Heim-Lorek supermodels.

Abstract

We first give a short review on supermodels in quantum mechanics where the main focus will be on selfsimilar supermodels, showing a point spectrum which consists of basic q-versions of the natural numbers. We want to relate these supermodels to discrete versions of Schrodinger operators which exhibit--at least partially--the same type of discrete point spectrum. To do so, the concept of strip discretizations is reviewed on basic linear grids. This type of discretization shows the typical point spectrum, consisting of basic q-versions of the natural numbers. Precisely the same type of spectrum is finally also presented in case of basic multigrid discretizations. We therefore obtain a unified discrete model of some Schrodinger equations which allow both, piecewise continuous solutions (Section 4) and sophisticated multigrid solutions (Section 5)--a scenario which plays already a great role in approaches established by A. Lorek and B. Heim.

AMS subject classification: 39A10, 39A13, 35Q40, 46N20.

Keywords: Schrodinger operator, quantum calculus, superpotential.

1. Schrodinger Equations and Superpotentials

Let us first review some basic facts which we had also stated in [7]. Supersymmetry is one of the most powerful tools being applied to problems of theoretical physics. In the last years, there were great achievements especially on the area of supersymmetry in quantum mechanics. For an excellent contribution to the topic see for instance the articles by M. Robnik [19] and M. Robnik and J. Liu [12]. The stationary one-dimensional version of Schrodinger's equation ([lambda] being a fixed value in R)

-[psi]"(x) + V (x)[psi](x) = [lambda][psi](x), x [member of] R (1.1)

can with some general success be factorized by using the concept of so-called superpotentials.

In Schrodinger theory, the following scenario is of particular interest: Given two Schrodinger equations with different potentials [V.sub.1] and [V.sub.2]. Under some circumstances, it is possible to write them in the form

[B.sup.+][B.sup.[psi]] = [lambda][psi], B[B.sup.+][psi] = [mu][psi],

where B and [B.sup.+] are formally adjoint to each other, being defined on some common domain in [L.sup.2](R). Let us shortly review the method how to address the stated factorization problem.

The first step is the construction of a so-called superpotential W such that one can express the partner potentials [V.sub.1], [V.sub.2] as follows:

[V.sub.1] = 1/2 ([W.sup.2] - [square root of 2]W'), [V.sub.2] = 1/2([W.sup.2] + [square root of 2]W').

The superpotential W is fixed by assuming that the potential [V.sub.1] allows 0 as an eigenvalue, the corresponding eigenfunction [psi] being positive. This leads to the condition

-[psi]" (x) + 1/2 ([W.sup.2](x) - [square root of 2]W'(x))[psi](x) = 0, x[member of] R.

A solution to this equation is given by

W(x) = -[square root of 2](ln [psi])'(x), x [member of] R.

The aimed factorization is now achieved by the equality

[H.sub.1] = [B.sup.+]B, [H.sub.2] = B[B.sup.+],

where the differential operators [H.sub.1], [H.sub.2] are specified by the supersymmetric ladder operators 

B := 1/[square root of 2] (W + [square root of 2]d/dx), [B.sup.+] := 1/[square root of 2] (W - [square root of 2]d/dx).

To illustrate this formalism, let us consider the two potentials

[V.sub.1](x) = [x.sup.2]/4 - 1/2, [V.sub.2](x) = [x.sup.2]/4 + 1/2, x[member of] R.

As for the superpotential W, we obtain just W(x) = [square root of 2]x, leading to the well-understood conventional ladder operator formalism

[H.sub.1] = [B.sup.+]B, [H.sub.2] = B[B.sup.+],

B = 1 [square root of 2] (x + [square root of 2] d/dx), [B.sup.+] = 1/[square root of 2] (x - [square root of 2] d/dx).

The key message is now that one can determine the point spectrum of [H.sub.1],[H.sub.2] completely by using the operators B,[B.sup.+]. A further, more illustrative example is the so-called Rosen-Morse potential, in which a real parameter y occurs:

[V.sub.1](x, y) = [y.sup.2] - y(y + 1)/[cosh.sup.2](x), x [member of] R.

Assuming that the equation

-[psi]" (x) + [V.sub.1](x, y)[psi](x) = 0

has a solution [psi] [member of] [L.sup.2](R), we are first led to the superpotential

W(x) = [square root of 2]y tanh(x), x [member of] R

as well as to the potential

[V.sub.2](x, y) = [y.sup.2] - y(y - 1)/[cosh.sup.2](x), x [member of] R.

The difference of the two partner potentials is given by

[V.sub.2](x, y) - [V.sub.1](x, y - 1) = 2y - 1/2, x[member of] R.

Generalizing the observations made so far, the two potentials [V.sub.1], [V.sub.2] are called form-invariant if the following identity holds for different values of x, [y.sub.1], [y.sub.2], the expression R([y.sub.1]) being a continuous function of [y.sub.1]:

[V.sub.2](x, [y.sub.1]) = [V.sub.1](x, [y.sub.2]) + R([y.sub.1]). (1.2)

According to the above construction, the pair B,[B.sup.+] specifies two different Schrodinger equations, which together are referred to as a supersymmetric model or just as a supermodel.

2. Selfsimilar Supermodels

In many important applications, it follows from the defining equation for form invariance 

[V.sub.2](x, [y.sub.1]) = [V.sub.1](x, [y.sub.2]) + R([y.sub.1]) (2.1)

that [y.sub.2] = [y.sub.1]+h where h is a real constant. A completely new class of form invariant potentials has been proposed in [11], where potentials were constructed whose parameters are related to each other by

[y.sub.2] = q[y.sub.1], 0 < q < 1. (2.2)

In order to make apparent what kind of possible point spectrum is generated by the property (2.2), we follow the basic outline in [10] where the superpotential is expanded as

W(x, y) = [[infinity].summation over (j=0)] [g.sub.j] (x)[y.sup.j]

for some suitable parameter y [member of] R. The function R from (1.2) is assumed to be given by an analytic ansatz 

R(y) = [[infinity].summation over (j=0)] [R.sub.j][y.sup.j]. (2.3)

Plugging this ansatz into formula (2.1) yields

R(y) = [V.sub.2](x, y) - [V.sub.1](x, qy) = [W.sub.2](x, y) - [W.sub.2](x, qy) + [square root of 2](([[partial derivative].sub.x]W)(x, y) + ([[partial derivative].sub.x]W)(x, qy)).

Inserting now the expansion (2.3) for the function R, one obtains by comparing the coefficients

[R.sub.n] = 1/2 [[infinity].summation over (i=1)] (1 + [q.sup.n-i])(1 - [q.sup.i])[g.sub.i][g.sunb.n-i] + [square root of 2](1 + [q.sup.n])[g'.sub.n], n [member of] N

and the value [R.sub.0] being given by [R.sub.0] = [g'.sub.0]. With the abbreviations

[r.sub.n] := [R.sub.n]/1 - [q.sub.n], [d.sub.n] := 1 - [q.sup.n]/1 + [q.sup.n], n [member of] N

one is led to nonlinear integral equations, given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where restrictions of the solutions of these equations are put by the conditions

[R.sub.0] = 0, [g.sub.0](x) = 0, [r.sub.n] = z[[delta].sub.n1], n [member of] N,

[[delta].sub.n1] denoting the Kronecker symbol and z being a positive parameter. This nonlinear integral equation allows now the solutions

R(y) = [R.sub.1]y = R, [g.sub.n](x) = 1/[square root of 2][[beta].sub.n][x.sup.2n-1], x [member of] R, n [member of] N,

where the coefficients [[beta].sub.n] are fixed by the recurrence formula

[[beta].sub.1] = 2[R.sub.1]/1 + q, [[beta].sub.0] = 0, [[beta].sub.n] = -dn/2n - 1 [n-1.summation over (i=1)][[beta].sub.i][[beta].sub.n-i], n [member of] N \ {1}.

The superpotential now reads

W(x, y) = [[infinity].summation over (j=1)][[beta].sub.j][y.sup.j] [(x/[square root of 2].sup.2j-1], x [member of] R.

The formal groundstate, belonging to [V.sub.1], is given by the formula

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Direct calculation leads to the formula

W(x, [y.sub.2]) = [square root of q]W([square root of q]x, [y.sub.1]), x [member of] R

showing some selfsimilar property. This type of superpotential is therefore also referred to by the name selfsimilar superpotential. Applying a generalized version of the ladder operator formalism, one is led to the energies of the operator [H.sub.1], being given by

[[lambda].sub.n] = R [n-1.summation over (j=0)] [q.sup.j] = R 1 - [q.sup.n]/1 - q, n [member of] [N.sub.0]. (2.4)

Let us now arrive at an interesting isospectrality scenario in context of the q-discretization, where we are interested in discrete versions of Schrodinger operators which allow (2.4) as their (at least partial) point spectrum.

3. Using Heim-Lorek Discretizations

A. Lorek investigated in her PhD thesis from 1995 several quantum mechanical toy models in which the phase space is discrete and which link in an elegant way continuous solutions and discrete solutions of q-difference equations in context of quantum mechanics, [14]. This is a fancy propagation of original ideas by B. Heim on discretized versions of fundamental equations in mathematical physics, see [8]. We speak therefore in the sequel of the Heim-Lorek discretization of Schrodinger operators: We address Schrodinger q-difference equations where we study piecewise continuous solutions to these equations, having support on some kind of strip structures which are generated by the symmetries of the lattice {+[q.sup.n], -[q.sup.n]|n [member of] Z}and also sophisticated related discrete solutions to these equations. The concept of strip discretizations is reviewed on basic linear grids: A brief review on related results by N. Garbers and A. Ruffing, see [7], will happen in Section 4. Already these results reveal a new and challenging aspect of isospectrality. In Section 5, we will address the concept of basic multigrid discretizations which shed new light on applications of basic linear grids within numerical analysis. Again, it is isospectrality which turns out to play a key role in understanding the related spectra. The main surprise is that there exist in parallel piecewise continuous solutions (Section 4) and multigrid discrete solutions to the same Schrodinger q-difference equations (Section 5). This perfectly matches the original scenario of the Heim-Lorek discretization.

4. Strip Discretizations and Isospectrality

Throughout this section, we refer to the recently obtained results by N. Garbers and A. Ruffing on strip discretizations, see [7]. As stated above, we will generalize these results in Section 5 to multigrid discretizations.

Let us refer throughout the sequel to a parameter 0 < q < 1, as it was motivated by the investigation of selfsimilar superpotentials in the previous section.

The following result reveals that the discrete Schrodinger equation with an oscillator potential on [OMEGA] shows similar properties than its classical analog does.

Definition 4.1. [Strip Discretization] Let [OMEGA] [subset or equal to] R \ {0} be a nonempty closed set with Lebesgue measure [mu]([OMEGA]) > 0 as well as

[for all]x [member of] [OMEGA] : qx [member of] [OMEGA], [q.sup.-1]x [member of] [OMEGA], -x [member of] [OMEGA].

We call the time scale [OMEGA] a homogeneous strip discretization or just strip discretization of the configuration space. The Hilbert space of the strip discretization is introduced by the requirement

[L.sup.2]([OMEGA]) := {f [member of] [L.sup.2](R) | f = f [??] [[chi square].sub.[OMEGA]]},

and the scalar product of two functions f, g [member of] [L.sup.2]([OMEGA]) is introduced by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.1)

using the characteristic function [[chi].sub.[OMEGA]] of the time scale [OMEGA]. By construction, it is clear that [L.sup.2]([OMEGA]) is a Hilbert space over C, being a proper subspace of the square-integrable functions themselves, i.e., of [L.sup.2](R). In order to proceed, let us first review some facts on the Schrodinger equation with quadratic potential, given by

-[psi]" (x) + [x.sup.2][psi](x) = [lambda][psi](x), x [member of] R.

The following structure is one the most familiar facts within mathematical physics: Let the sequence of functions [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be recursively given by the requirement

[[psi].sub.n+1](x) := -[[psi]'.sub.n] (x) + x[[psi].sub.n](x), x [member of] R, n [member of] [N.sub.0],

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. We then have [[psi].sub.n] [member of] [L.sup.2](R) [intersection] [C.sup.2](R) for n [member of] [N.sub.0] and moreover

-[[psi]".sub.n] (x) + [x.sup.2][[psi].sub.n](x) = (2n + 1)[[psi].sub.n](x), x [member of] R, n [member of] [N.sub.0].

This result reflects the conventional ladder operator formalism. We now develop a result in discrete Schrodinger theory on strip structures which turns out to be a q-analog of the just described continuous situation. Let us therefore state in a next step some more useful tools for the strip discretization approach.

Definition 4.2. Let [OMEGA] [subset or equal to] R \ {0} be a nonempty closed set with the property [mu]([OMEGA]) > 0 as well as

[for all]x [member of] [OMEGA] : q x [member of] [OMEGA], [q.sup.-1]x [member of] [OMEGA], -x [member of] [OMEGA]. (4.2)

Let for any f : [??] R the right-shift resp. left-shift operation be defined by

(Rf)(x) := f (qx), (Lf)(x) := f ([q.sup-1]x), x [member of] [OMEGA].

Respectively, the right-hand resp. left-hand q-difference operation shall for any f : [OMEGA] [right arrow] R be given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let moreover [alpha] > 0 and let

g : [OMEGA] [right arrow] [R.sup.+], x [??] g(x) := [square root of ([psi](qx)] - [square root of ([psi](x)]/[square root of ([psi](x))](q - 1)x = [square root of (1 + [alpha](1 - q)[x.sup.2])] - 1/qx - x, (4.3)

where the positive even continuous function [psi] : [OMEGA] [right arrow] [R.sup.+] is chosen as a solution to the q-difference equation

[psi](qx) = (1 + [alpha](1 - q)[x.sup.2])[psi](x), x [member of] [OMEGA]. (4.4)

We are now able to define discrete ladder operators on strip structures.

The creation operation [A.sup.[dagger].sub.q] resp. annihilation operation [A.sub.q] are introduced by their actions on any [psi] : [OMEGA] [right arrow] R as follows:

[A.sup.[dagger].sub.q][psi] = (-[D.sub.q] + g(X)R)[psi], [A.sub.q][psi] = [q.sup.-1]([LD.sub.q] + [L.sub.g](X))[psi].

We refer to the discrete Schrodinger equation with an oscillator potential on [OMEGA] by

[q.sup.-1](-[D.sub.q] + g(X)R)([LD.sub.q] + [L.sub.g](X))[psi] = [lambda][psi]. (4.5)

The following result reveals that the discrete Schrodinger equation with an oscillator potential on [OMEGA] shows similar properties than its classical analog does.

Theorem 4.3. Let the time scale [OMEGA] be a strip discretization in the sense of Definition 4.1 and let the function [psi] be specified like in Definition 4.2, satisfying the q-difference equation (4.4) on [OMEGA]:

[psi](qx) = (1 + [alpha](1 - q)[x.sup.2])[psi](x), [psi](x) = [psi](-x) > 0, x [member of] [OMEGA].

For n [member of] [N.sub.0], the functions [[psi].sub.n] : [OMEGA] [right arrow] R, given by [[psi].sub.n](x) := ([([A.sup.[dagger].sub.q]).sup.n][square root of [psi]](x), x [member of] [OMEGA] are well defined in [L.sup.2]([OMEGA]) and solve the q-Schrodinger equation (4.5) in the following sense:

[q.sup.-1](-[D.sub.q] + g(X)R)([LD.sub.q] + Lg(X))[[psi].sub.n] = [alpha]/q [q.sup.n] - 1/q - 1 [[psi].sub.n]. (4.6)

Moreover, the linear maps [A.sub.q],[A.sup.[dagger].sub.q] act as ladder operators on the functions [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in the following sense (n [member of] [N.sub.0], [[psi].sub.-1] := 0):

[A.sup.[dagger].sub.q[psi]]n = [[psi].sub.n+1], [A.sub.q][[psi].sub.n] = [alpha]/q [q.sup.n] - 1/q - 1 [[psi].sub.n-1], [[psi].sub.n](x) = [H.sup.q.sub.n] (x)[[psi].sub.0](x), x [member of] [OMEGA], (4.7)

where for n [member of] [N.sub.0], the functions [H.sup.q.sub.n] : [OMEGA] [right arrow] R are given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and the functions [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] constitute an orthonormal system in [L.sup.2]([OMEGA]).

Proof. Let us for [psi] [member of] C(R) first consider the equation

[psi](qx)[x.sup.n] = (1 + [alpha](1 - q)[x.sup.2])[psi](x)[x.sup.n], x [member of] [OMEGA], n [member of] [N.sub.0],

which directly follows from (4.4). Using standard arguments, one can show that the function [psi] fulfilling (4.4) is in [L.sup.1](R). This implies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Using the substitution rule on the left-hand side, this directly implies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.8)

Because of (4.2) we have [[chi].sub.[OMEGA]]([q.sup.-1]t) = [[chi].sub.[OMEGA]](t) for any t [member of] R and therefore, (4.8) is equivalent to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Using the abbreviation [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for n [member of] [N.sub.0] we obtain the following result:

[[mu].sub.2n+2]([OMEGA]) = [q.sup.-2n-1] - 1/[alpha](1 - q) [[mu].sub.2n]([OMEGA]), [[mu].sub.2n+1]([OMEGA]) = 0, n [member of] [N.sub.0]. (4.9)

We have shown earlier that any probability density [psi] which generates moments of type (4.9) yields an orthogonality measure to the polynomials [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] which are for k [member of] N fixed through the recurrence relation 

[H.sup.q.sub.k+1](x) - [alpha][q.sup.k]x[H.sup.q.sub.k] (x) + [alpha] [q.sup.k] - 1/q - 1 [H.sup.q.sub.k-1](x) = 0, [H.sup.q.sub.0] (x) = 1, [H.sup.q.sub.1] (x) = [alpha]x,

the variable x being chosen in a suitable integration support. As a consequence of this general result, we obtain the orthogonality relation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with a sequence of positive numbers [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Direct calculations and induction show

[[psi].sub.n](x) := ([([A.sup.[dagger].sub.q]).sup.n][square root of [psi]] [??] [[chi].sub.[OMEGA]])(x) = ([H.sup.q.sub.n] (X)[square root of [psi]] [??] [[chi].sub.[OMEGA]])(x), x [member of] R, n [member of] [N.sub.0].

Let us from now on--without any restriction--refer to the special parameter choice [alpha] = 1. The functions [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] constitute an orthonormal system in [L.sup.2]([OMEGA]). Let us show next that the ladder property (4.7) is fulfilled. The first equation in (4.7) is trivial due to the definition of the functions [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. We remember that the function g is specified like in Definition 4.2. We obtain in the sense of the multiplication operator notation

([LD.sub.q] + Lg(X))([X.sup.n][[psi].sub.0]) = [LD.sub.q][X.sup.n][[psi].sub.0] + Lg(X)[X.sup.n][[psi].sub.0], n [member of] [N.sub.0]

which yields

([LD.sub.q]+Lg(X))([X.sup.n][[psi].sub.0]) = L([q.sup.n] - 1/q - 1 [X.sup.n-1]R[[psi].sub.0] + [X.sup.n][D.sub.q] [[psi].sub.0])+[L.sup.g](X)[X.sup.n][[psi].sub.0], n [member of] [N.sub.0].

This may be rewritten as

([LD.sub.q] + Lg(X))([X.sup.n][[psi].sub.0]) = [q.sup.n] - 1/q - 1 [q.sup.-n+1][X.sup.n-1][[psi].sub.0] + L[X.sup.n]([D.sub.q][[psi].sub.0] + g[[psi].sub.0]), n [member of] [N.sub.0].

Using now however the formulas in (4.3) for the function g, we obtain ([D.sub.q][[psi].sub.0]+g[[psi].sub.0]) = 0 and therefore

([LD.sub.q] + Lg(X))([X.sup.n][[psi].sub.0]) = [q.sup.n] - 1/q - 1 [q.sup.-n+1][X.sup.n-1][[psi].sub.0], n [member of] N.

For m [member of] [N.sub.0], the first m + 1 polynomials of the sequence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] can uniquely be generated by linear combinations of the first m+1 monomials of the sequence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. We therefore conclude

[A.sub.q][[psi].sub.n] = [n-1.summation over (j=0)] [c.sup.n.sub.j][[psi].sub.j], n [member of] N

with uniquely defined real numbers [c.sup.n.sub.j] where j = 0, ..., n - 1 with n [member of] N. Applying again standard substitution techniques to the scalar product integral (4.1), we can derive for any functions f, g [member of] [L.sup.2]([OMEGA]) which are both in the algebraic span of the functions [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] the following relation:

[([A.sup.[dagger].sub.q]f, g).sub.[OMEGA]] = [(f, Aq g).sub.[OMEGA]].

In particular, this result implies

[([A.sup.[dagger].sub.q][[psi].sub.m],[[psi].sub.n]).sub.[OMEGA]] = [([[psi].sub.m],[A.sub.q][[psi].sub.n]).sub.[OMEGA]], m, n [member of] [N.sub.0].

Using the first equation in (4.7) and because of the fact that the functions [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] constitute an orthogonal system in [L.sup.2]([OMEGA]), the second relation in (4.7) follows from standard methods of calculating the norms of the functions [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Equation (4.6) now follows immediately from the first two relations in (4.7). Taking all the steps of the proof together, this finally confirms the statements of Theorem 4.3. []

5. Basic Multigrid Discretizations and Isospectrality

In this last section, we concentrate our interest to isospectrality results in context of some kind of special type of multigrid discretization. We will generalize the results from Section 4 to purely discrete versions of Heim-Lorek discretizations. To do so, we first provide the following.

Definition 5.1. [Basic Multigrid Discretizations] Let for a fixed value n [member of] N the pairwise different real numbers [c.sub.1], ..., [c.sub.n] be given and consider a finite set M [subset or equal to] Z\{0} such that all

[[OMEGA].sup.j.sub.m] := {+[c.sub.j][q.sup.k/m], -[c.sub.j][q.sup.k/m] |k [member of] Z},

where m [member of] M and j [member of] {1, ..., n}, are pairwise disjoint. Let moreover

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The Hilbert space of the basic multigrid discretization is denoted by [L.sup.2]([OMEGA]) and introduced as the set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For any f : [OMEGA] [right arrow] C, the shift operators shall--as previously--be defined by

(Rf)(x) := f (qx), (Lf)(x) := f ([q.sup.-1]x), x [member of] [OMEGA].

And again, the q-difference operations shall for any f : [OMEGA] [right arrow] C be given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Definition 5.2. [Discrete Ladder Operators and Basic Multigrids] Let [alpha] > 0 and let [OMEGA] be a basic multigrid discretization in the sense of the last definition. Let moreover

g : [OMEGA] [right arrow] [R.sup.+], x [??] g(x) := [square root of ([psi](qx)] - [square root of ([psi](x))]/[square root of ([psi](x))](q - 1)x = [square root of (1 + [alpha](1 - q)[x.sup.2])] - 1/qx - x,

where the positive even continuous function [psi] : [OMEGA] [right arrow] [R.sup.+] is chosen as a solution to the q-difference equation

[psi](qx) = (1 + [alpha](1 - q)[x.sup.2])[psi](x), [psi](x) = [psi](-x) > 0, x [member of] [OMEGA].

The creation operation [A.sup.[dagger]] resp. annihilation operation A are again introduced by their actions on any [psi] : [OMEGA] [right arrow] R as follows:

[A.sup.[dagger]][psi] = [(-[D.sub.q] + g(X)R).sub.[psi]], [A.sub.[psi]] = [q.sup.-1][([LD.sub.q] + [L.sub.g](X)).sub.[psi]].

Like in the strip discretization case, we refer to the discrete Schrodinger equation with an oscillator potential on [OMEGA] by

[q.sup.-1](-[D.sub.q] + g(X)R)[([LD.sub.q] + Lg(X)).sub.[psi]] = [lambda][psi]. (5.1)

Evaluating the moments of the density under consideration, we see--and this is a certain kind of surprise--that the density generates--up to a constant factor--the same moments that the corresponding density in the strip discretization case generates. Using a similar argumentation than in the strip discretization case from the last subsection, this directly leads to the basic multigrid analogon of Theorem 4.3.

Theorem 5.3. [Solutions of the Schrodinger Oscillator Equation] Let [OMEGA] be a basic multigrid discretization in the sense of Definition 5.1 and let the function [psi] be specified like in Definition 5.2, satisfying the q-difference equation (4.4) on [OMEGA]. Under these assertions, for n [member of] [N.sub.0], the functions [[psi].sub.n] : [OMEGA] [right arrow] R, given by [[psi].sub.n](x) := ([([A.sup.[dagger]]).sup.n][square root of [psi]])(x), x [member of] [OMEGA] are well defined in [L.sup.2]([OMEGA]) and solve the q-Schrodinger equation (5.1)

[q.sup.-1](-[D.sub.q] + g(X)R)([LD.sub.q] + Lg(X))[[psi].sub.n] = [alpha][q.sup.n] - 1/q - 1 [[psi].sub.n], n [member of] [N.sub.0].

In analogy to the strip discretization case, the linear maps A, [A.sup.[dagger]] act as ladder operators on the functions [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in the following sense (n [member of] [N.sub.0], [[psi].sub.-1] := 0):

[A.sup.[dagger]][[psi].sub.n] = [[psi].sub.n+1], A[[psi].sub.n] = [q.sup.n] - 1/q - 1[[psi].sub.n-1], [[psi].sub.n](x) = [H.sup.q.sub.n] (x)[[psi].sub.0](x),

where for n [member of] [N.sub.0], the functions [H.sup.q.sub.n] : [OMEGA] [right arrow] R are given by

[H.sup.q.sub.n+1](x) - [alpha][q.sup.n]x[H.sup.q.sub.n] (x) + [alpha][q.sup.n] - 1/q - 1 [H.sup.q.sub.n-1](x) = 0, [H.sup.q.sub.0] (x) = 1, [H.sup.q.sub.1] (x) = [alpha]x,

the recurrence relations applying again for x [member of] [OMEGA] and n [member of] [N.sub.0], where we have chosen [[psi].sub.-1] := 0, [H.sup.q.sub.-1] := 0. In the sense of the Hilbert space scalar product, we have also here the general observation

[([A.sup.[dagger]][[psi].sub.m],[[psi].sub.n]).sub.[OMEGA]] = [([[psi].sub.m],A[[psi].sub.n]).sub.[OMEGA]], m, n [member of] [N.sub.0],

the functions [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] constituting an orthonormal system in [L.sup.2]([OMEGA]).

Received January 23, 2007; Accepted June 12, 2007

References

[1] M. Arik and D. D. Coon, Hilbert spaces of analytic functions and generalized coherent states, J. Mathematical Phys., 17(4):524-527, 1976.

[2] R. Askey and S.K. Suslov, The q-harmonic oscillator and an analogue of the Charlier polynomials, J. Phys. A, 26(15):L693-L698, 1993.

[3] R. Askey and S.K. Suslov, The q-harmonic oscillator and the Al-Salam and Carlitz polynomials, Lett. Math. Phys., 29(2):123-132, 1993.

[4] Christian Berg and Andreas Ruffing, Generalized q-Hermite polynomials, Comm. Math. Phys., 223(1):29-46, 2001.

[5] Jacob Stordal Christiansen, The moment problem associated with the q-Laguerre polynomials, Constr. Approx., 19(1):1-22, 2003.

[6] V.V. Dodonov, "Nonclassical" states in quantum optics: a "squeezed" review of the first 75 years, J. Opt. B Quantum Semiclass. Opt., 4(1):R1-R33, 2002.

[7] Nicole Garbers and Andreas Ruffing, Using supermodels in quantum optics, Adv. Difference Equ., pages Art. ID 72768, 14, 2006.

[8] Burkhard Heim, Elementarstrukturen der Materie, Resch Verlag, Innsbruck, Austria, 1984.

[9] Mourad E.H. Ismail and Mizan Rahman, The q-Laguerre polynomials and related moment problems, J. Math. Anal. Appl., 218(1):155-174, 1998.

[10] H. Kalka and G. Soff, Supersymmetrie, Edition Teubner, Stuttgart, Germany, 1997.

[11] A. Khare and U.P. Sukhatme, New shape-invariant potentials in supersymmetric quantum mechanics, J. Phys. A, 26(18):L901-L904, 1993.

[12] J. Liu and Marko Robnik, Supersymmetric quantum mechanics based on higher excited states II: a few new examples of isospectral partner potentials, 1997. Preprint CAMTP.

[13] X.-M. Liu and C. Quesne, Even and odd q-deformed charge coherent states and their nonclassical properties, Phys. Lett. A, 317(3-4):210-222, 2003.

[14] A. Lorek, Dissertation, PhD thesis, Universitat Munchen, 1995.

[15] K.A. Penson and A.I. Solomon, New generalized coherent states, J. Math. Phys., 40(5):2354-2363, 1999.

[16] C. Quesne, Completeness of photon-added squeezed vacuum and one-photon states and of photon-added coherent states on a circle, Phys. Lett. A, 288(5-6):241-250, 2001.

[17] C. Quesne, New q-deformed coherent states with an explicitly known resolution of unity, J. Phys. A, 35(43):9213-9226, 2002.

[18] C. Quesne, K.A. Penson, and V.M. Tkachuk, Maths-type q-deformed coherent states for q > 1, Phys. Lett. A, 313(1-2):29-36, 2003.

[19] Marko Robnik, Supersymmetric quantum mechanics based on higher excited states, J. Phys. A, 30(4):1287-1294, 1997.

[20] Marko Robnik and Valery G. Romanovski, Two-dimensional hydrogen atom in a strong magnetic field, J. Phys. A, 36(29):7923-7951, 2003.

[21] Andreas Ruffing and Moritz Simon, Analytic properties of a special q-exponential function, Progr. Theoret. Phys. Suppl., (150):401-410, 2003. Let's face chaos through nonlinear dynamics (Maribor, 2002).

[22] Barry Simon, The classical moment problem as a self-adjoint finite difference operator, Adv. Math., 137(1):82-203, 1998.

[23] Moritz Simon and Andreas Ruffing, Power series techniques for a special Schrodinger operator and related difference equations, Adv. Difference Equ., (2):109-118, 2005.

[24] V.P. Spiridonov, The factorization method, self-similar potentials and quantum algebras, In Special functions 2000: current perspective and future directions (Tempe, AZ), volume 30 of NATO Sci. Ser. II Math. Phys. Chem., pages 335-364. Kluwer Acad. Publ., Dordrecht, 2001.

[25] V.P. Spiridonov and A.S. Zhedanov, Generalized eigenvalue problem and a new family of rational functions biorthogonal on elliptic grids, In Special functions 2000: current perspective and future directions (Tempe, AZ), volume 30 of NATO Sci. Ser. II Math. Phys. Chem., pages 365-388. Kluwer Acad. Publ., Dordrecht, 2001.

[26] Sergei K. Suslov, An introduction to basic Fourier series, volume 9 of Developments in Mathematics, Kluwer Academic Publishers, Dordrecht, 2003. With a foreword by Mizan Rahman.

Nicole Garbers and Andreas Ruffing

Fakultat fur Mathematik, Technische Universitat Munchen,

Boltzmannstrasse 3, D-85747 Garching, Germany

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