1 Introduction

Dispersive Properties for Discrete Schrodinger Equations

Abstract.

In this paper we prove dispersive estimates for the system formed by two coupled discrete Schrödinger equations. We obtain estimates for the resolvent of the discrete operator and prove that it satisfies the limiting absorption principle. The decay of the solutions is proved by using classical and some new results on oscillatory integrals.

Key words and phrases:
Discrete Schrödinger equation, dispersion and Strichartz inequalities, oscillatory integrals.
2000 Mathematics Subject Classification. 35J10, 35C, 42B20.

1. Introduction

Let us consider the linear Schrödinger equation (LSE):

(1.1)

Linear equation (1.1) is solved by , where is the free Schrödinger operator. The linear semigroup has two important properties. First, the conservation of the -norm:

(1.2)

and a dispersive estimate of the form:

(1.3)

The space-time estimate

(1.4)

due to Strichartz [13], is deeper. It guarantees that the solutions of system (1.1) decay as becomes large and that they gain some spatial integrability. Inequality (1.4) was generalized by Ginibre and Velo [3]. They proved the mixed space-time estimates, well known as Strichartz estimates:

(1.5)

for the so-called admissible pairs :

(1.6)

Similar results can be stated in any space dimension but it is beyond the scope of this article. These estimates have been successfully applied to obtain well-posedness results for the nonlinear Schrödinger equation (see [2], [14] and the reference therein).

Let us now consider the following system of difference equations

(1.7)

where is the discrete laplacian defined by

Concerning the long time behavior of the solutions of system (1.7) in [11] the authors have proved that a decay property similar to the one obtained for the continuous Schrödinger equation holds:

(1.8)

The proof of (1.8) consists in writing the solution of (1.7) as the convolution between a kernel and the initial data and then estimate by using Van der Corput’s lemma. For the linear semigroup , Strichartz like estimates similar to those in (1.5) have been obtained in [11] for a larger class of pairs :

(1.9)

We also mention [5] and [6] where the authors consider a similar equation on by replacing by and analyze the same properties in the context of numerical approximations of the linear and nonlinear Schrödinger equation.

A more thorough analysis has been done in [9] and [10] where the authors analyze the decay properties of the solutions of equation where , with a real-valued potential. In these papers and estimates for have been obtained where is the spectral projection to the absolutely continuous spectrum of and are weighted -spaces.

In what concerns the Schödinger equation with variable coefficients we mention the results of Banica [1]. Consider a partition of the real axis as follows: and a step function where are positive numbers. The solution of the Schrödinger equation

satisfies the dispersion inequality

where constant depends on and on sequence . We recall that in [4] the above result was used in the analysis of the long time behavior of the solutions of the linear Schödinger equation on regular trees. In the case of discrete equations the corresponding model is given by

(1.10)

where the infinite matrix is symmetric with a finite number of diagonals nonidentically vanishing. Once a result similar to [1] will be obtained for discrete Schrödinger equations with non-constant coefficients we can apply it to obtain dispersive estimates for discrete Schrödinger equations on trees. But as far as we know the study of the decay properties of solutions of system (1.10) in terms of the properties of is a difficult task and we try to give here a partial answer to this problem. In the case when is a diagonal matrix these properties are easily obtained by using the Fourier transform and classical estimates for oscillatory integrals.

The main goal of this article is to analyze a simplified model which consists in coupling two DSE by Kirchhoff’s type condition:

(1.11)

In the above system and have been artificially introduced to couple the two equations on positive and negative integers. The third condition in the above system requires continuity along the interface and the fourth one can be interpreted as the continuity of the flux along the interface.

The main result of this paper is given in the following theorem.

Theorem 1.1.

For any there exists a unique solution of system (1.11). Moreover, there exists a positive constant such that

(1.12)

holds for all .

Using the well-known results of Keel and Tao [7] we obtain the following Strichartz-like estimates for the solutions of system (1.11).

Theorem 1.2.

For any the solution of system (1.11) satisfies

for all pairs satisfying (1.9).

The paper is organized as follows: In section 2 we present some discrete models, in particular system (1.11) in the case and show how it is related with problem (1.7). In addition, a system with a dynamic coupling along the interface is presented. In section 3 we present some classical results on oscillatory integrals and make some improvements that we will need in the proof of Theorem 1.1. In section 4 we obtain an explicit formula for the resolvent associated with system (1.11). We prove a limiting absorption principle and we give the proof of the main result of this paper. Finally we present some open problems.

2. Some discrete models

In this section in order to emphasize the main differences and difficulties with respect to the continuous case when we deal with discrete systems we will consider two models. In the first case we consider system (1.11) with the two coefficients in the front of the discrete laplacian equal. In the following we denote

Theorem 2.1.

Let us assume that . For any there exists a unique solution of system (1.11). Moreover there exists a positive constant such that

(2.1)

holds for all .

In the particular case considered here we can reduce the proof of the dispersive estimate (2.1) to the analysis of two problems: one with Dirichlet’s boundary condition and another one with a discrete Neumann’s boundary condition.

Before starting the proof of Theorem 2.1 let us recall that in the case of system (1.7) its solution is given by where is the standard convolution on and

In [11] a simple argument based on Van der Corput’s lemma has been used to show that for any real number the following holds:

(2.2)
Proof of Theorem 2.1.

The existence of the solutions is immediate since operator defined in (2.7) is bounded in . We prove now the decay property (2.1). Let us restrict for simplicity to the case .

For solution of system (1.11) let us set

Observe that and can be recovered from and as follows

Writing the equations satisfied by and we obtain that and solve two discrete Schrödinger equations on with Dirichlet, respectively Neumann boundary conditions:

(2.3)

and

(2.4)

Making an odd extension of the function and using the representation formula for the solutions of (1.7) we obtain that the solution of the Dirichlet problem (2.3) satisfies

(2.5)

A similar even extension of function permits us to obtain the explicit formula for the solution of the Neumann problem (2.4)

(2.6)

Using the decay of the kernel given by (2.2) we obtain that and decay as and then the same property holds for and . This finishes the proof of this particular case. ∎

Observe that our proof has taken into account the particular structure of the equations. When the coefficients and are not equal we cannot write an equation verified by functions or .

We now write system (1.11) in matrix formulation. Using the coupling conditions at system (1.11) can be written in the following equivalent form

where , , and

(2.7)

In the particular case the operator can be decomposed as follows

However, we do not know how to use the dispersive properties of and the particular structure of in order to obtain the decay of the new semigroup .

Another model of interest is the following one inspired in the numerical approximations of LSE. Set

Using the following discrete derivative operator

we can introduce the second order discrete operator

In this case we have to analyze the following system

(2.8)

In matrix formulation it reads where , and the operator is given by the following one

(2.9)

Observe that in the case the results of [11] give us the decay of the solutions.

Regarding the long time behavior of the solutions of system (2.8) we have the following result.

Theorem 2.2.

For any there exists a unique solution of system (2.8). Moreover, there exists a positive constant such that

holds for all .

The proof of this result is similar to the one of Theorem 1.1 and we will only sketch it at the end of Section 4.

3. Oscillatory integrals

In this section we present some classical tools for oscillatory integrals and we give an improvement of Van der Corput’s Lemma that is in some sense similar to the one obtained in [8]. First of all let us recall Van der Corput’s lemma(see for example [12], p. 332).

Lemma 3.1.

(Van der Corput) Let be an integer, and such that for all , and monotone in the case .
Then

A first improvement has been obtained in [8] where the authors analyze the smoothing effect of some dispersive equations. We will present here a particular case of the results in [8], that will be sufficient for our purposes. In the sequel will be a bounded interval. We consider class of real functions satisfying the following conditions:
1) Set is finite,
2) If then there exist constants and such that for all ,

3) has a finite number of changes of monotonicity.

Lemma 3.2.

Let be a bounded interval, and

Then for any

(3.1)

where depends only on the constants involved in the definition of class .

Remark 1.

The results of [8] are more general that the one we presented here allowing functions with vertical asymptotics, finite union of intervals or infinite domains.

As a corollary we also have [8]:

Corrolary 3.1.

If then

holds for all .

In the proof of our main result we will need a result similar to Lemma 3.2 but with instead of in the definition of . We define class of real functions satisfying the following conditions:
1) Set is finite,
2) If then there exist constants and such that for all ,

(3.2)

3) has a finite number of changes of monotonicity.

Lemma 3.3.

Let be a bounded interval, and

Then for any

(3.3)

where depends only on the constants involved in the definition of class .

In the following we will write if there exists a positive constant such that . Similar for . Also we will write if for some positive constants and .

Proof.

We observe that since is bounded we only need to consider the case when is large.

Case 1: .
We apply Van der Corput’s Lemma with to the phase function and to . Then

Since has a finite number of changes of monotonicity we deduce that changes the sign finitely many times and then

Hence

Case 2: .
Using the assumptions on we can assume that there exists only one point such that . Notice that if , then any translation and any linear perturbation of (i.e. ) is still in and the conditions in the definition of set are verified with the same constants as . Therefore we can assume that and . Moreover let us assume that as , and for some numbers and .

We distinguish now two cases depending on the behavior of near . If then as for and, in particular . The case cannot appear since then and does not vanish at . For , , as and the third derivative satisfies as for some positive integer . This last case occurs for example when . In all cases .

We split as follows

Since is the only point where the third derivative vanishes we have that outside an interval that contains the origin does not vanish. Thus can be treated as in the first case.

Let us now estimate the first term . We define , as follows

In the case of we use that for some , the third derivative of satisfies for . We get

where the last inequality holds since and .

In the case of the integral on we assume that since otherwise has measure zero. Observe that for we have

which implies that

Since we have that . Then and

Applying Van der Corput’s Lemma with and using that changes the sign finitely many times we obtain that

Since on , , there exists a positive constant such that

which gives us the desired estimates on the integral on .

Now, we estimate the integral on . Observe that we have to consider the case , otherwise . In particular, for , we have . Integrating by parts the integral on satisfies

(3.4)

In the following we obtain upper bounds for all terms in the right hand side of (3.4). Since on , , there exists a positive constant such that

In the case of the first term

(3.5)

since and .

The second term satisfies