- Research Article
- Open Access
Some Basic Difference Equations of Schrödinger Boundary Value Problems
Advances in Difference Equations volume 2009, Article number: 569803 (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.
Let be twice partially differentiable in its three variables. Let moreover be a piecewise continuous function, denoting the space of piecewise continuous functions with values in . The linear map , given by
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).
Let the Schrödinger equation (1.1) be given, fulfilling the assertions of Definition 1.1 where In addition, the function will have the property
where are continuous. Let now be such that there exist eigenfunctions with
Then the function , given by
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.
Let as well as . The set
denotes the basic discrete quadratic grid where . For , we abbreviate and . Define the set
as well as the set by
The boundary conditions on the functions we need for the discretized version of Schrödinger's equation are then given by the requirement
where the scalar product for two suitable functions will be specified by
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.
Before presenting discrete versions of the Schrödinger equation on basic quadratic grids, let us first come back to the situation of Lemma 1.2 where we now assume that the potential is given by the requirement
Hence, it is sufficient to look at the one-dimensional Schrödinger equation
For the convenience of the reader, let us refer to the following fact. let the sequence of functions be recursively given by the requirement
where We then have for and moreover
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.
Let and let be a nonempty closed set with the properties
Let for any the right-shift resp. left-shift operation be fixed through
respectively, the right-hand resp. left-hand basic difference operation will for any function be given by
Let moreover and let
where the positive even function is chosen as a solution to the basic difference equation
The creation operationresp.annihilation operation are then introduced by their actions on any as follows:
We refer to the discrete Schrödinger equation with an oscillator potential on by
The following result reveals that the discrete Schrödinger equation with an oscillator potential on shows similar properties than its classical analog does.
Let the function be specified like in Definition 1.4, satisfying the basic difference equation (1.19) on with . Let moreover . For the functions , given by (while ) are well defined in and solve the basic Schrödinger equation (1.21) in the following sense:
These relations apply for and where one set . Moreover, the corresponding moments of the orthogonality measure for the polynomials , arising from (1.19), are given by
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.
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 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
Let and as well as an even positive solution of
on the basic discrete grid . Let the sequences of functions be given by shifted versions of the -function as follows:
The finite linear complex span of precisely all the functions and is then dense in .
Let be a positive and even solution to
One can easily show that an -solution with these properties uniquely exists, up to a positive factor, moreover all the functions defined by (2.4) are well defined in . Let us refer by the sequence to all the orthonormal polynomials which arise from the Gram-Schmidt procedure with respect to the function . They satisfy a three-term recurrence relation
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
The following observation is essential. the -linear finite span of all functions given by
is the same than the -linear finite span of all functions specified by
As a consequence of (2.6), we conclude that the functions fulfill the recurrence relation:
How ever (2.9) can be brought into the standard form which is of relevance for considering the corresponding Jacobi operator,
where the coefficients are given by
The representation (2.10) results from the fact that the functions constitute a system of orthonormal functions and due to the fact that , acting as a multiplication operator, requires to be a formally symmetric linear operator on the finite linear span of the orthonormal system . Let us now consider the Hilbert space:
As for the definition range of in , let us choose as a densely defined linear operator in where we assume that
Let the expansion for a possible eigenvector of the adjoint be written as , the eigenvalue equation being . Note that the type of moment problem behind is related to the situation that has deficiency indices This also implies that any constitutes an eigenvalue of , hence the point spectrum of is . According to the deficiency index structure of the operator , let us now choose the particular self-adjoint extension of which allows a prescribed real-eigenvalue The corresponding situation for the eigensolution may be written as
where the sequence converges to 0 in the sense of the canonical -norm.
The element is in the finite linear space of all functions . Applying the powers of the shift operator (being given by for any function ) to (2.13) now leads to the fact that we can construct all eigenfunctions of the operator belonging to , as a consequence of
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 .
Summing up the stated facts, we see that the self-adjoint operator , interpreted now as the multiplication operator, acting on a dense domain in , has precisely the point spectrum in the sense of
the functions with norm 1 being fixed by
Let us recall what we had stated at the beginning: the -linear finite span of all functions
is the same than the -linear finite span of all functions:
Taking the observations together, we conclude that the -linear span of all functions
is dense in the original Hilbert space .
We finally want to show that the -linear span of precisely all the functions in (2.19) is dense in . This can be seen as follows. taking away one of the functions or would already remove the completeness of the smaller Hilbert space . According to the property that the functions from (2.19) are dense in , it follows, for instance, that for any , there exists a double sequence such that in the sense of the canonically induced -norm:
Suppose that there exists a specific such that for all .
Let us consider the following expression where and :
The last expression now may be rewritten as
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.
let be a positive symmetric polynomial, that is,
Let with finite moments. Then
is called the real polynomial hull of .
Let and moreover , , . Then is not dense in .
For the th moment of can be calculated from the prerequisites of Theorem 2.3, namely;
where —written in short:
Discretization and integration on the basic grid gives
and, if is changed into on the left-hand side,
Define for :
Then, we have
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.
Let us make use again of the lattice
We define the restriction of on by
and we will use again the Hilbert space
Then the discrete analog of is
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 .
Let us start from a function given by
and define generally for :
We denote from now on the respective multiplication operator again by and use to see that for a suitable operator-valued function the following holds:
Using , choose such that obeys:
that is, we choose .
Note that for the definition of , we need to consider the characterization of the corresponding measure.
Now, rewrite this as
Therefore, there exist such that
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.
Define the operators by
For the operator has the same eigenvectors as and we receive
Therefore, it follows with :
is a bounded operator on since as . Let us state that
that is, the domain of may be chosen as the entire span of . Therefore, is a bounded operator in the space of .
For topological reasons, it follows that
is also a bounded operator.
holds. Further, we have
Defining for to rewrite this as
and defining for we obtain
Applying to the results in
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 ?
The are pairwise orthogonal, that is, from
it follows that
and therefore we have
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.
Let be a differentiable positive even solution to
Let moreover be the continuous solutions to the recurrence relation
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 .
According to the assertions of the theorem, the inverse of the function , given by
is differentiable and fulfills the basic difference equation
The function , 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,
the sequence of complex numbers being uniquely fixed, denoting a multiplicative constant. Hence, the function given by
is also holomorphic. Inserting the corresponding power series:
we end up with the statement
Any monomial may be written as
with a double sequence of uniquely fixed real numbers . The polynomial functions of degree will be chosen such that
for but . Polynomials which fulfill these properties are, for instance, those fixed by (2.60), see also . We may rewrite (2.66) as follows:
Generalizing this result, we see that there exists a threefold sequence of real numbers such that the classical continuous Hermite functions, given by
have the following representation
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.
For , let us now consider the sequences of functions , given by
According to what we have shown it follows that each of the functions converges pointwise to the functions given by
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.
By the moments of a given orthogonality measure, we understand—like in the previous sections—the numbers
Let us now proceed to the main result of this section.
There exists a positive symmetric -solution to the basic difference equation
not being a solution to
but generating the same moments and therefore yielding an orthogonality measure to the polynomials from (2.60) in the following sense:
The proof to establish will be a step beyond the already known orthogonality results for the polynomials under consideration.
Let us consider first the special basic difference equations:
Obviously, any positive -solution of (3.8) satisfies (3.9). Moreover, one can show that these -solutions of (3.8) are in . Therefore, the set of positive symmetric solutions to (3.9) which are in is nonempty. Let now be such that
In the sequel, we are going to use the moments
Remember that for as was assumed to act symmetrically on the real axis. Equation (3.10) may therefore now be rewritten—in terms of the numbers —as
Let now be a positive symmetric solution to (3.8). Then, similar integration like in (3.10) shows that the corresponding moments
indeed satisfy (3.12), in particular they obey
Hence, provides an orthogonality measure to the discrete basic Hermite polynomials under consideration—see . The main issue to address now is the following. we want to show that there are positive symmetric functions such that satisfies (3.9) but not (3.8) and such that the moments, given by (3.11) and (3.12), satisfy
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.
Let us denote first by the -linear subspace of such that all are in the kernel of the map , given by
Let moreover be the -linear subspace of containing all the functions which are in the kernel of the linear map , given by
We will also make use of the -linear subspace which we choose as the maximal common domain on which the following two linear functionals are well defined:
It is easy to see that is continuous. to verify this, we consider the expression in the sense of the -norm for any . We directly obtain
But as is assumed to be an element of and hence of , we may rewrite the expression in the denominator and give the following estimate (with a positive constant ):
Hence is a bounded linear map and therefore continuous. In the same way, we show that is continuous.
We now continue as follows.
Using the terminology of characteristic functions, we first look at
It is possible to choose continuous even -functions being in and fulfilling
Let now with . The function given by
obeys by construction always but never as we have chosen such that and as vanish by construction on .
We now choose an intermediate value argumentation. According to the continuity of resp., we can choose parameters with such that fulfills the moment property
Let now be a positive and even function such that . Note that we have in particular as well as and . The function (3.23) is by construction also in . According to the construction of the function , we now may choose sufficiently large such that the positive continuous even function
finally fulfills the required properties, namely,
as well as the moment conditions
From (3.27) and by integrating
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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