Some Basic Difference Equations of Schrödinger Boundary Value Problems
© Andreas Ruffing et al. 2009
Received: 1 April 2009
Accepted: 28 August 2009
Published: 11 October 2009
We consider special basic difference equations which are related to discretizations of Schrödinger 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 Schrödinger 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.
It is well known that solving Schrödinger's equation is a prominent -boundary value problem. In this article, we want to become familiar with some of the dynamic equations that arise in context of solving the Schrödinger equation on a suitable time scale where the expression time scale is in the context of this article related to the spatial variables.
The Schrödinger equation is the partial differential equation
where the function : yields information on the corresponding physically relevant potential. The solutions of the Schrödinger equation play a probabilistic role, being modeled by suitable -functions. For the convenience of the reader, let us first cite some of the fundamental facts on Schrödinger's equation. To do so, let us denote by all complex-valued functions which are defined on and which are twice differentiable in each of their variables.
is called Schrödinger Operator in .
The following lemma makes a statement on the separation ansatz of the conventional Schrödinger partial differential equation where we throughout the sequel assume
Lemma 1.2 (Separation Ansatz).
is a solution to Schrödinger'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 Schrödinger 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 the purely discrete case, that is, we are going to focus on a so-called basic discrete quadratic grid resp. on its closure which is a special time scale with fascinating symmetry properties.
In this context, we assume that for all . By construction, it is clear that is a Hilbert space over as it is a weighted sequence space, one of its orthogonal bases being given by all functions which are specified by with and .
Already now, we can say that the separation ansatz for the discretized Schrödinger equation will lead us to looking for eigensolutions of a given Schrödinger operator in the threefold product space .
Hence we come to the conclusion that in case of the separation ansatz for the Schrödinger equation, the following rationale applies:
The solutions of a Schrödinger 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.
This result reflects the celebrated so-called Ladder Operator Formalism. We first review a main result in discrete Schrödinger 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.
The following result reveals that the discrete Schrödinger equation with an oscillator potential on shows similar properties than its classical analog does.
The proof for the lemma is straightforward and obeys the techniques in .
The following central question concerning the functions spaces behind the Schrödinger equation (1.21) is open and shall be partially attacked in the sequel.
1.1. Central Problem
What are the relations between the linear span of all functions arising from Lemma 1.5 and the function space ?
In contrast to the fact that the corresponding question in the Schrödinger 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.
which originated in context of basic discrete ladder operator formalisms. We are going to investigate the rich analytic structure of its solutions in Section 2 and are going to exploit new facts on the corresponding moment problem in Section 3 of this article.
Let us remark finally that we will—throughout the presentation of our results in this article—repeatedly make use of the suffix basic. 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 Schrödinger equations on time scales (that have the structure of adaptive grids) is a wide new research area. A lot of work still has to be invested into this direction.
2. Completeness and Lack of Completeness
In the sequel, we will make use of the basic discrete grid:
and we will consider the Hilbert space
The finite linear complex span of precisely all the functions and is then dense in .
where for the coefficients may be determined by standard methods through the moments resulting from (2.5). From the basic difference equation (2.5) we may also conclude that the polynomials are subject to an indeterminate moment problem, we come back to this in Section 3.
For and the functions given by may now be normalized, let us denote their norms by where is running in . Let us for moreover denote the normalized versions of the functions by
where the coefficients are given by
where the sequence converges to 0 in the sense of the canonical -norm.
Note that we have used in (2.14) the commutation behavior which is satisfied for any fixed and in addition the fact that the sequence again converges to 0 in the sense of the canonical -norm for any . An analogous result is obtained in the case when we start with the eigenvalue .
is dense in the original Hilbert space .
Suppose that there exists a specific such that for all .
Successive application of to (2.21) resp. (2.22) with resp. shows that the existence of such a specific for all would finally imply that . This however would lead to a contradiction. Therefore, it becomes apparent that the complex finite linear span of precisely all the functions resp. (where is running in ) is dense in the Hilbert space . Summing up all facts, the basis property stated in the theorem finally follows according to (2.15).
Let us now focus on the following situation to move on towards the second main result of this article.
is called the real polynomial hull of .
Let and moreover , , . Then is not dense in .
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 type and on the other hand the moments of type and by comparing (2.26) and (2.30), 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.
In order to show that is not dense in , we will construct a linear operator being bounded in the space but unbounded when restricted to .
that is, we choose .
Note that for the definition of , we need to consider the characterization of the corresponding measure.
represent the eigenfunctions of (not necessarily orthogonal since was not required to be symmetric) and are the eigenvalues. However since the eigenvalues are unbounded, this implies that is an unbounded operator. Let us choose its domain as the algebraic span of the occurring eigenfunctions.
that is, the domain of may be chosen as the entire span of . Therefore, is a bounded operator in the space of .
is also a bounded operator.
as . Thus is defined on any and generates infinitely many "rods" on the left-hand side of going toward 0. Therefore, is well defined on any and, therefore, it is well defined on any finite linear combination of the .
By hypothesis, is dense in . Then for, there exists a sequence in which approximates to any degree of accuracy in the sense of the -norm.
Now the question arises: looking at all is there a lower bound for ?
as . It follows that is an unbounded operator, a contradiction. Therefore, is not dense in , implying that is not dense in .
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.
with initial conditions for all . The closure of the finite linear span of all these continuous functions is a Hilbert space For any element in the finite linear span of the conventional (continuous) Hermite functions, there exists a sequence which converges pointwise to .
We recall that the closure of the finite linear complex span of all functions is a Hilbert space, call it , which is a proper subspace of hence being not dense in —see Theorem 2.3.
3. Basic Difference Equations and Moment Problems
Let us make first some more general remarks on the special type of polynomials (2.60) we are considering. In literature, see, for instance, the internet reference to the Koekoek-Swarttouw online report on orthogonal polynomials http://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 symbol while is a nonnegative integer, can be succesively transformed (scaling the argument and renormalizing the coefficients) into the one and same form which is given by
with initial conditions for all . Note that is chosen as a fixed positive real number. Here, the number may range in the set of all positive real numbers, without the number 1—the case being reserved for the classical conventional Hermite polynomials. Depending on the choice of , the two different types of discrete basic Hermite polynomials can be found. The case corresponds to the discrete basic Hermite polynomials of type II, the case 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 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 to (3.1) with 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 polynomials (3.1) with , yields moments being given by
In , it was shown that there exist continuous and piecewise continuous solutions to the difference equation
leading to the same moments (3.2). Such a behavior of the discrete basic Hermite polynomials of type II, hence being related to the scenario (3.1) with , was quite unexpected. Vice versa, once moments with nonnegative integer of a given weight function are given through (3.2), it can immediately be said that the weight function provides an orthogonality measure for the discrete basic Hermite polynomials of type II, related to (3.1) with .
The question however remains whether all weight functions for the discrete basic Hermite polynomials of type II, being related to (3.1) with 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 and We first put forward the following definition.
Let us now proceed to the main result of this section.
The proof to establish will be a step beyond the already known orthogonality results for the polynomials under consideration.
In other words, we have to prove that there exist orthogonality measures to the discrete basic Hermite polynomials which stem from a solution to (3.9) but not from a solution to (3.8).
We proceed in a constructive way.
Hence is a bounded linear map and therefore continuous. In the same way, we show that is continuous.
We now continue as follows.
obeys by construction always but never as we have chosen such that and as vanish by construction on .
now it follows for that the moments satisfy (3.15) and therefore also (3.12). In particular, the function yields therefore an orthogonality measure to the discrete basic Hermite polynomials, given by (2.60). Note that by construction, the function now fulfills all the assertions of Theorem 3.2. Hence, Theorem 3.2 holds in total.
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