Zero energy scattering for one-dimensional Schrödinger operators and applications to dispersive estimates
By Iryna Egorova, Markus Holzleitner, and Gerald Teschl
Abstract
We show that for a one-dimensional Schrödinger operator with a potential, whose $(j+1)$-th moment is integrable, the $j$-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 $V$ contained in one of the spaces $L^1_{\sigma }=L^1_{\sigma }({\mathbb{R}})$,$\sigma \in {\mathbb{R}}$, associated with the norms
We recall (e.g. Reference 15 or Reference 16, §9.7) that for $V\in L^1_1$ the operator $H$ has a purely absolutely continuous spectrum on $[0,\infty )$ and a finite number of eigenvalues in $(-\infty ,0)$. Associated with the absolutely continuous spectrum is the scattering matrix
which maps incoming to outgoing states at a given energy $\omega =k^2\ge 0$. Here $T$ is the transmission coefficient and $R_\pm$ are the reflection coefficients with respect to right and left incident. At the edge of the continuous spectrum $k=0$ the scattering matrix generically looks like
More precisely, this happens when the zero energy is non-resonant, that is, if the equation $H f_0 =0$ has no bounded (distributional) solution. In the resonant situation the behavior of the scattering matrix is more delicate. For $V\in L^1_1$ it is already non-trivial to establish continuity of the scattering matrix at $k=0$ (for $V\in L^1_2$ 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 $V\in L_2^1$ in Reference 6, are valid for $V\in L_1^1$. 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 $V\in L^1_{j+1}$ with $j>0$, then, away from $0$, one can take derivatives of the scattering matrix up to order $j$, and in the non-resonant case it is again easy to see that they are continuous at $k=0$. This clearly raises the question about continuity at $k=0$ of the $j$-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 $V$ decays exponentially, then $\mathcal{S}(k)$ is analytic in a neighborhood of $k=0$ 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:
This theorem is an improvement of an earlier result by Goldberg Reference 10 who established it for $V \in L_4^1({\mathbb{R}})$. If there is no resonance (i.e. no bounded solution) this result (with $P_0=0$) was shown for $V \in L_2^1({\mathbb{R}})$ 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 $H$ with $V\in L_{j+1}^1$,$j\geq 0$. To this end we introduce the Banach algebra ${\mathcal{A}}$ of Fourier transforms of integrable functions,
with the norm $\|f\|_{{\mathcal{A}}_1}= |c|+\|\hat{g}\|_{L^1}$. We also use the fact, which is known as Wiener’s lemma Reference 17, that if $f\in {\mathcal{A}}_1\setminus {\mathcal{A}}$ and $f(k)\not =0$ for all $k\in {\mathbb{R}}$, then $f^{-1}(k)\in {\mathcal{A}}_1$.
First we recall a few facts from scattering theory Reference 6Reference 15. If $V\in L^1_1$ there exist Jost solutions $f_\pm (x,k)$ of $H f=k^2 f$,$k\in \overline{{\mathbb{C}}_+}$, which asymptotically behave like $f_\pm (x,k)\sim \mathrm{e}^{\pm \mathrm{i}kx}$ as $x\to \pm \infty$. These solutions are given by
Here and throughout the rest of this paper a prime will always denote a derivative with respect to the spatial variable $x$. As an immediate consequence of the estimates Equation 2.4 and Equation 2.5 we have the following strengthening of Equation 2.7.
The fact that $f_\pm (x,-k)$ also solve $H f=k^2 f$ for $k\in {\mathbb{R}}$ leads to the scattering relations
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.
To complete the proof of Theorem 2.2 we need the following result which is an extension of Reference 9, Lemma 2.2.
For later use we note that in the resonant case the Jost solutions are dependent at $k=0$. If we define $\gamma$ via
In particular, all three quantities are real-valued since $f_\mp (x,0)\in {\mathbb{R}}$ and hence $\gamma \in {\mathbb{R}}$.
To establish Theorem 1.1 we need the following generalization of Lemma 2.1.
Similarly,
Combining the last two lemmas we obtain:
3. Dispersive decay estimates
In this section we prove the integrable dispersive decay estimate Equation 1.2 for the Schrödinger equation Equation 1.1 in the resonant case. For the one-parameter group of Equation 1.1 the spectral theorem and Stone’s formula imply
where ${\mathcal{R}}(\omega )=(H-\omega )^{-1}$ is the resolvent of the Schrödinger operator $H$ and the limit is understood in the strong sense Reference 16, Problem 4.3. Given the Jost solutions we can express the kernel of the resolvent ${\mathcal{R}}(\omega )$ for $\omega =k^2\pm \mathrm{i}0$,$k>0$, as Reference 16, Lemma 9.7
for all $x\leq y$ (and the positions of $x,y$ reversed if $x>y$). Therefore, in the case $x\le y$, the integral kernel of $\mathrm{e}^{-\mathrm{i}tH}P_{ac}$ is given by
where the integral has to be understood as an improper integral. Another result that we need in order to obtain our decay estimates is the following variant of the van der Corput lemma Reference 9, Lemma 5.4.
Now we come to the proof of our main Theorem 1.1. We first give an alternate representation of our projection operator $(4 \pi \mathrm{i}t)^{-\frac{1}{2}}P_0$.
Finally we have all the ingredients needed to obtain Theorem 1.1.
Acknowledgments
The first author is indebted to the Department of Mathematics at the University of Vienna for its hospitality and support during the fall of 2014, where some of this work was done. The authors thank Fritz Gesztesy for discussions on this topic.
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Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria — and — International Erwin Schrödinger Institute for Mathematical Physics, Boltzmanngasse 9, 1090 Wien, Austria
Show rawAMSref\bib{3450570}{article}{
author={Egorova, Iryna},
author={Holzleitner, Markus},
author={Teschl, Gerald},
title={Zero energy scattering for one-dimensional Schr\"odinger operators and applications to dispersive estimates},
journal={Proc. Amer. Math. Soc. Ser. B},
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date={2015},
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review={3450570},
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