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Hindawi Publishing Corporation Advances in Difference Equations Volume 2009, Article ID 569803, 22 pages doi:10.1155/2009/569803 Research Article Some Basic Difference Equations of Schr ¨ odinger Boundary Value Problems Andreas Ruffing, 1, 2 Maria Meiler, 2 and Andrea Bruder 3 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, ruffing@ma.tum.de 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. Copyright q 2009 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 L 2 -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 the spatial variables. The Schr ¨ odinger equation is the partial differential equation t ∈ R  − ∂ 2 ∂x 2 − ∂ 2 ∂y 2 − ∂ 2 ∂z 2  V  x, y,z   ψ  x, y,z, t   i ∂ ∂t ψ  x, y,z, t  , 1.1 where the function V : Ω ⊆ R 3 → R yields information on the corresponding physically relevant potential. The solutions of the Schr ¨ odinger equation play a probabilistic role, being modeled by suitable L 2 -functions. For the convenience of the reader, let us first cite some 2 Advances in Difference Equations of the fundamental facts on Schr ¨ odinger’s equation. To do so, let us denote by C 2 Ω all complex-valued functions which are defined on Ω and 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, PΩ denoting the space of piecewise continuous functions with values in C. The linear map H : C 2 Ω → PΩ,given by  Hψ  x, y,z  :  − ∂ 2 ∂x 2 − ∂ 2 ∂y 2 − ∂ 2 ∂z 2  V  x, y,z   ψ  x, y,z  1.2 is called Schr ¨ odinger Operator in C 2 Ω. The following lemma makes a statement on the separation ansatz of the conventional Schr ¨ odinger partial differential equation where we throughout the sequel assume ΩR 3 . Lemma 1.2 Separation Ansatz. Let the Schr ¨ odinger equation 1.1 be given, fulfilling the assertions of Definition 1.1 where ΩR 3 . In addition, the function V will have the property V : R 3 −→ R,  x, y,z  −→ V  x, y,z   f  x   g  y   h  z  , 1.3 where f, g, h are continuous. Let now λ 1 ,λ 2 ,λ 3 ∈ R be such that there exist eigenfunctions ψ 1 ,ψ 2 ,ψ 3 ∈L 2 R ∩ C 2 R with −ψ  1  x   f  x  ψ 1  x   λ 1 ψ 1  x  ,x∈ R, −ψ  2  y   g  y  ψ 2  y   λ 2 ψ 2  y  ,y∈ R, −ψ  3  z   h  z  ψ 3  z   λ 3 ψ 3  z  ,z∈ R. 1.4 Then the function ϕ ∈ L 2 R 3  × C 2 R ∩ C 2 R 4 , given by  x, y,z, t  −→ ϕ  x, y,z, t  : ψ 1  x  ψ 2  y  ψ 3  z  e −iλ 1 t−iλ 2 t−iλ 3 t 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 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 T with fascinating symmetry properties. Advances in Difference Equations 3 Definition 1.3. Let 0 <q<1 as well as c 1 ,c 2 ,c 3 ∈ R.Theset R  c 1 ,c 2 ,c 3  q :  c 1 q n  c 2 q −n  c 3 | n ∈ Z  1.6 denotes the basic discrete quadratic grid where c 2 1  c 2 2 /  0. For n ∈ Z, we abbreviate γ n : c 1 q n  c 2 q −n  c 3 and −R c 1 ,c 2 ,c 3  q : {t ∈ R |−t ∈ R c 1 ,c 2 ,c 3  q } . Define the set T : R  c 1 ,c 2 ,c 3  q ∪−R  c 1 ,c 2 ,c 3  q 1.7 as well as the set T ∗ 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 L 2  T ∗  :  f : T ∗ −→ C |  f, f  < ∞  , 1.9 where the scalar product f, g for two suitable functions f, g : T ∗ → R will be specified by  f, g  : ∞  n−∞   γ n1 − γ n    f  γ n  g  γ n   f  −γ n  g  −γ n   . 1.10 In this context, we assume that |γ n1 − γ n | /  0 for all n ∈ Z. By construction, it is clear that L 2 T ∗  is a Hilbert space over C as it is a weighted sequence space, one of its orthogonal bases being given by all functions e σ n : T \{0}→{0, 1} which are specified by e σ n τq m  : δ mn δ στ with m, n ∈ Z and σ, 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 space L 2 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 following rationale applies: 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 Lemma 1.2 where we now assume that the potential is given by the requirement f  x   x 2 ,g  y   y 2 ,h  z   z 2 ,  x, y,z  ∈ R 3 . 1.11 Hence, it is sufficient to look at the one-dimensional Schr ¨ odinger equation −ψ   x   x 2 ψ  x   λψ  x  . 1.12 4 Advances in Difference Equations For the convenience of the reader, let us refer to the following fact. let the sequence of functions ψ n  n∈N 0 be recursively given by the requirement ψ n1  x  : −ψ  n  x   xψ n  x  ,x∈ R,n∈ N 0 , 1.13 where ψ 0 : R → R,x → ψ 0 x : e −  1/2  x 2 . We then have ψ n ∈L 2 R ∩ C 2 R for n ∈ N 0 and moreover −ψ  n  x   x 2 ψ n  x    2n  1  ψ n  x  ,x∈ R,n∈ N 0 . 1.14 This result reflects the celebrated so-called Ladder 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<1andletT be a nonempty closed set with the properties ∀x ∈ T : qx ∈ T,q −1 x ∈ T, −x ∈ T. 1.15 Let for any f : T → R the right-shift resp. left-shift operation be fixed through  Rf   x  : f  qx  ,  Lf   x  : f  q −1 x  ,x∈ T, 1.16 respectively, the right-hand resp. left-hand basic difference operation will for any function f : T ∗ → R be given by  D q f   x  : f  qx  − f  x  qx − x ,  D q −1 f   x  : f  q −1 x  − f  x  q −1 x − x ,x∈ T \ { 0 } . 1.17 Let moreover α>0andlet g : T \ { 0 } −→ R,x−→ g  x  :  ϕ  qx  −  ϕ  x   ϕ  x   q − 1  x   1  α  1 − q  x 2 − 1 qx − x , 1.18 where the positive even function ϕ : T → R  is chosen as a solution to the basic difference equation ϕ  qx    1  α  1 − q  x 2  ϕ  x  ,x∈ T. 1.19 The creation operation A ∗ resp. annihilation operation A are t hen introduced by their actions on any ψ : T ∗ → R as follows: A ∗ ψ   −D q  g  X  R  ψ, Aψ  q −1  LD q  Lg  X   ψ. 1.20 Advances in Difference Equations 5 We refer to the discrete Schr ¨ odinger equation with an oscillator potential on T ∗ by q −1  −D q  g  X  R  LD q  Lg  X   ψ  λψ. 1.21 The following result reveals that the discrete Schr ¨ odinger equation with an oscillator potential on T ∗ 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 equation 1.19  on R c 1 ,0,0 q with c 1 /  0. Let more over ψ 0 : R c 1 ,0,0 q → R,x → ψ 0 x :  ϕx. For n ∈ N 0 , the functions ψ n : R c 1 ,0,0 q → R, given by ψ n x :A ∗  n ψ 0 x (while x ∈ R c 1 ,0,0 q ) are well defined in L 2 R c 1 ,0,0 q  and solve the basic Schr ¨ odinger equation 1.21 in the following sense: q −1  −D q  g  X  R  LD q  Lg  X   ψ n  q n − 1 q − 1 ψ n , A ∗ ψ n  ψ n1 ,Aψ n  q n − 1 q − 1 ψ n−1 ,ψ n  x   H q n  x  ψ 0  x  , H q n1  x  − αq n xH q n  x   α q n − 1 q − 1 H q n−1  x   0,H q 0  x   1,H q 1  x   αx. 1.22 These relations apply for x ∈ R c 1 ,0,0 q and n ∈ N 0 where one set ψ −1 : 0,H q −1 : 0. Moreover, the corresponding moments of the orthogonality measure for the polynomials H q n  n∈N 0 , arising from 1.19, are given by μ 2n2  q −2n−1 − 1 α  1 − q  μ 2n ,μ 2n1  0,n∈ N 0 . 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 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 ψ n ,n ∈ N 0 arising from Lemma 1.5 and the function space L 2 R c 1 ,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 ϕ  qx    1  α  1 − q  x 2  ϕ  x  ,x∈ T 1.24 6 Advances in Difference Equations 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 Schr ¨ odinger 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. 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: R q :  ±q n | n ∈ Z  , 2.1 and we will consider the Hilbert space L 2  R q  :  f : R q −→ C | ∞  n−∞ q n  f  q n  f  q n   f  −q n  f  −q n   < ∞  . 2.2 Theorem 2.1. Let 0 <q<1 and α>0 as well as ϕ an even positive solution of ϕ  qx    1  α  1 − q  x 2  ϕ  x  2.3 on the basic discrete grid R q . Let the sequences of functions ϕ m  m∈Z be given by shifted versions of the ϕ-function as follows: ϕ m ∈L 2  R q  ϕ m : R q −→ R,x−→ ϕ m  x  : ϕ  q m x   ϕ  −q m x  ,m∈ Z, ψ m ∈L 2  R q  ψ m : R q −→ R,x−→ ψ m  x  : xϕ  q m x  ,m∈ Z. 2.4 The finite linear complex span of precisely all the functions ϕ m  m∈Z and ψ m  m∈Z is then dense in L 2 R q . Advances in Difference Equations 7 Proof. Let ϕ ∈L 2 R q  be a positive and even solution to ϕ  qx    1  α  1 − q  x 2  ϕ  x  ,x∈ R q . 2.5 One can easily show that an L 2 R q -solution with these properties uniquely exists, up to a positive factor, moreover all the functions defined by 2.4 are well defined in L 2 R q .Letus refer by the sequence P n  n∈N 0 to all the orthonormal polynomials P n : R q → R which arise from the Gram-Schmidt procedure with respect to the function ϕ 2 . They satisfy a three-term recurrence relation P n1  x  − α n xP n  x   β n P n−1  x   0,P −1  x   0,x ∈ R q ,n∈ N 0 , 2.6 where for n ∈ N 0 the coefficients α n ,β n may be determined by standard methods through the moments resulting from 2.5.Fromthebasicdifference equation 2.5 we may also conclude that the polynomials P n  n∈N 0 are subject to an indeterminate moment problem, we come back to this in Section 3. For n ∈ N 0 and x ∈ R q , the functions given by P n xϕx may now be normalized, let us denote their norms by ρ n where n is running in N 0 .Letusforn ∈ N 0 moreover denote the normalized versions of the functions P n ϕ by u n . The following observation is essential. the C-linear finite span of all functions given by  R m ϕ   x  ,x  R m ϕ   x  ,x∈ R q ,m∈ Z 2.7 is the same than the C-linear finite span of all functions specified by  R m u n  x  ,x  R m u n  x  ,x∈ R q ,m∈ Z,n∈ N 0 . 2.8 As a consequence of 2.6, we conclude that the functions u n  n∈N 0 fulfill the recurrence relation: ρ n1 u n1  x  − α n ρ n xu n  x   β n ρ n−1 u n−1  x   0,u −1  x   0,x ∈ R q ,n∈ N 0 . 2.9 How ever 2.9 can be brought into the standard form which is of relevance for considering the corresponding Jacobi operator, Xu n  a n1 u n1  a n u n−1 ,n∈ N 0 , 2.10 where the coefficients are given by a 0  0,a n1   β n1 /α n α n1 ,n∈ N 0 . The representation 2.10 results from the fact that the functions u n  n∈N 0 constitute a system of orthonormal functions and due to the fact that X, acting as a multiplication 8 Advances in Difference Equations operator, requires to be a formally symmetric linear operator on the finite linear span of the orthonormal system u n  n∈N 0 . Let us now consider the Hilbert space: H :  ∞  n0 c n u n | ∞  n0 | c n | 2 < ∞,c n ∈ C,n∈ N 0  . 2.11 As for the definition range of X in H, let us choose X as a densely defined linear operator in H where we assume that D  X  :  k  n0 c n u n | c n ∈ C,k ∈ N 0  . 2.12 Let the expansion for a possible eigenvector of the adjoint X ∗ be written as  ∞ j0 c j ψ j ,the eigenvalue equation being X ∗  ∞ j0 c j ψ j  λ  ∞ j0 c j ψ j . Note that the type of moment problem behind is related to the situation that X : DX ⊆H→Hhas deficiency indices 1, 1. This also implies that any λ ∈ C constitutes an eigenvalue of X ∗ , hence the point spectrum of X ∗ is C. According to the deficiency index structure 1, 1 of the operator X, let us now choose the particular self-adjoint extension Y of X which allows a prescribed real-eigenvalue λ  1 /  0. The corresponding situation for the eigensolution may be written as Y n  j0 c j u j  n  j0 c j u j  w n ,n∈ N 0 , 2.13 where the sequence w n  n∈N 0 converges to 0 in the sense of the canonical L 2 R q -norm. The element  n j0 c j u j is in the finite linear space of all functions u j ,j ∈ N 0 . Applying the powers R k ,k ∈ Z of the shift operator R being given by Rvx : vqx for any function v ∈L 2 R q ,x ∈ R q  to 2.13 now leads to the fact that we can construct all eigenfunctions of the operator Y belonging to q k ,k ∈ Z, as a consequence of R k Y n  j1 c j u j  λq k R k n  j1 c j u j  R k w n ,n∈ N 0 . 2.14 Note that we have used in 2.14 the commutation behavior R k X  q k XR k which is satisfied for any fixed k ∈ Z and in addition the fact that the sequence R k w n  n∈N 0 again converges to 0 in the sense of the canonical L 2 R q -norm for any k ∈ Z. An analogous result is obtained in the case when we start with the eigenvalue λ  −1 /  0. Summing up the stated facts, we see that the self-adjoint operator Y , interpreted now as the multiplication operator, acting on a dense domain in L 2 R q , has precisely the point spectrum {q n , −q n | n ∈ Z} in the sense of Ye σ n  σq n e σ n ,n∈ Z,σ∈ { 1, −1 } , 2.15 Advances in Difference Equations 9 the functions e σ n ,n∈ Z with norm 1 being fixed by e σ n  τq m  : 1 q n/2  1 − q δ στ δ mn ,m,n∈ Z,σ,τ∈ { 1, −1 } . 2.16 Let us recall what we had stated at the beginning: the C-linear finite span of all functions  R m ϕ   x  ,x  R m ϕ   x  ,x∈ R q ,m∈ Z 2.17 is the same than the C-linear finite span of all functions:  R m u n  x  ,x  R m u n  x  ,x∈ R,m∈ Z,n∈ N 0 . 2.18 Taking the observations together, we conclude that the C-linear span of all functions ϕ m  x    R m ϕ   x  ,ψ m  x   x  R m ϕ   x  ,x∈ R q ,m∈ Z 2.19 is dense in the original Hilbert space L 2 R q . We finally want to show that the C-linear span of precisely all the functions in 2.19 is dense in L 2 R q . This can be seen as follows. taking away one of the functions R n ϕ or XR n ϕ n ∈ N 0  would already remove the completeness of the smaller Hilbert space H⊆L 2 R q . According to the property that the functions from 2.19 are dense in L 2 R q ,it follows, for instance, that for any k ∈ Z, there exists a double sequence c k j  k,j∈Z such that in the sense of the canonically induced L 2 R q -norm: lim l−→∞       e 1 k1  e −1 k1 − l  j−l c k j ϕ j       L 2  R q  0 ,k∈ Z . 2.20 Suppose that there exists a specific i ∈ Z such that c k i  0 for all k ∈ Z. Let us consider the following expression where l ∈ N and −l<i<l: Re 1 k1  Re −1 k1 − l  j−l c k j Rϕ j  e 1 k  e −1 k − l  j−l c k j ϕ j1 ,k∈ Z. 2.21 The last expression now may be rewritten as e 1 k  e −1 k − l1  j−l2 c k j−1 ϕ j ,k∈ Z. 2.22 Successive application of R m to 2.21 resp. 2.22 with m ∈ N 0 resp. −m ∈ N 0 shows that the existence of such a specific c k i  0 for all k ∈ Z would finally imply that For all j, k ∈ Z : c k j  0. 10 Advances in Difference Equations This however would lead to a contradiction. Therefore, it becomes apparent that the complex finite linear span of precisely all the functions R m ϕ resp. XR m ϕ where m is running in Z is dense in the Hilbert space L 2 R q . 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 P be a positive symmetric polynomial, that is, P : R −→ R  ,x−→ P  x   P  −x  > 0,P  x   N  j0 γ j x 2j ,γ j > 0, 2.23 where N ∈ N 0 . Definition 2.2. Let f ∈L 1 R ∩ C 1 R with finite moments. Then P f : ⎧ ⎨ ⎩ N  j0 c j X j f | c j ∈ R,N ∈ N 0 ⎫ ⎬ ⎭ 2.24 is called the real polynomial hull of f. Theorem 2.3. Let 0 <q<1 and moreover fqxPxfx, x ∈ R, f ∈ C 1 R.ThenP √ f is not dense in L 2 R. Proof. For n ∈ N 0 , the nth moment μ n of f can be calculated from the prerequisites of Theorem 2.3, namely;  ∞ −∞ x n f  qx  dx   ∞ −∞ x n N  j0 γ j x 2j f  x  dx, 2.25 where N ∈ N 0 —written in short: q −n−1 μ n  N  j0 γ j μ n2j . 2.26 Discretization and integration on the basic grid R q : {±q k | k ∈ Z} gives ∞  k−∞ q k  q k  n f  q k1  ∞  k−∞ q k  q k  n N  j0 γ j  q k  2j f  q k  . 2.27 and, if k is changed into k  1 on the left-hand side, ∞  k−∞ q k−1  q k−1  n f  q k  N  j0 γ j ∞  k−∞ q k  q k  n2j f  q k  . 2.28 [...]... set of all positive real numbers, 18 Advances in Difference Equations without the number 1—the case q 1 being reserved for the classical conventional Hermite polynomials Depending on the choice of q, 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 case q > 1 corresponds to the discrete basic. .. 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... sequences of functions um,k k∈N0 , given by ⎛ k x −→ um,k x : ⎝ um,k : R −→ R, n ⎞ n,m ∗q cj Hj x ⎠Φ x , k ∈ N0 , 2.72 n 0j 0 According to what we have shown it follows that each of the functions um,k ∈ F converges pointwise to the functions hm : R → R, given by hm : R −→ R, x −→ hm x : Hm x e −1/2 x , 2 m ∈ N0 , 2.73 as k → ∞ 3 Basic Difference Equations and Moment Problems Let us make first some more... 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. .. Difference Equations 11 Define for n ∈ N0 : ∞ qk qk μn : n f qk 2.29 k −∞ Then, we have q−n−1 μn N γj μn 2j 2.30 j 0 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 μn and on the other hand the moments of type... H1 x αx for all x ∈ R The closure of the finite linear span q of all these continuous functions Hn Φ, n ∈ N0 is a Hilbert space F ⊆ L2 R For any element v in the finite linear span of the conventional (continuous) Hermite functions, there exists a sequence um,k k,m∈N0 ⊆ F which converges pointwise to v Proof According to the assertions of the theorem, the inverse Φ−1 of the function Φ : R → R, given... Such a behavior of the discrete basic Hermite polynomials of type II, hence being related to the scenario 3.1 with 0 < q < 1, was quite unexpected Vice versa, once moments νm with nonnegative integer m 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... 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: ψ qx ψ q2 x α 1 − q x2 ψ x , 1 α 1 − q x2 1 x ∈ R, α 1 − q q2 x2 ψ x , 1 3.8 x ∈ R 3.9 Obviously, any positive C R -solution ψ of 3.8 satisfies 3.9 Moreover, one can show that these C R -solutions of 3.8 are... harmonic Schrodinger ¨ oscillator using unitary linear lattices,” Journal of Computational and Applied Mathematics, vol 153, no 1-2, pp 395–410, 2003 10 B Simon, “The classical moment problem as a self-adjoint finite difference operator,” Advances in Mathematics, vol 137, no 1, pp 82–203, 1998 11 M Simon and A Ruffing, “Power series techniques for a special Schrodinger operator and related ¨ difference equations, ”... N Xj f X 2.41 j 0 represent the eigenfunctions of L P X not necessarily orthogonal since L P X was not required to be symmetric and q−N are the eigenvalues However since the eigenvalues are unbounded, this implies that L P X is an unbounded operator Let us choose its domain as the algebraic span U of the occurring eigenfunctions Advances in Difference Equations 13 Now consider q−N ϕN L P X ϕN 2.42 . Corporation Advances in Difference Equations Volume 2009, Article ID 569803, 22 pages doi:10.1155/2009/569803 Research Article Some Basic Difference Equations of Schr ¨ odinger Boundary Value Problems Andreas. Schr ¨ odinger’s equation is a prominent L 2 -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. show that solving the boundary value problems of Schr ¨ odinger 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