (52)† † †⎡⎣ ⎤⎦where I is the unit operator. Finally, in complete analogy with the one-variable case one candefine a bivariate number operator as

= = + − + ∂∂

+ ∂∂

− + s t bs t

bA A1

2( ) (1 ) (1 ). (53)b( ) † 2 2

2⎜ ⎟

⎡⎣⎢⎢

⎛⎝

⎞⎠

⎤⎦⎥⎥

However, from (50) and (51) one then obtains that the number operator is not diagonal

ψ ψ

ψ ψ

= + +

+ +− +− +

+ −+ −

s t b b n m s t b

nd b

d bs t b m

d b

d bs t b

( , ; ) (1 ) ( ) ( , ; )

( )

( )( , ; )

( )

( )( , ; ) .

bn m n m

n m

n mn m

n m

n mn m

( ), ,

1, 1

,1, 1

1, 1

,1, 1

⎡⎣⎤⎦⎥⎥

Nevertheless, the bivariate operators b( ) and b( ) do commute, as can be readily verified byusing explicit forms (53) and (30) of the operators b( ) and b( ), respectively. This

J. Phys. A: Math. Theor. 48 (2015) 075201 I Area et al

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circumstance is a consequence of the degeneracy of the spectrum of the bivariate second-order differential Hamiltonian operator b( ).

Observe also that, as it is evident from (49), the bivariate number operator b( ) com-mutes with the bivariate Fourier transform b, i.e., = [ , ] 0b

b( ) . Moreover, using the

relations (47) it is not difficult to verify that the Hamiltonian operator b( ) defined in (30) alsocommutes with the bivariate Fourier transform, = [ , ] 0b

b( ) , thus confirming that this

bivariate Fourier transform has been defined consistently. All these expressions reproduce thewell-known ordinary 2D Fourier transform when b = 0.

6. Concluding comments and outlook

We have studied a one-parameter family of 2D extensions of the bivariate Hermite functionsand explicitly constructed a corresponding bivariate Fourier transform which reproducesthem. Raising and lowering operators for the eigenfunctions of this bivariate Fourier trans-form are also explicitly found.

We may surmise that the existence of raising and lowering operators hints at the exis-tence of an su(1,1) dynamical algebra in 2D space. It also seems possible to extend thisconstruction to a fractionalization of the bivariate Fourier transform (34) by means of bilineargenerating functions in the spirit of Namias [22], or within a larger symplectic group, such aswas developed by Moshinsky and Quesne [23] through an integral transform representation.Both of these lines of research will be pursued in future work.

Acknowledgments

We are grateful to the first referee for bringing out to our attention the references [7, 8] and wethank the second referee for helpful comments on an earlier version of this presentation.Research of the first and third authors has been partially supported by the Ministerio deMinisterio de Economìa y Competitividad of Spain under grant MTM2012-38794-C02-01,co-financed by the European Community fund FEDER. The work of the second and fourthauthors has been supported by UNAM-DGAPA-PAPIIT project 101115 ‘Óptica Matemá-tica’. The first author thanks the hospitality of the Instituto de Matemáticas, Unidad Cuer-navaca, UNAM, where the essential part of this research was performed during his visit inSeptember 2014. We thank the help of Guillermo Krötzsch in producing the figure.

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