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On a discrete number operator and its eigenvectors associated with the 5D discrete Fourier transform

On a discrete number operatorand its eigenvectors associated withthe 5D discrete Fourier transform

M K Atakishiyeva(a), N M Atakishiyev(b) and J Méndez Franco(b)

(a) Facultad de Ciencias, Universidad Autónoma del Estado de Morelos,62250 Cuernavaca, Morelos, México

(b) Instituto de Matemáticas, Unidad Cuernavaca, Universidad NacionalAutónoma de México, 62251 Cuernavaca, Morelos, México


1 Introduction

2 Raising and lowering difference operators for eigenvectors of the5D DFT

3 Eigenvalues and eigenvectors of the discrete number operator

4 Eigenvectors of the number operator N (5) versus the Hermitefunctions ψn(x)

5 Concluding remarks

6 References


Abstract

We construct an explicit form of a difference analogue of thequantum number operator in terms of the raising and loweringoperators that govern eigenvectors of the 5D discrete (finite)Fourier transform. Eigenvalues of this difference operator arerepresented by distinct nonnegative numbers so that it can beused to sistematically classify, in complete analogy with thecase of the continuous classical Fourier transform, eigenvectorsof the 5D discrete Fourier transform, thus resolving theambiguity caused by the well-known degeneracy of theeigenvalues of the discrete Fourier transform.

On a discrete number operator and its eigenvectors associated with the 5D discrete Fourier transformIntroduction

We are to begin by recalling first a few well-known facts about theclassical Fourier transform (FT) and its finite analogue, discreteFourier transform (DFT). It is known that the Hermite functions

ψn(x) := c−1n Hn(x) exp

(−x2/2

), cn =

√√π 2n n! ,

where Hn(x) are the classical Hermite polynomials, represent animportant explicit example of an orthonormal and complete systemin the Hilbert space L2(R, dx) of square-integrable functions onthe full real line x ∈ R. It is further well known that the functionsψn(x) possess the simple transformation property with respect tothe Fourier transform: they are eigenfunctions of the Fouriertransform, associated with the eigenvalues i n,

(F ψn) (x) ≡ 1√2π

∫Re i x y ψn(y) dy = i n ψn(x) .


The question is then can be posed whether there is a way ofderiving the eigenfunctions of the Fourier transform, which doesnot presuppose a knowledge of the analytic formula to be proved.

Since mutually commuting operators have the same set ofeigenfunctions, one may solve this problem by defining such aself-adjoint differential operator with simple spectrum of distincteigenvalues, that commutes with the FT operator F . Then theeigenfunctions of that differential operator can be found by solvinga corresponding to this case differential equation and they will beat same time the eigenfunctions of the F . So in this way onereduces a problem of finding eigenfunctions of the FT operator Fto one of solving some differential equation.


To illustrate how to find such differential operator let us start withthe first-order differential operator d

dx and evaluate its action onthe Fourier integral transform:

d

dx

∫Re i x y f(y) dy = i

∫Re i x y y f(y) dy ,

where f(x) ∈ L2(R, dx). Consequently, from the right side of theabove relation one deduces that the next step should be to evaluate

x

∫Re i x y f(y) dy = − i

∫R

(d e i x y

dy

)f(y) dy

= i∫Re i x y d f(y)

dydy ,

upon integrating by parts the middle term in it.


From these two relations it thus follows that(x ± d

dx

) ∫Re i x y f(y) dy = ± i

∫Re i x y

(y ± d

dy

)f(y) dy .

In the operator form these identities can be written as intertwiningrelations

aF = iF a , a†F = − iFa† ,

where a := 1√2

(x+ d

dx

)and a† := 1√

2

(x− d

dx

)are the lowering

and raising first-order differential operators, which obey thestandard Heisenberg commutation relation[

a , a†]

:= a a† − a† a ≡[d

dx, x

]= I .


The final step for finding the desired differential operator isactually revealed by the intertwining relations because one readilyconcludes that

a†aF = i a†Fa = Fa†a

on account of both identities in the intertwining relations.Consequently, the self-adjoint second-order differential numberoperator N := a†a does commute with the FT operator F andit only remains to resolve the eigenproblem Nfn(x) = λnfn(x)for this operator N. It is not difficult to show then that theeigenfunctions of the number operator N are the Hermitefunctions ψn(x) (up to the arbitrariness in the choice of anormalization constant factor), whereas the correspondingeigenvalues are λn = n, n = 0, 1, 2, ... .


Turning to the discrete Fourier transform Φ (N), we recall that it isbased on N points and represented by the N ×N unitarysymmetric matrix with matrix elements

Φ (N)m, n = 1√

Nexp

(2πiN

mn

)≡ 1√

Nqm n ,

where q := e2πiN and m, n ∈ {0, 1, . . . , N−1}. Given a vector ~v

with components {vk}N−1k=0 , one can compute another vector ~u

with components

um =N−1∑n=0

Φ (N)m, n vn ,

referred to as the discrete (finite) Fourier transform of the vector ~v.


Those vectors ~fk, which are solutions of the standard equations

N−1∑n=0

Φ (N)m, n

(~fk

)n

= λk

(~fk

)m

k ∈ {0, 1, . . . , N−1} ,

then represent eigenvectors of the DFT operator Φ (N), associatedwith the eigenvalues λk.Since the fourth power of Φ(N) is the unit matrix, the only fourdistinct eigenvalues among λk’s are ± 1 and ± i.


Although there exists a plethora of discussion in the literature oneigenvectors of the DFT, the problem of deriving eigenvectors ofDFT analytically still remains to be solved.Recently, M.Atakishiyeva and N.Atakishiyev have proposed astrategy for resolving this problem by constructing a self-adjointdifference operator N (N) (with distinct nonnegative eigenvalues) interms of the difference raising and lowering operators, which aredefined by the intertwining relations

bN Φ(N) = i Φ(N) bN, bTN Φ(N) = − i Φ(N) bT

N.

The ability to solve a difference equation for eigenvectors of thisdiscrete number operator N (N), which commutes with the DFToperator Φ(N), then enables one to define an analytical form of thedesired set of eigenvectors for the latter operator.


An important aspect to observe at this point is that althoughthe idea of making use an analogy with the continuous case forderiving eigenvectors of the DFT is not new (see, for example,[3, 4]), it seems, however, that there never was consistent attemptto find out how symmetry properties of the continuous Fouriertransform might best be transferred to the discrete case.The limited aim of this presentation is to restrict our attention tothe 5D DFT and give a detailed account of how one can solve theeigenproblem for the discrete number operator N (5) by using thedifference raising and lowering operators that govern eigenvectorsof the 5D discrete Fourier transform Φ(5).


The motivation for selecting this special dimension N = 5 of thegeneral discrete Fourier transform Φ(N) is twofold. First, thisdimension is large enough to contain a multiple eigenvalue andtherefore one has to handle the same degeneracy problem as in themore general case. Second, this dimension is small enough in orderto have calculational advantages that appear in the process ofresolving the eigenproblem for the discrete number operator N (5).We hope that this study will deepen our understanding of the casewith an arbitrary ND discrete Fourier transform and help us toprovide some rigorous proofs, still needed for general values of N .

On a discrete number operator and its eigenvectors associated with the 5D discrete Fourier transformRaising and lowering difference operators for eigenvectors of the 5D DFT

We recall that the 5D discrete (finite) Fourier transform (DFT) istraditionally represented by an 5× 5 unitary symmetric matrixΦ(5) =

(Φ (5)

m, n

)with elements

Φ (5)m, n = 1√

5exp

(2πi5 mn

)≡ 1√

5qm n ,

where q = e2πi5 is the 5th root of unity and indices m,n run in the

interval {0, 1, 2, 3, 4}. So the matrix form of Φ(5) is

(Φ (5)

m, n

)= 1√

5

1 1 1 1 11 q q2 q3 q4

1 q2 q4 q q3

1 q3 q q4 q2

1 q4 q3 q2 q

.


In the above-mentioned paper by Atakishiyeva and Atakishiyev itwas shown how to construct the difference lowering and raisingoperators b5 and bT

5 for eigenvectors of the DFT operator Φ(5),which satisfy ‘proper’ intertwining relations with the Φ(5) of theform

b5 Φ(5) = i Φ(5) b5, bT5 Φ(5) = − i Φ(5) bT

5 ,

where bT5 is the matrix transpose of b5. Let us draw attention

here to these intertwining relations, which evidently imply that if avector ~fk is the eigenvector of the DFT operator Φ(5), associatedwith the eigenvalue ik, 0 ≤ k ≤ 3, then the vectors bT

5~fk and b5 ~fk

are also the eigenvectors of the same operator Φ(5), associatedwith the eigenvalues ik+1 and ik−1, respectively.


But note carefully that this does not necessarily mean that thosevectors bT

5~fk and b5 ~fk essentially coincide (to within constant

factors) with the eigenvectors ~fk+1 and ~fk−1 of the DFT operatorΦ(5), respectively, as it does happen to be the case with theFourier transform operator F . Later we detail the action of theoperators b5 and bT

5 on eigenvectors of the DFT operator Φ(5).The operators b5 and bT

5 are explicitly given as

b5 := c(2 S + T(+) −T(−)

),

bT5 := c

(2 S−T(+) + T(−)

), c = 1

4

√5π,

where the operator S represents the diagonal matrix with elementsSkl := sin(kθ)δkl, θ := 2π/5, 0 ≤ k, l ≤ 4 and a pair of the shiftoperators T(±) are defined as T (±)

kl := δk±1,l with δ−1,l ≡ δ4,l andδ5,l ≡ δ0,l.


We also display the matrix form of the lowering and raisingoperators b5 and bT

5 , respectively:

((b5) m,m′

)= c

0 1 0 0 −1−1 2 sin θ 1 0 00 −1 2 sin 2θ 1 00 0 −1 −2 sin 2θ 11 0 0 −1 −2 sin θ

,

((bT

5) m,m′

)= c

0 −1 0 0 11 2 sin θ −1 0 00 1 2 sin 2θ −1 00 0 1 −2 sin 2θ −1−1 0 0 1 −2 sin θ

.


Note that the determinants of both matrices are equal to 0,det(b5) = det(bT

5) = 0; therefore they are not invertible.Observe also that both of these matrices are of ‘almost’ tridiagonalform: they have ±1 elements in the upper-right and lower-leftcorners but otherwise are tridiagonal. Since those ±1 elements canbe regarded as cyclic extensions of the subdiagonal and thesuperdiagonal elements, these type of matrices are referred to asextended-tridiagonal matrices [Clary and Mugler].Moreover, another confirmation of the ‘cyclic’ properties of theoperators b5 and bT

5 is revealed by the identities(b5)5

+ 5 c4τ b5 = 0 ,(bT

5

)5+ 5 c4τ bT

5 = 0 ,

where τ is the golden ratio, τ := (√

5 + 1)/2 = −2 cos 2θ.


This particular irrational number τ is known to turn out frequentlyin geometry, particularly in figures with pentagonal symmetry; so itis not surprising that it appears here as well.Since the successive powers of τ obey the Fibonacci recurrenceτn+1 = τn + τn−1 , n ≥ 0, this characteristic property of thegolden ratio allows any polynomial in τ to be reduced to a linearexpression in τ .In the sequel, it proves therefore convenient to parametrize theoperators b5 and bT

5 in terms of the golden ratio τ and itsconjugate τ−1 := (

√5− 1)/2 = 2 cos θ = τ − 1. Taking into

account that 2 sin θ = κτ1/2 and 2 sin 2θ = κτ−1/2, whereκ := (5)1/4, one rewrites matrix elements of the operators b5 andbT

5 as


((b5) m,m′

)= c

0 1 0 0 −1−1 κτ1/2 1 0 00 −1 κτ−1/2 1 00 0 −1 −κτ−1/2 11 0 0 −1 −κτ1/2

,

((bT

5) m,m′

)= c

0 −1 0 0 11 κτ1/2 −1 0 00 1 κτ−1/2 −1 00 0 1 −κτ−1/2 −1−1 0 0 1 −κτ1/2

.


From definition of the lowering b5 and raising bT5 operators it

follows that their commutator

K :=[b5,bT

5

]−≡ b5 bT

5 − bT5 b5

is equal toK = 4 c2

[T(+) −T(−), S

]−.

Its explicit matrix form in terms of the golden ratio τ is

((K) m,m′

)= 2κ

√τ c2

0 1 0 0 11 0 τ − 2 0 00 τ − 2 0 2(1− τ) 00 0 2(1− τ) 0 τ − 21 0 0 τ − 2 0

.

To compare the commutator K with the continuous case recallthat the lowering and raising first-order differential operators aand a† obey the Heisenberg commutation relation [ a,a†] = I.


We close this section by emphasizing that it was intuitivelyunderstood much earlier that probably the extended-tridiagonaltype matrices lie at the core of the adequate description ofeigenvectors of the general ND discrete Fourier transform[Dickinson and Steiglitz, Clary and Mugler].But it seems that this particular band structure was impreciselyattributed to those operators with distinct eigenvalues, whichcommute with the associated DFT operators and can be thereforeused for the unambiguous classification of the eigenvectors of thelatter ones. Only recently it has become clear that it is thelowering bN and raising bT

N operators for the eigenvectors of theDFT operator Φ(N) that are of the extended-tridiagonal type.Defined by the standard intertwining relations

bN Φ(N) = i Φ(N) bN, bTN Φ(N) = − i Φ(N) bT

N,

these operators bN and bTN do not commute with the Φ(N).


Nevertheless, from the same defining identities for the bN and bTN

it follows at once that their product bTN bN does commute with

the DFT operator Φ(N).Moreover, although the operator N (N) := bT

N bN is not of theextended-tridiagonal type, it turns out to be quite sufficient forfinding explicit forms of all mutually orthogonal eigenvectors of theDFT operator Φ(N) in a systematic and unambiguous way.As we shall see in the next section, this briefly outlined abovealgebraic approach to solving the eigenproblem for the operatorN (N) can be effectively employed in the particular case of the 5DDFT.

On a discrete number operator and its eigenvectors associated with the 5D discrete Fourier transformEigenvalues and eigenvectors of the discrete number operator

Let us study in detail a discrete number operator N (5), whosematrix elements are defined as(

N (5))

m,m′=(bT

5b5)

m,m′

= c2

2 −κτ1/2 −1 −1 −κτ1/2

−κτ1/2 4 + τ κτ−3/2 −1 −1−1 κτ−3/2 5− τ 2κτ−1/2 −1−1 −1 2κτ−1/2 5− τ κτ−3/2

−κτ1/2 −1 −1 κτ−3/2 4 + τ

.

As a product of a matrix and its transpose, this defining matrix issymmetric and all of its eigenvalues are nonnegative. Moreover,since the determinant of this matrix is equal to zero, at least oneof the eigenvalues should have zero value as well; but this lowesteigenvalue turns out to be unique and all eigenvalues of this matrixare actually distinct.


Before entering into further details about explicit forms of theeigenvalues and eigenvectors of the operator N (5), we may recallfirst the following important facts, associated with thiseigenproblem.In this particular case under study, when the operator N (5) is justrepresented by a 5× 5 matrix, one can use some computerprogram in order to evaluate the eigenvalues and eigenvectors ofthe discrete number operator N (5).For instance, this is what one gets by using Mathematica :


Eigenvalues of the N (5) are c2 νk, 0 ≤ k ≤ 4, where νk’s, arrangedin the descending order, are given by

ν4 = 12(15−

√5)

+√

5− 2√

5 = κ[κ(3τ − 2) + τ−3/2

],

ν3 = 12(5 +√

5)

+√

5 + 2√

5 = κ(κ+ τ1/2

)τ,

ν2 = 12(15−

√5)−√

5− 2√

5 = κ[κ(3τ − 2)− τ−3/2

],

ν1 = 12(5 +√

5)−√

5 + 2√

5 = κ(κ− τ1/2

)τ, ν0 = 0;


Eigenvectors ~yk of N (5), associated with these eigenvalues νk,0 ≤ k ≤ 4, have the following components(

~y4)4

k=0={

0, κ+ τ12 , τ−

12 , − τ−

12 , −κ− τ

12},

(~y3)4

k=0={

2(1− τ), 1, 1, 1, 1},(

~y2)4

k=0={

2,−(τ+2κτ

12), 2κτ

12 +3τ−2, 2κτ

12 +3τ−2,−

(τ+2κτ

12)},(

~y1)4

k=0={

0, τ12 − κ, τ−

12 , −τ−

12 , κ− τ

12},(

~y0)4

k=0={

2τ + κτ12 , 1 + κτ−

12 , 1, 1, 1 + κτ−

12}.


Since the discrete number operator N (5) commutes with the DFToperator Φ(5), the above eigenvectors of the N (5) are at the sametime eigenvectors of the Φ(5): two of them, ~y0 and ~y2, areassociated with the same eigenvalue i0 = 1 of the Φ(5), whilethe eigenvectors ~y4, ~y3 and ~y1 correspond to the eigenvalues i,i2 = −1 and i3 = −i of the Φ(5), respectively.Obviously, these multiplicities corresponding to the eigenvalues ik,0 ≤ k ≤ 3, of the 5D DFT operator Φ(5), are the particular N = 5cases of the general explicit expressions for the multiplicitiesmk(ik) of the eigenvalues of the ND DFT,

m0(1) =[N

4]

+ 1, m1(i) =[N + 1

4],

m2(−1) =[N + 2

4], m3(−i) =

[N + 34

]− 1 ,

where the symbol [X] stands for the greatest integer in X.


It is important also to realize that the eigenvectors ~yk, 0 ≤ k ≤ 4,are distinct from those eigenvectors of the DFT operator Φ(5),which have appeared before in the literature. For instance,Matveev evaluated explicit forms of the eigenvectors of the NDDFT operator Φ(N) for the values of N from N = 2 to N = 8 bycombining the technique of spectral projectors for the operatorΦ(N) with the Gramm–Schmidt orthogonalization algorithm. Inparticular, Matveev’s eigenvectors ~vn, 1 ≤ n ≤ 5, of the operatorΦ(5) are interrelated with the eigenvectors ~yk, 0 ≤ k ≤ 4, in thefollowing way: the eigenvectors ~v1 and ~v2 are some linearcombinations of the ~y0 and ~y2, whereas the eigenvectors ~v3, ~v4,and ~v5 coincide with the ~y3, ~y4, and ~y1, respectively, up tonormalization by constant factors:

~v3 = ~y3 , ~v4 = κ

2 ~y4 , ~v5 = −κ2 ~y1 .


Finally, the incentive for making those extended comments, givenabove, is just to emphasize that this set of eigenvectors of thediscrete number operator N (5), produced by Mathematica, is stillambiguous until a rule is given for ordering those eigenvectors andassociated eigenvalues of the operator N (5).

We return now to a study of the eigenvectors and eigenvalues ofthe operator N (5) in a systematic algebraic way. Since the lowesteigenvalue of the N (5) is 0, its lowest eigenvector ~f0 is defined as

N (5) ~f0 = 0 .

Moreover, an explicit form of the same eigenvector ~f0 can befound from the simpler equation

b5 ~f0 = 0 .


Since the symmetric matrix (N (5))m,m′ clearly exhibits additionalsymmetry among the entries of the all antidiagonals, it is evidentthat all eigenvectors of the operator N (5) must be either ‘even’ or‘odd’ with respect to that particular reflection symmetry about thesubantidiagonal {(N (5))5−k,k}4k=1 of the matrix (N (5))m,m′ , thatis, (

~fn

)k

= (−1)n(~fn

)5−k

, 0 ≤ n, k ≤ 4 .

This means that we should look for a lowest eigenvector ~f0, whosecomponentwise structure is of the form(

~f0)4

k=0=(α, β, γ, γ, β

).

Employing an explicit form of the matrix b5, one obtains only twolinearly-independent equations for the three unknowns α, β, and γ:

α = β κ τ1/2 + γ , β = γ (1 + κ τ−1/2) .


Taking into account that κ2 = 2τ − 1, one concludes thatα = γ (2τ + κ τ1/2) and the lowest eigenvector ~f0 with thecomponents(

~f0)4

k=0= γ

(2τ + κ τ1/2, 1 + κ τ−1/2, 1, 1, 1 + κ τ−1/2

)is thus determined by its defining equation up to normalizationby a constant γ. Notice that ~f0 = γ ~y0, thus the ~f0 is also theeigenvector of the DFT operator Φ(5) coresponding to theeigenvalue i0 = 1.Also, to normalize the lowest eigenvector ~f0 to have length one, itsuffices to choose normalization constant as γ = γ0 ≡ (ν2ν3)−1/2,and we shall employ in what follows the same notation ~f0 for theunit-length lowest eigenvector with the components(

~f0)4

k=0= γ0

(2τ + κ τ1/2, 1 + κ τ−1/2, 1, 1, 1 + κ τ−1/2

).


In order to find next eigenvectors of the number operator N (5), wefirst define 4 vectors of the form

~fk := dk

c k

(bT

5

)k~f0 , 1 ≤ k ≤ 4 ,

where ~f0 is the lowest eigenvector of the N (5) and dk’s are somenormalization scalar factors. Since ~f0 is the eigenvector of the DFToperator Φ(5) also and it coresponds to the eigenvalue i0 = 1, fromthe second intertwining relation it follows at once that all vectors~fk, 1 ≤ k ≤ 4, are, in effect, the eigenvectors of the Φ(5),

Φ(5) ~fk = dk

c kΦ(5)

(bT

5

)k~f0 = i dk

c kbT

5 Φ(5)(bT

5

)k−1~f0 = ...

= ik dk

c k

(bT

5

)kΦ(5) ~f0 = ik dk

c k

(bT

5

)k~f0 = ik ~fk ,

corresponding to the eigenvalues ik, respectively.


Moreover, it actually turns out that all vectors ~fk, 0 ≤ k ≤ 4, areat the same time the eigenvectors of the number operator N (5).Indeed, since the operators Φ(5) and N (5) commute, one checkseasily that

Φ(5)N (5) ~fk = N (5) Φ(5) ~fk = ikN (5) ~fk

is valid for all integer values of k between 0 and 4. This meansthat for any integer k ∈ [0, 4] both vectors ~fk and N (5) ~fk areassociated with the same eigenvalues ik of the 5D DFT operatorΦ(5). Consequently, three vectors ~fk, 1 ≤ k ≤ 3, do representthe eigenvectors of the number operator N (5) because thecorresponding multiplicities m(ik) = 1 for those integer valuesof 1 ≤ k ≤ 3.


As for the last vector ~f4, one readily verifies that

N (5) ~f4 ' bT5 b5

(bT

5

)4~f0 ' bT

5b5 bT5~f3 ' bT

5

(N (5) +K

)~f3

' bT5~f3 ' ~f4 ,

where the symbol A ' B indicates that A is equal to B multipliedby a nonzero scalar constant factor and we employed the fact thatthe vector ~f3 is an eigenvector of the operator K also, despite thenoncommutativity of the operators K and N (5).


Since the vector ~f0 has been already defined as the lowesteigenvector of the operator N (5), one does conclude thatthe 5 orthonormal vectors ~fk, 0 ≤ k ≤ 4, explicitly given as

~f0 = 1√ν2ν3

~y0, ~f1 = 12

√τ

ν3~y4, ~f2 =

√τ

2κ ~y3,

~f3 = 12

√τ

ν1~y1, ~f4 = 1

2√ν2ν3~y2,

do represent the desired set of the eigenvectors for the numberoperator N (5),

N (5) ~fk = λk~fk, 0 ≤ k ≤ 4,

associated with the eigenvalues

λ0 = 0, λ1 = c2ν4, λ2 = c2ν3, λ3 = c2ν1, λ4 = c2ν2,

respectively.


The explicit analytical form of the spectrum of the discrete numberoperator N (5) can be thus represented as

λk = c2[5(1− δk0) + 4

((τ − 1)sin kθ + cos kθ

)sin 2kθ

],

where θ = 2π/5 and 0 ≤ k ≤ 4.

Our first graph compares these eigenvalues λk, 0 ≤ k ≤ 4, of thediscrete number operator N (5) and the first 5 eigenvalues of thestandard quantum number operator N = a†a.


The next step is to clarify how these eigenvectors transform underthe action of the raising bT

5 and lowering b5 difference operatorsand then to compare them with the behaviour of their continuouscounterparts ψn(x), which satisfy the well-known relations

aψn(x) =√nψn−1(x), a† ψn(x) =

√n+ 1ψn+1(x).

But observe first that from definition of vectors ~fk it follows atonce that

~fk+1 = dk+1c dk

bT5~fk , 0 ≤ k ≤ 3 , d0 = 1 .


Therefore the required relations can be explicitly written for eachappropriate value of the index k as

bT5~f0 = 4κ c2

√τλ4

~f1 ≡ η√λ1 ~f1,

bT5~fk =

√λk+1 ~fk+1, 1 ≤ k ≤ 3,

whereη := 4κ/

√τν2ν4 ≡ 4/

√5τ + 21.


It remains only to evaluate the last identity

bT5~f4 = 1

4c4κ2 (bT5)5 ~f0 = − 5τ

4κ2 bT5~f0 = −

√(1− η2)λ1 ~f1,

which is a direct consequence of the second ‘cyclic’ identity andthe relation

~f4 = 14c4κ2 (bT

5)4 ~f0,

readily obtained by the successive use of all entries in the abovechain of relations for vectors ~fk . The last formulas are thusdiscrete analogues of those, which are collected in the secondidentity of the continuous case.


As for the action of the lowering difference operator b5, thesituation here is slightly different. The point is that already atthe first step one evaluates that

b5 ~f1 = δ b5bT5~f0 = δ

(K +N (5)

)~f0 = δK ~f0 = δ (α~f0 + β ~f4) ,

where δ :=(η√λ1)−1

and coefficients α and β can be explicitlydefined by the following easy algebra:

(~f0,b5 ~f1) = δα = (bT5~f0, ~f1) = δ−1(~f1, ~f1) = δ−1 = η

√λ1,

(~f4,b5 ~f1) = δβ = (bT5~f4, ~f1) = −

√(1− η2)λ1(~f1, ~f1) = −

√(1− η2)λ1.


From last two relations one thus concludes that

α = η2λ1, β = − η√

1− η2 λ1,

and the action of lowering operator b5 on the eigenvector ~f1 nowexplicitly reads

b5 ~f1 =√λ1[η ~f0 −

√1− η2 ~f4

].

The evaluation of the action of the lowering operator b5 on theremaining three eigenvectors ~fn, n = 2, 3, 4, requires less efforts forthe following reason. As we have already remarked above, threevectors ~f1, ~f2 and ~f3, which are associated with the eigenvaluesik(k = 1, 2, 3) with multiplicities 1, are actually commoneigenvectors of the number operator N (5) and the operator K(although these operators do not commute).


Moreover, the explicit form of corresponding eigenvalues of theoperator K,

K ~fn = (λn+1 − λn) ~fn , n = 1, 2, 3,

is a direct consequence of the evident intertwining relationN (5)bT

5 = bT5

(N (5) +K

). Therefore, for 2 ≤ k ≤ 4 one readily

derives that

b5 ~fk = 1√λk

b5bT5~fk−1 = 1√

λk

(K +N (5)

)~fk−1 =

√λk

~fk−1 .


We are now in a position to write down all matrix elements of theraising and lowering operators bT

5 and b5 in the basis, built overthe eigenvectors ~fn, 0 ≤ n ≤ 4:

( (~fk ,bT

5~fl

) )=

0 0 0 0 0

η√λ1 0 0 0 −

√(1− η2)λ1

0√λ2 0 0 0

0 0√λ3 0 0

0 0 0√λ4 0

,

( (~fk ,b5 ~fl

) )=

0 η

√λ1 0 0 0

0 0√λ2 0 0

0 0 0√λ3 0

0 0 0 0√λ4

0 −√

(1− η2)λ1 0 0 0

.


To close this section, we point out that as a consistency check onemay verify that both of these matrix realizations of the raising andlowering operators bT

5 and b5 in the ~fn-basis, respectively, dopossess the same ‘cyclic’ properties as indicated in identities,displayed in Section 2. Indeed, a direct computation of the 5thpower of the matrix b5 shows that( (

~fk ,b5 ~fl

) )5+[(1− η2)λ1λ2λ3λ4

]1/2( (~fk ,b5 ~fl

) )= 0,

where[(1− η2)λ1λ2λ3λ4

]1/2= c4

[(1− η2)ν1ν2ν3ν4

]1/2= 5 c4τ,

upon using definitions of the eigenvalues νk, 1 ≤ k ≤ 4, given atthe beginning of Section 3. Finally, since the operator bT

5 is thematrix transpose of b5, the former one has the same‘cyclic’property as the latter.

On a discrete number operator and its eigenvectors associated with the 5D discrete Fourier transformEigenvectors of the number operator N (5) versus the Hermite functions ψn(x)

From the outset the ND discrete Fourier transform Φ(N) wasconceived as a finite (discrete) analogue of the Fourier transformF and the N eigenvectors ~fk, 0 ≤ k ≤ N − 1, of the formertransform operator were therefore required to converge to thecorresponding Hermite functions ψn(x) in the limit as N →∞.So the question is : how the eigenvectors ~fk, 0 ≤ k ≤ N − 1, ofa ND DFT Φ(N) with a fixed integer value of N can be relatedto the Hermite functions ψn(x), 0 ≤ n <∞ ?Note first that a ND discrete Fourier transform is actually adiscrete (finite) image of the N -dimensional subspace of theinfinite dimensional Hilbert space L2(R, dx), spanned by thefirst N basis functions ψn(x), 0 ≤ n ≤ N − 1, in this space,rather than of the whole Hilbert space L2(R, dx) itself.


Consequently, one should find out how the eigenvectors ~fk matchthe first N Hermite functions ψn(x), 0 ≤ n ≤ N −1. In this regard,it is important to take into account the following fundamentalproperties of the Hermite functions ψn(x): ψn(−x) = (−1)nψn(x)and each function ψn(x) has exactly n alternations in its sign.As we have already remarked above, the eigenvectors ~fk do exhibitthe same type of the reflection symmetry as

ψn(−x) = (−1)nψn(x)

in the continuous case, so that it remains only to verify that eacheigenvector ~fk has the same number of alternations in itscomponents

(~fk

)l, 0 ≤ l ≤ N − 1, as a Hermite function ψn(x),

associated with it.


But the careful examination of the eigenvectors ~fn under studyindicates that their components are not appropriately structuredin order to enable one to match them with the first 5 Hermitefunctions ψn(x). It has been then realized that one actually needsto rearrange components of the eigenvectors ~fk and introduceanother set of centered vectors ~f (c)

k with the components(~f

(c)k

)2

l=−2, defined as(

~f(c)

k

)l−2

=(( U )l,m

) (~fk

)m, 0 ≤ k, l,m ≤ 4,

where U is the unitary operator, U UT = UT U=I , with thematrix elements

((U) m,m′

)=

0 0 0 1 00 0 0 0 11 0 0 0 00 1 0 0 00 0 1 0 0

.


Their explicit componentwise forms are(~f

(c)0

)2

l=−2= 1√ν2 ν3

{1, 1 + κτ−1/2, 2τ + κτ1/2, 1 + κτ−1/2, 1

},

(~f

(c)1

)2

l=−2= 1

2

√τ

ν3

{− τ−1/2, −κ− τ1/2, 0, κ+ τ1/2, τ−1/2

},

(~f

(c)2

)2

l=−2=√τ

2κ{

1, 1, 2(1− τ), 1, 1},

(~f

(c)3

)2

l=−2= 1

2

√τ

ν1

{− τ−1/2, κ− τ1/2, 0, τ1/2 − κ, τ−1/2

},

(~f

(c)4

)2

l=−2= 1

2√ν2ν3

{2κτ1/2 + 3τ −2,−

(τ + 2κτ1/2

), 2, II, I

}.

In next 5 graphs we compare the centered vectors ~f (c)n and the

first Hermite functions ψn(x), 0 ≤ n ≤ 4, respectively.


It is to be emphasized that thus introduced centered vectors ~f (c)n

are actually eigenvectors of the centered discrete number operatorN (5;c),

N (5;c) := UN (5) UT,

associated with the same eigenvalues λn: N (5;c) ~f(c)

n = λn~f

(c)n .

Matrix elements of this operator N (5;c) are explicitly given as(N (5;c)

)m,m′

=((b(c)

5 )Tb(c)5

)m,m′

= c2

5− τ κτ−3/2 −1 −1 2κτ−1/2

κτ−3/2 4 + τ −κτ1/2 −1 −1−1 −κτ1/2 2 −κτ1/2 −1−1 −1 −κτ1/2 4 + τ κτ−3/2

2κτ−1/2 −1 −1 κτ−3/2 5− τ

,

where the centered lowering and raising operators b(c)5 and (b(c)

5 )Tare defined through the operators b5 and bT

5 , respectively, by thesame similarity transformation with the operator U.


Furthemore, the centered vectors ~f (c)n turn out to be also

eigenvectors of the centered discrete Fourier transform operatorΦ(5;c),

Φ(5;c) := U Φ(5) UT ,

corresponding to the respective eigenvalues in. The eigenproblemfor the centered DFT operator Φ(5;c) can be thus written in thematrix form as

2∑n=−2

Φ (5;c)m, n

(~f

(c)k

)n

= λk

(~f

(c)k

)m, 0 ≤ k ≤ 4, −2 ≤ m ≤ 2,

and associated matrix for this eigenproblem has elements

(Φ (5;c)

m, n

)= 1√

5

q4 q2 1 q3 qq2 q 1 q4 q3

1 1 1 1 1q3 q4 1 q q2

q q3 1 q2 q4

.


A word of explanation regarding the present results is in order atthis point, but let us recall first the following. Square matrices Aand B are said to be similar, if there is a nonsingular matrix C(which is referred to as a transforming matrix of B to A) suchthat A = CBC−1. In the case when the transforming matrix C isa unitary matrix U, U U† = I, then B is unitarily similar to A(see, for example, [Lancaster and Tismenetsky]).Taking that into account note that in the process of establishinghow the eigenvectors ~fn are related to the Hermite functionsψn(x), we actually arrived at another form of the eigenproblem forthe 5D DFT operator, which appear to be different from thetraditional one. This form of DFT is also well known, but lessfrequently used. However, some authors have even suggested thatit is better not to use the standard Fourier matrix that represents adiscretization of the Fourier transform, but rather to use a‘centered’ version of it [Clary and Mugler 2].


But it is now clear that Φ(5) and Φ(5;c) are actually unitarilysimilar, with the transforming matix U, introduced above, andthey represent therefore the same linear transformation after achange of basis.Nonetheless, it is true that there is considerable merit in workingwith the non-standard 5D DFT operator Φ(5;c) because it explicitlydisplays all those symmetry properties in the DFT eigenproblem,which are so characteristic of the continuous Fourier transform.As a direct consequence of the appealing feature of this form, thematrix form of Φ(5;c) clearly exhibits its remarkable symmetry: thematrix Φ(5;c) is a centrosymmetric matrix [Dickinson and Steiglitz].


This means that it is reproduced by the similarity transformationwith the transforming matrix J,

Φ(5;c) = J Φ(5;c) J ,

where J is the 5× 5 matrix with ones on the antidiagonal,

(Jm, n

)=

0 0 0 0 10 0 0 1 00 0 1 0 00 1 0 0 01 0 0 0 0

.

Finally, the operator N (5;c), which commutes with Φ(5;c), is alsoof the centrosymmetric type.

On a discrete number operator and its eigenvectors associated with the 5D discrete Fourier transformConcluding remarks

To summarize, we have constructed an explicit form of a differenceanalogue of the quantum number operator in terms of the raisingand lowering operators that govern eigenvectors of the 5D discrete(finite) Fourier transform. The main algebraic properties of this ope-rator have been examined in detail. The eigenvalues of this discretenumber operator are represented by distinct nonnegative numbersso that this operator has been used to sistematically classify, incomplete analogy with the case of the continuous classical Fouriertransform, eigenvectors of the 5D DFT, thus resolving the ambigu-ity caused by the well-known degeneracy of the eigenvalues of thediscrete Fourier transform. We hope that this particular knowledgewill help us to extend our results to the case of an arbitrary NDdiscrete Fourier transform.We are grateful to Vladimir Matveev, Timothy Gendron and ÍvanArea for illuminating discussions and thank Fernando González forgraphical support of our results and the computation of the eigen-values (3.2) and eigenvectors (3.3) with the aid of Mathematica.

On a discrete number operator and its eigenvectors associated with the 5D discrete Fourier transformReferences

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Atakishiyev N M 2006 On q-extensions of Mehta’seigenvectors of the finite Fourier transform Int. J. Mod. Phys.A 21 4993–5006Atakishiyeva M K and Atakishiyev N M 2015 On the raisingand lowering difference operators for eigenvectors of the finiteFourier transform J. Phys.: Conference Series 597 012012

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