Zero energy scattering for one-dimensional Schrödinger operators and applications to dispersive estimates

By Iryna Egorova, Markus Holzleitner, Gerald Teschl

Abstract

We show that for a one-dimensional Schrödinger operator with a potential, whose -th moment is integrable, the -th derivative of the scattering matrix is in the Wiener algebra of functions with integrable Fourier transforms. We use this result to improve the known dispersive estimates with integrable time decay for the one-dimensional Schrödinger equation in the resonant case.

1. Introduction

This paper is concerned with the one-dimensional Schrödinger equation

with a real-valued potential contained in one of the spaces , , associated with the norms

We recall (e.g. Reference 15 or Reference 16, §9.7) that for the operator has a purely absolutely continuous spectrum on and a finite number of eigenvalues in . Associated with the absolutely continuous spectrum is the scattering matrix

which maps incoming to outgoing states at a given energy . Here is the transmission coefficient and are the reflection coefficients with respect to right and left incident. At the edge of the continuous spectrum the scattering matrix generically looks like

More precisely, this happens when the zero energy is non-resonant, that is, if the equation has no bounded (distributional) solution. In the resonant situation the behavior of the scattering matrix is more delicate. For it is already non-trivial to establish continuity of the scattering matrix at (for de l’Hospital’s rule suffices). This question arose around 1985 in an attempt to clarify whether the low-energy asymptotics of the scattering matrix, obtained for in Reference 6, are valid for . The problem was solved independently by Guseĭnov Reference 12 and Klaus Reference 13 (for a refined version see Reference 1 and Reference 9, Theorem 2.1). It also plays an important role in semiclassical analysis; see Reference 5 and the references therein. Furthermore, if with , then, away from , one can take derivatives of the scattering matrix up to order , and in the non-resonant case it is again easy to see that they are continuous at . This clearly raises the question about continuity at of the -th derivative in the resonant case. We will establish this as one of our main results in Theorem 2.2. This result is new even for the first derivative. We remark that if decays exponentially, then is analytic in a neighborhood of and the full Taylor expansion can be obtained Reference 2Reference 3.

It is important to emphasize that this question is not only of interest in scattering theory but also plays a role in the solution of the Korteweg–de Vries equation via the inverse scattering transform (see e.g. Reference 11 where continuity of higher derivatives is needed) and in deriving dispersive estimates for Equation 1.1. The latter case has attracted considerable interest (e.g. Reference 9Reference 10 and the references therein) due to its importance for proving asymptotic stability of solitons for the associated non-linear evolution equations (see e.g. Reference 4Reference 14).

As an application of our results we will establish the following dispersive decay estimate with integrable time decay in the resonant case:

Theorem 1.1.

Suppose and suppose there is a bounded solution of satisfying the normalization . Denote by the operator given by the kernel . By we denote the projector onto the absolutely continuous subspace of . Then extends to a bounded operator satisfying the following decay estimate:

This theorem is an improvement of an earlier result by Goldberg Reference 10 who established it for . If there is no resonance (i.e. no bounded solution) this result (with ) was shown for in Reference 9, Theorem 1.2. For extensions to discrete one-dimensional Schrödinger equations (Jacobi operators) see Reference 7Reference 8.

2. Low energy scattering

In this section we establish some properties of the scattering data for our operator with , . To this end we introduce the Banach algebra of Fourier transforms of integrable functions,

with the norm , plus the corresponding unital Banach algebra ,

with the norm . We also use the fact, which is known as Wiener’s lemma Reference 17, that if and for all , then .

First we recall a few facts from scattering theory Reference 6Reference 15. If there exist Jost solutions of , , which asymptotically behave like as . These solutions are given by

where are real-valued and satisfy (see Reference 6, §2 or Reference 15, §3.1)

with

Since we conclude that

Here and throughout the rest of this paper a prime will always denote a derivative with respect to the spatial variable . As an immediate consequence of the estimates Equation 2.4 and Equation 2.5 we have the following strengthening of Equation 2.7.

Lemma 2.1.

Let . Then , are contained in for . Moreover, for , the -norms of these expressions do not depend on .

The fact that also solve for leads to the scattering relations

where the transmission coefficient and the reflection coefficients can be expressed in terms of Wronskians. To this end we denote by

the usual Wronskian and set

Then

The transmission and reflection coefficients are elements of the Wiener algebra, which was established in Reference 9, Theorem 2.1. Here we extend this result to the derivatives of the scattering data.

Theorem 2.2.

If , then and for .

Proof.

We only focus on the resonant case, since the other case is straightforward.

First of all, we abbreviate , . Then Equation 2.3 leads us to

Following Reference 9 we introduce

where are given by Equation 2.3. Again is denoted by and similarly for . In Reference 9, Theorem 2.1, the following equation for was obtained:

where

Moreover, satisfy an estimate similar to Equation 2.4 as we will show in Lemma 2.3 below. As a consequence, and its derivatives up to order will be in the Wiener algebra.

Next a straightforward computation (cf. Reference 9) shows

and since for all , and as , we conclude that for . Analogously,

and hence has the claimed properties.

To complete the proof of Theorem 2.2 we need the following result which is an extension of Reference 9, Lemma 2.2.

Lemma 2.3.

Let be given by Equation 2.15. For the following estimate is valid:

with some constant and given by Equation 2.6. Moreover, for defined by Equation 2.14 and for we have

Proof.

It suffices to prove the estimate for . The Marchenko equation (§3.5 in Reference 15) states that the kernels solve the equations