Some Basic Difference Equations of Schrödinger Boundary Value Problems

Advances in Difference Equations Volume 2009, Article ID 569803,22pages doi:10.1155/2009/569803

Research Article

Some Basic Difference Equations of Schr ¨odinger

Boundary Value Problems

Andreas Ruffing,

_{Maria Meiler,}

_{and Andrea Bruder}

1_{Center for Applied Mathematics and Theoretical Physics (CAMTP), University of Maribor,}

Krekova Ulica 2, 2000 Maribor, Slovenia

2_{Department of Mathematics, Technische Universit¨at M ¨unchen, Boltzmannstraße 3,}

85747 Garching, Germany

3_{Department of Mathematics, Baylor University, One Bear Place 97328, Waco, TX 76798-7328, USA}

Correspondence should be addressed to Andreas Ruffing,ruffi[email protected]

Received 1 April 2009; Accepted 28 August 2009

Recommended by Alberto Cabada

We consider special basic difference equations which are related to discretizations of Schr ¨odinger equations on time scales with special symmetry properties, namely, so-called basic discrete grids. These grids are of an adaptive grid type. Solving the boundary value problem of suitable Schr ¨odinger equations on these grids leads to completely new and unexpected analytic properties of the underlying function spaces. Some of them are presented in this work.

Copyrightq2009 Andreas Ruffing et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

It is well known that solving Schr ¨odinger’s equation is a prominent L2_{-boundary value}

problem. In this article, we want to become familiar with some of the dynamic equations that arise in context of solving the Schr ¨odinger equation on a suitable time scale where the expression time scale is in the context of this article related to thespatial variables.

The Schr ¨odinger equation is the partial differential equationt∈R

− ∂2

∂x2 −

∂2

∂y2 −

∂2

∂z2 V

x, y, z

ψx, y, z, ti∂ ∂tψ

x, y, z, t, 1.1

where the function V :Ω ⊆ R3 _{→ R yields information on the corresponding physically}

of the fundamental facts on Schr ¨odinger’s equation. To do so, let us denote by C2_{Ω all}

complex-valued functions which are defined onΩand which are twice differentiable in each of their variables.

Definition 1.1. Let ψ : Ω → C be twice partially differentiable in its three variables. Let

moreover V : Ω → R be a piecewise continuous function, PΩ denoting the space of piecewise continuous functions with values inC. The linear mapH:C2_{Ω → PΩ, given}

by

Hψx, y, z:

− ∂2

∂x2 −

∂2

∂y2 −

∂2

∂z2 V

x, y, z

ψx, y, z 1.2

is calledSchr¨odinger OperatorinC2Ω.

The following lemma makes a statement on the separation ansatz of the conventional Schr ¨odinger partial differential equation where we throughout the sequel assumeΩ R3_.

Lemma 1.2 Separation Ansatz. Let the Schr¨odinger equation 1.1 be given, fulfilling the assertions of Definition1.1whereΩ R3_{.In addition, the functionV will have the property}

V :R3 _{−→R, x, y, z−→Vx, y, zfx gyhz, 1.3}

where f, g, h are continuous. Let now λ1, λ2, λ3 ∈ R be such that there exist eigenfunctions

ψ1, ψ2, ψ3∈ L2R∩C2Rwith

−ψ₁x fxψ1x λ1ψ1x, x∈R,

−ψ₂ygyψ2

yλ2ψ2

y, y∈R,

−ψ₃z hzψ3z λ3ψ3z, z∈R.

1.4

Then the functionϕ∈L2_R3_×C2_R∩C2_R4_{, given by}

x, y, z, t−→ϕx, y, z, t:ψ1xψ2

yψ3ze−iλ1t−iλ2t−iλ3t 1.5

is a solution to Schr¨odinger’s equation 1.1, revealing a completely separated structure of the

variables.

A fascinating topic which has led to the results to be presented in this article is discretizing the Schr ¨odinger equation on particular suitable time scales. This might be of importance for applications and numerical investigations of the underlying eigenvalue and spectral problems. Let us therefore restrict to thepurely discrete case, that is, we are going to focus on a so-calledbasic discrete quadratic gridresp. on its closure which is a special time scale

Definition 1.3. Let 0< q <1 as well asc1, c2, c3∈R. The set

Rc1,c2,c3

q :

c1qnc2q−nc3|n∈Z

1.6

denotes thebasic discrete quadratic gridwherec2₁c2₂/0. Forn∈Z, we abbreviateγn :c1qn

c2q−nc3and−Rqc1,c2,c3:{t∈R | −t∈Rqc1,c2,c3}. Define the set

T:Rc1,c2,c3

q ∪ −Rqc1,c2,c3 1.7

as well as the setT∗by

T∗_{T\ {0}. 1.8}

The boundary conditions on the functions we need for the discretized version of Schr ¨odinger’s equation are then given by the requirement

L2_T∗_:f:T∗_{−→C|f, f<∞, 1.9}

where the scalar productf, gfor two suitable functionsf, g:T∗ → Rwill be specified by

f, g: ∞

n−∞

γn1−γn f

γn

gγn

f−γn

g−γn

. 1.10

In this context, we assume that |γn1−γn|/0 for alln ∈ Z. By construction, it is clear that

L2_T∗_{is a Hilbert space overCas it is a weighted sequence space, one of its orthogonal bases} being given by all functionseσ

n :T\ {0} → {0,1}which are specified byeσnτqm:δmn δστ

withm, n∈Zandσ, τ∈ {1,−1}.

Already now, we can say that the separation ansatz for the discretized Schr ¨odinger equation will lead us to looking for eigensolutions of a given Schr ¨odinger operator in the threefold product spaceL2_T∗_{× L}2_T∗_{× L}2_T∗_.

Hence we come to the conclusion that in case of the separation ansatz for the Schr ¨odinger equation, the followingrationaleapplies:

The solutions of a Schr¨odinger equation on a basic discrete quadratic grid are directly related to the spectral behavior of the Jacobi operators acting in the underlying weighted sequence spaces.

Before presenting discrete versions of the Schr ¨odinger equation on basic quadratic grids, let us first come back to the situation of Lemma1.2where we now assume that the potential is given by the requirement

fx x2, gyy2, hz z2, x, y, z∈R3. 1.11

Hence, it is sufficient to look at the one-dimensional Schr ¨odinger equation

For the convenience of the reader, let us refer to the following fact. let the sequence of functionsψn_n∈N0be recursively given by the requirement

ψn1x:−ψnx xψnx, x∈R, n∈N0, 1.13

whereψ0 :R → R, x→ ψ0x: e−1/2x

2

.We then haveψn ∈ L2R∩C2Rforn∈N0and

moreover

−ψ_nx x2ψnx 2n1ψnx, x∈R, n∈N0. 1.14

This result reflects the celebrated so-calledLadder Operator Formalism. We first review a main result in discrete Schr ¨odinger theory that is a basic analog of the just described continuous situation. Let us therefore state in a next step some more useful tools for the discrete description.

Definition 1.4. Let 0< q <1 and letTbe a nonempty closed set with the properties

∀x∈T: qx∈T, q−1x∈T,−x∈T. 1.15

Let for anyf:T → Rthe right-shift resp. left-shift operation be fixed through

Rfx:fqx, Lfx:fq−1_{x, x∈T, 1.16}

respectively, the right-hand resp. left-hand basic difference operation will for any function f:T∗ → Rbe given by

Dqf

x: f

qx−fx qx−x ,

Dq−1fx:

fq−1_x−fx

q−1_x−x , x∈T\ {0}. 1.17

Let moreoverα >0 and let

g:T\ {0} −→R, x−→gx:

ϕqx−ϕx

ϕxq−1x

1α1−qx2₋₁

qx−x , 1.18

where the positive even functionϕ: T → R is chosen as a solution to the basic difference equation

ϕqx1α1−qx2ϕx, x∈T. 1.19

Thecreation operationA∗resp.annihilation operationAare then introduced by their actions on

anyψ :T∗ → R as follows:

A∗ψ −DqgXR

ψ, Aψq−1LDqLgX

We refer to the discrete Schr ¨odinger equation with an oscillator potential onT∗by

q−1−DqgXR

LDqLgX

ψ λψ. 1.21

The following result reveals that the discrete Schr ¨odinger equation with an oscillator potential onT∗shows similar properties than its classical analog does.

Lemma 1.5. Let the function ϕ be specified like in Definition 1.4, satisfying the basic difference equation1.19 onRc1,0,0

q with c1/0. Let moreover ψ0 : Rqc1,0,0 → R, x → ψ0x :

ϕx. Forn∈N0,the functionsψn:Rqc1,0,0 → R, given by ψnx: A∗nψ0x(whilex∈Rqc1,0,0)

are well defined inL2_Rc1,0,0

q and solve the basic Schr¨odinger equation1.21in the following sense:

q−1−DqgXR

LDqLgX

ψn

qn₋₁

q−1 ψn,

A∗ψn ψn1, Aψn

qn₋₁

q−1 ψn−1, ψnx H

q

nxψ0x,

H_nq₁x−αqnxHnqx α

qn₋₁

q−1 H

q

n−1x 0, H q

0x 1, H q

1x αx.

1.22

These relations apply forx ∈ Rc1,0,0

q and n ∈ N0 where one set ψ−1 : 0, H₋q1 : 0. Moreover,

the corresponding moments of the orthogonality measure for the polynomialsHnqn∈N0, arising from

1.19, are given by

μ2n2

q−2n−1₋₁

α1−qμ2n, μ2n10, n∈N0. 1.23

The proof for the lemma is straightforward and obeys the techniques in 1.

The following central question concerning the functions spaces behind the Schr ¨odinger equation1.21is openand shall be partially attacked in the sequel.

1.1. Central Problem

What are the relations between the linear span of all functions ψn, n ∈ N0 arising from

Lemma1.5and the function spaceL2_Rc1,0,0

q ?

In contrast to the fact that the corresponding question in the Schr ¨odinger differential equation scenario is very well understood, the basic discrete scenario reveals much more structure which is going to be presented throughout the sequel of this article.

All the stated questions are closely connected to solutions of the equation

which originated in context of basic discrete ladder operator formalisms. We are going to investigate the rich analytic structure of its solutions in Section2 and are going to exploit new facts on the corresponding moment problem in Section3of this article.

Let us remark finally that we will—throughout the presentation of our results in this article—repeatedly make use of the suffixbasic. The meaning of it will always be related to the basic discrete grids that we have introduced so far.

The following results will shed some new light on function spaces which are behind basic difference equations. They are not only of interest to applications in mathematical physics but their functional analytic impact will speak for itself. The results altogether show that solving the boundary value problems of Schr ¨odinger equations on time scalesthat have the structure of adaptive grids is a wide new research area. A lot of work still has to be invested into this direction.

For more physically related references on the topic, we invite the interested reader to consider also the work in 2–5.

For the more mathematical context, see, for instance, 1,6–12.

2. Completeness and Lack of Completeness

In the sequel, we will make use of the basic discrete grid:

Rq :

±qn_{|n∈Z, 2.1}

and we will consider the Hilbert space

L2_R q

:

f:Rq−→C|

∞

n−∞

qnfqnfqn_f−qn_f−qn_<∞

. 2.2

Theorem 2.1. Let0< q <1andα >0 as well asϕan even positive solution of

ϕqx1α1−qx2ϕx 2.3

on the basic discrete gridRq. Let the sequences of functionsϕmm∈Zbe given by shifted versions of

theϕ-function as follows:

ϕm∈ L2

Rq

ϕm:Rq−→R, x−→ϕmx:ϕ

qmxϕ−qmx, m∈Z,

ψm∈ L2

Rq

ψm:Rq−→R, x−→ψmx:x ϕ

qm_{x, m∈Z.} 2.4

The finite linear complex span of precisely all the functionsϕmm∈Z andψmm∈Z is then dense in

Proof. Letϕ∈ L2_R

qbe a positive and even solution to

ϕqx1α1−qx2_{ϕx, x∈R}

q. 2.5

One can easily show that anL2_R

q-solution with these properties uniquely exists, up to a

positive factor, moreover all the functions defined by2.4are well defined inL2_R

q. Let us

refer by the sequencePnn∈N0to all the orthonormal polynomialsPn :Rq → Rwhich arise from the Gram-Schmidt procedure with respect to the functionϕ2_{. They satisfy a three-term}

recurrence relation

Pn1x−αnxPnx βnPn−1x 0, P−1x 0, x∈Rq, n∈N0, 2.6

where forn∈N0the coefficientsαn, βnmay be determined by standard methods through the

moments resulting from2.5. From the basic difference equation2.5we may also conclude that the polynomialsPnn∈N0are subject to an indeterminate moment problem, we come back to this in Section3.

Forn∈N0andx∈Rq, the functions given byPnxϕxmay now be normalized, let

us denote their norms byρnwherenis running inN0. Let us forn∈N0moreover denote the

normalized versions of the functionsPnϕbyun.

The following observation is essential. theC-linear finite span of all functions given by

Rmϕx, xRmϕx, x∈Rq, m∈Z 2.7

is the same than theC-linear finite span of all functions specified by

Rmunx, xRmunx, x∈Rq, m∈Z, n∈N0. 2.8

As a consequence of 2.6, we conclude that the functions unn∈N0 fulfill the recurrence relation:

ρn1un1x−αnρnxunx βnρn−1un−1x 0, u−1x 0, x∈Rq, n∈N0. 2.9

How ever2.9can be brought into the standard form which is of relevance for considering the corresponding Jacobi operator,

Xunan1un1anun−1, n∈N0, 2.10

where the coefficients are given bya00, an1

βn1/αnαn1, n∈N0.

operator, requires to be a formally symmetric linear operator on the finite linear span of the

As for the definition range ofXinH, let us chooseXas a densely defined linear operator in

Hwhere we assume that

DX: also implies that anyλ∈Cconstitutes an eigenvalue ofX∗, hence the point spectrum ofX∗is

C. According to the deficiency index structure1,1of the operatorX, let us now choose the particular self-adjoint extensionY ofXwhich allows a prescribed real-eigenvalueλ 1/0. The corresponding situation for the eigensolution may be written as

Y

Note that we have used in2.14the commutation behaviorRk_{X q}k_XRk_{which is satisfied}

for any fixedk∈Zand in addition the fact that the sequenceRk_w

nn∈N0again converges to 0 in the sense of the canonicalL2_R

q-norm for anyk∈Z. An analogous result is obtained in

the case when we start with the eigenvalueλ−1/0.

Summing up the stated facts, we see that the self-adjoint operatorY, interpreted now as the multiplication operator, acting on a dense domain inL2_R

q, has precisely the point

spectrum{qn_,−qn_{|n∈Z}in the sense of}

the functionseσ

n, n∈Z with norm 1 being fixed by

eσ_nτqm: 1

qn/2_1−q δστδmn, m, n∈Z, σ, τ ∈ {1,−1}. 2.16

Let us recall what we had stated at the beginning: theC-linear finite span of all functions

Rmϕx, xRmϕx, x∈Rq, m∈Z 2.17

is the same than theC-linear finite span of all functions:

Rmunx, xRmunx, x∈R, m∈Z, n∈N0. 2.18

Taking the observations together, we conclude that theC-linear span of all functions

ϕmx

Rmϕx, ψmx x

Rmϕx, x∈Rq, m∈Z 2.19

is dense in the original Hilbert spaceL2_R q.

We finally want to show that the C-linear span ofprecisely allthe functions in2.19 is dense in L2_R

q. This can be seen as follows. taking away one of the functions Rnϕ

or XRn_{ϕ n ∈ N}

0would already remove the completeness of the smaller Hilbert space

H ⊆ L2_R

q. According to the property that the functions from2.19are dense inL2Rq, it

follows, for instance, that for anyk ∈Z, there exists a double sequenceck

j_k,j∈Zsuch that in

the sense of the canonically inducedL2_R

q-norm:

Suppose that there exists a specifici∈Zsuch thatck

i 0 for allk∈Z.

The last expression now may be rewritten as

e_k1e−_k1−

l1

j−l2

ck_j−1ϕj, k∈Z. 2.22

Successive application ofRm_{to2.21resp.2.22withm∈N}

0resp.−m∈N0shows that the

This however would lead to a contradiction. Therefore, it becomes apparent that the complex finite linear span of precisely all the functions Rm_ϕresp.XRm_{ϕ wheremis running inZ}

is dense in the Hilbert spaceL2_R

q. Summing up all facts, the basis property stated in the

theorem finally follows according to2.15.

Let us now focus on the following situation to move on towards the second main result of this article.

letPbe a positive symmetric polynomial, that is,

P :R−→R, x−→Px P−x>0, Px

is called thereal polynomial hulloff.

Define forn∈N0:

If two densities generate the same moments then the induced orthogonal polynomials are the same. this is an isometry situation. According to the constructions of the two different types of moments, namely, on the one, hand side, the moments of typeμn and on the other hand

the moments of typeμnand by comparing2.26and2.30, we see that here, the mentioned

isometry situation is matched—provided the initial conditions for the respective moments are chosen in the same way.

We use this observation now to proceed with the conclusions. Let us make use again of the lattice

Rq :

and we will use again the Hilbert space

L2_R

Then the discrete analog ofPf is

P_f:

q, we will construct a linear operator being

bounded in the spaceP√_f but unbounded when restricted toP

and define generally forψ∈ L2_R:

Lψx:ψq−1x. 2.36

We denote from now on the respective multiplication operator again byX and use LX q−1_{XLto see that for a suitable operator-valued functionf,the following holds:}

LQX

Note that for the definition of Q, we need to consider the characterization of the corresponding measure.

represent the eigenfunctions ofLPX not necessarily orthogonal sinceLPXwas not required to be symmetricandq−N_{are the eigenvalues. However since the eigenvalues are}

Now consider

L

PXϕNq−NϕN. 2.42

Define the operatorsTmby

Tm:

For topological reasons, it follows that

In particular,

asn → ∞. ThusTis defined on anyenand generates infinitely many “rods” on the left-hand

side of en going toward 0. Therefore, T is well defined on anyen and, therefore, it is well

defined on any finite linear combination of theen.

By hypothesis,P

f is dense inL 2_R

q. Then for,n∈Zthere exists a sequence inP_f

which approximatesento any degree of accuracy in the sense of theL2Rq-norm.

Now the question arises: looking at alln∈Z,is there a lower bound forTe n 2

? Theenmare pairwise orthogonal, that is, from

it follows that

_Te

n 2

en2

≥ α2nen12

en2

α2_n·q

n1

qn q·α 2

n, αn:

Pqn−1_, 2.57

and therefore we have

_Te

n 2

en2

≥q·Pqn−1−→ ∞ 2.58

asn → −∞. It follows thatTis an unbounded operator, a contradiction. Therefore,P

f is not

dense inL2_R

q, implying thatP√_f is not dense inL2R.

However note that the result on the lack of completeness stated in the previous theorem should not be confusing with the fact that pointwise convergence may occur as the following theorem reveals.

Theorem 2.4. LetΦ:R → Rbe a differentiable positive even solution to

Φqx1α1−q x2Φx, x∈R. 2.59

Let moreoverHnq:R → R, n∈N0be the continuous solutions to the recurrence relation

H_nq₁x−αqnxHnqx α

qn₋₁

q−1H

q

n−1x 0, n∈N0, 2.60

with initial conditionsH₀qx 1, H₁qx αx for allx ∈ R. The closure of the finite linear span of all these continuous functionsHnqΦ, n ∈ N0 is a Hilbert spaceF ⊆ L2R.For any element v

in the finite linear span of the conventional (continuous) Hermite functions, there exists a sequence

um,kk,m∈N0⊆ Fwhich converges pointwise tov.

Proof. According to the assertions of the theorem, the inverseΦ−1_{of the functionΦ:R → R,}

given by

∀x∈R: Φ−1Φx x 2.61

is differentiable and fulfills the basic difference equation

The function Φ−1_{, being extended to the whole complex plane, can be interpreted as a}

holomorphic function due to its growth behavior, in particular, it allows a product expansion in the whole complex plane,

∀z∈C: Φ−1z a ∞

k0

z−ak, 2.63

the sequenceakk∈Zof complex numbers being uniquely fixed,adenoting a multiplicative

constant. Hence, the functionh:C → Cgiven by

hz e−1/2z2Φ−1z, z∈C, 2.64

is also holomorphic. Inserting the corresponding power series:

hz ∞

n0

bnzn, z∈C. 2.65

we end up with the statement

e−1/2z2hzΦz

_∞

n0

bnzn

Φz, z∈C. 2.66

Any monomialzn_{, n∈N}

0may be written as

zn

⎛ ⎝n

j0

cn_jH_j∗qz

⎞

⎠_{, z∈C 2.67}

with a double sequence of uniquely fixed real numbersc_jn

j,n∈N0. The polynomial functions H_j∗q, j∈N0of degreejwill be chosen such that

_∞

−∞H ∗

i q_xH∗

j

q_{xΦxdx0 2.68}

fori, j ∈N0 buti /j. Polynomials which fulfill these properties are, for instance, those fixed

by2.60, see also 1. We may rewrite2.66as follows:

e−1/2z2

⎛ ⎝∞

n0 n

j0

cn_jH_j∗qz

⎞

Generalizing this result, we see that there exists a threefold sequence c_jn,m

j,m,n∈N0 of real numbers such that the classical continuous Hermite functions, given by

Hmxe−1/2x

2

, x∈R, m∈N0, 2.70

have the following representation

Hmze−1/2z

2

⎛ ⎝∞

n0 n

j0

cn,m_j H_j∗qz

⎞

⎠_{Φz, z∈C. 2.71}

We recall that the closure of the finite linear complex span of all functionsHn∗qΦ, n ∈ N0

is a Hilbert space, call itF, which is aproper subspace ofL2_{R,hence being not dense in}

L2_{R—see Theorem2.3.}

Form∈N0, let us now consider the sequences of functionsum,kk∈N0, given by

um,k:R−→R, x−→um,kx:

⎛ ⎝k

n0 n

j0

cn,m_j H_j∗qx

⎞

⎠_{Φx, k∈N0, 2.72}

According to what we have shown it follows that each of the functionsum,k ∈ Fconverges

pointwise to the functionshm:R → R,given by

hm:R−→R, x−→hmx:Hmxe−1/2x

2

, m∈N0, 2.73

ask → ∞.

3. Basic Difference Equations and Moment Problems

Let us make first some more general remarks on the special type of polynomials2.60we are considering. In literature, see, for instance, the internet reference to the Koekoek-Swarttouw online report on orthogonal polynomialshttp://fa.its.tudelft.nl/∼koekoek/askey/there are listed two types of deformed discrete generalizations of the classical conventional Hermite polynomials, namely, the discrete basic Hermite polynomials of type I and the discrete basic Hermite polynomials of type II. These polynomials appear in the mentioned internet report under citations 3.28 and 3.29 . Both types of polynomials, specified under the two respective citations by the symbolhn whilenis a nonnegative integer, can be succesively transformed scaling the argument and renormalizing the coefficientsinto the one and same form which is given by

H_nq₁x−αqnxHnqx α

qn₋₁

q−1H

q

n−1x 0, n∈N0, 3.1

without the number 1—the caseq1 being reserved for the classical conventional Hermite polynomials. Depending on the choice ofq, the two different types of discrete basic Hermite polynomials can be found. The case 0 < q < 1 corresponds to the discrete basic Hermite polynomials of type II, the caseq >1 corresponds to the discrete basic Hermite polynomials of type I. Up to the late 1990 years, the perception was that both type of discrete basic Hermite polynomials have only discrete orthogonality measures. This is certainly true in the case of q > 1 since the existence of such an orthogonality measure was shown explicitly and since the moment problem behind the discrete basic Hermite polynomials of type I is uniquely determined.

However, it could be shown that beside the known discrete orthogonality measure, specified in the aforementioned internet report, the discrete basic Hermite polynomials of type II, hence being connected to3.1with 0< q <1 allow also orthogonality measures with continuous support.

Let us look at this phenomenon in some more detail.

It is known as a conventional result that a symmetric orthogonality measure with discrete support for the polynomials3.1with 0< q <1, yields moments being given by

ν2m2

q−2m−1₋₁

α1−q ν2m, ν2m10, m∈N0. 3.2

In 1, it was shown that there exist continuous and piecewise continuous solutions to the difference equation

ψqx1α1−qx2ψx, x∈R 3.3

leading to the same moments3.2. Such a behavior of the discrete basic Hermite polynomials of type II, hence being related to the scenario3.1with 0< q <1, was quite unexpected. Vice versa, once momentsνm with nonnegative integermof a given weight function are given

through3.2, it can immediately be said that the weight function provides an orthogonality measure for the discrete basic Hermite polynomials of type II, related to3.1with 0< q <1.

The question however remains whether all weight functions for the discrete basic Hermite polynomials of type II, being related to 3.1with 0 < q < 1 must fulfill a basic difference equation of type 3.3. We develop now an answer to this question which goes beyond the results known so far.

Let throughout the sequel 0 < q < 1 andα > 0.We first put forward the following definition.

Definition 3.1. By the moments of a given orthogonality measure, we understand—like in the

previous sections—the numbers

μm:

_∞

−∞x

m_{ψxdx, m∈N}

0. 3.4

Theorem 3.2. There exists a positive symmetricCR-solutionψto the basic difference equation

ψq2x1α1−qx21α1−qq2x2ψx, x∈R, 3.5

not being a solution to

ψqx1α1−qx2ψx, x∈R, 3.6

but generating the same moments and therefore yielding an orthogonality measure to the polynomials

Hnqn∈N0from2.60in the following sense:

_∞

−∞H

q

mxHnqxψxdxvqnδmn, m, n∈N0. 3.7

The proof to establish will be a step beyond the already known orthogonality results for the polynomials under consideration.

Proof. Let us consider first the special basic difference equations:

ψqx1α1−qx2ψx, x∈R, 3.8

ψq2x1α1−qx21α1−qq2x2ψx, x∈R. 3.9

Obviously, any positiveCR-solutionψof3.8satisfies3.9. Moreover, one can show that theseCR-solutions of3.8are inL1_{R. Therefore, the set of positive symmetric solutions}

to3.9which are inL1_{R∩CRis nonempty. Let nowψ∈ L}1_{R∩CRbe such that}

_∞

−∞x

2m_ψq2_xdx

_∞

−∞x

2m_11−qαx2_11−qαq2_x2_{ψxdx, m∈N} 0.

3.10

In the sequel, we are going to use the moments

μm:

_∞

−∞x

m_{ψxdx, m∈N}

0. 3.11

Remember thatμ2m10 form∈N0asψwas assumed to act symmetrically on the real axis.

Equation3.10may therefore now be rewritten—in terms of the numbersμ2m—as

q−2m−1_μ

2mμ2mα

1q2_1−qμ

2m2α2q2

Let nowϕ∈ L1_{R∩CRbe a positive symmetric solution to3.8. Then, similar integration}

like in3.10shows that the corresponding moments

νm:

_∞

−∞x

m_{ϕxdx, m∈N}

0, 3.13

indeed satisfy3.12, in particular they obey

ν2m2

q−2m−1₋₁

α1−q ν2m, ν2m10, m∈N0, 3.14

Hence,ϕprovides an orthogonality measure to the discrete basic Hermite polynomials under consideration—see 1. The main issue to address now is the following. we want to show that there are positive symmetric functionsψ ∈ L1_{R∩CRsuch thatψ satisfies3.9but not} 3.8and such that the moments, given by3.11and3.12, satisfy

μ2m2

q−2m−1₋₁

α1−q μ2m, μ2m10, m∈N0. 3.15

In other words, we have to prove that there exist orthogonality measures to the discrete basic Hermite polynomialsHnqn∈N0which stem from a solution to3.9but not from a solution to

3.8.

We proceed in a constructive way.

Let us denote first byVtheC-linear subspace ofL1R∩CRsuch that allψ∈Vare in the kernel of the mapS:V → CR, given by

ψ−→Sψx:ψqx−11−q αx2ψx, x∈R. 3.16

Let moreoverWbe theC-linear subspace ofL1_{R∩CRcontaining all the functions which}

are in the kernel of the linear mapT :W → CR, given by

ψ−→Tψx:ψq2_{x−11−qαx}2_11−qq2_αx2_{ψx, x∈R. 3.17}

We will also make use of the C-linear subspace U ⊆ W which we choose as the maximal common domain on which the following two linear functionals are well defined:

s:U−→C, ψ−→sψ:

_∞

−∞x

2_ψxdx,

t:U−→C, ψ−→tψ:

_∞

−∞ψxdx.

It is easy to see thatsis continuous. to verify this, we consider the expression|sψ|/ψ₁in

But asψis assumed to be an element ofUand hence ofW, we may rewrite the expression in the denominator and give the following estimatewith a positive constantγ:

sψ

Hences :U → Cis a bounded linear map and therefore continuous. In the same way, we show thatt:U → Cis continuous.

We now continue as follows.

Using the terminology of characteristic functions, we first look at

u1:R−→R, x−→u1x:χq2_,q3/2x

It is possible to choose continuous even L1_{R-functions v}

1, v2 : R → Rbeing in Uand

also inL1_{R. According to the construction of the functionv, we now may choosen ∈ N}

sufficiently large such that the positive continuous even function

ψ:nϕv 3.25

finally fulfills the required properties, namely,

SψSnϕvSv /0, TψTnϕvn, TϕTv0 3.26

as well as the moment conditions

sψ n1μ2, t

ψ n1μ0. 3.27

From3.27and by integrating

x2mψq2xx2m11−qαx211−qαq2x2ψx, x∈R, m∈N0 3.28

now it follows form∈N0that the momentsμm:

"_∞

−∞xmϕxdxsatisfy3.15and therefore also 3.12. In particular, the function ψ yields therefore an orthogonality measure to the discrete basic Hermite polynomials, given by2.60. Note that by construction, the function ψnow fulfills all the assertions of Theorem3.2. Hence, Theorem3.2holds in total.

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