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 -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

where the functions are absolutely continuous with and

with from Equation 2.6. Now the calculations in Reference 9, Lemma 2.2 lead to the following integral equation for :

The estimates Equation 2.4 and Equation 2.18 imply

Now, for a given potential fix such that

where is given by Equation 2.18. Then we can rewrite Equation 2.19 in the form

From Equation 2.13 and the estimates Equation 2.4Equation 2.5 we deduce . We also have

by the boundedness of , the estimate Equation 2.20, and monotonicity of . Applying to Equation 2.21 the method of successive approximations, we get

with . This implies Equation 2.16. The rest follows from Equation 2.14.

For later use we note that in the resonant case the Jost solutions are dependent at . If we define via

then a straightforward calculation using the scattering relations Equation 2.8 as well as shows

In particular, all three quantities are real-valued since and hence .

To establish Theorem 1.1 we need the following generalization of Lemma 2.1.

Lemma 2.4.

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

Proof.

Using Equation 2.3 and Fubini, a little calculation shows

Now, after differentiating with respect to , the claim follows by Equation 2.4. For the second item we can proceed exactly in the same way.

Similarly,

Lemma 2.5.

Let with . Then for .

Proof.

This follows as in the previous lemma using the estimate for from Lemma 2.3.

Combining the last two lemmas we obtain:

Theorem 2.6.

Let with . Then and are elements of for .

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 is the resolvent of the Schrödinger operator 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 for , , as Reference 16, Lemma 9.7

for all (and the positions of reversed if ). Therefore, in the case , the integral kernel of 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.

Lemma 3.1.

Consider the oscillatory integral , where is a real-valued function. If in and , then we have for , where is the optimal constant from the van der Corput lemma.

Now we come to the proof of our main Theorem 1.1. We first give an alternate representation of our projection operator .

Lemma 3.2 (Reference 10).

The integral kernel of , which is (per definition) given by , can also be written in the form

Proof.

It is clear that and by our normalization . Using Equation 2.23 we have and hence . Moreover, Equation 2.24 implies and using the claim follows.

Finally we have all the ingredients needed to obtain Theorem 1.1.

Proof of Theorem 1.1.

For the kernel of we use Equation 3.3. Then by Lemma 3.2, the kernel of can now be written as

Integrating this formula by parts, we obtain

where

Now we apply Lemma 3.1 to get the desired time-decay. So in order to finish our proof, it remains to bound the -norm of which follows from the lemma below.

Lemma 3.3.

Assume . Then

Proof.

We assume w.l.o.g. that and distinguish the cases (i) , (ii) and (iii) . Introduce the function . Then can be written as

The -norm of is bounded by and that of its derivative by . So it remains to consider the -norms of and , and the -norm of , where

We start with case (i). Then with -norm independent of and . After applying the product rule, Lemmas 2.1 and 2.6 imply . Moreover,

Taking here the derivative with respect to and invoking Lemma 2.1, Lemma 2.4, and Theorem 2.6, we are done in this case. In the cases (ii) and (iii) we use the scattering relations Equation 2.8 to see that the following representations are valid:

Thus has an -norm independent of and , since for any function and any real we have with the norm independent of . If we take the derivative with respect to , again everything is contained in by Lemma 2.1 and Theorem 2.6, however we get additional terms from the derivatives of and . So it follows that the -norm of is at most proportional to or respectively. Finally, let us have a look at . We observe that in case (ii) one can represent as

Here again every summand is an element of by Lemma 2.1, Lemma 2.4, and Theorem 2.6. Since the derivative of also occurs here, we conclude that the -norm of is at most proportional to . From the same reasons in case (iii) this derivative will be proportional to .

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.

Mathematical Fragments

Equation (1.1)
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:

Equation (2.3)
Equations (2.4), (2.5)
Equation (2.6)
Equation (2.7)
Lemma 2.1.

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

Equation (2.8)
Theorem 2.2.

If , then and for .

Equation (2.13)
Equation (2.14)
Equation (2.15)
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

Equation (2.18)
Equation (2.19)
Equation (2.20)
Equation (2.21)
Equation (2.23)
Equation (2.24)
Lemma 2.4.

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

Theorem 2.6.

Let with . Then and are elements of for .

Equation (3.3)
Lemma 3.1.

Consider the oscillatory integral , where is a real-valued function. If in and , then we have for , where is the optimal constant from the van der Corput lemma.

Lemma 3.2 (Reference 10).

The integral kernel of , which is (per definition) given by , can also be written in the form

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Article Information

MSC 2010
Primary: 34L25 (Scattering theory, inverse scattering), 35Q41 (Time-dependent Schrödinger equations, Dirac equations)
Secondary: 81U30 (Dispersion theory, dispersion relations), 81Q15 (Perturbation theories for operators and differential equations)
Keywords
  • Schrödinger equation
  • scattering
  • resonant case
  • dispersive estimates
Author Information
Iryna Egorova
B. Verkin Institute for Low Temperature Physics, 47, Lenin ave, 61103 Kharkiv, Ukraine
iraegorova@gmail.com
MathSciNet
Markus Holzleitner
Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria
amhang1@gmx.at
Gerald Teschl
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
Gerald.Teschl@univie.ac.at
Homepage
MathSciNet
Additional Notes

This research was supported by the Austrian Science Fund (FWF) under Grants No. Y330 and W1245.

Communicated by
Joachim Krieger
Journal Information
Proceedings of the American Mathematical Society, Series B, Volume 2, Issue 4, ISSN 2330-1511, published by the American Mathematical Society, Providence, Rhode Island.
Publication History
This article was received on , revised on , and published on .
Copyright Information
Copyright 2015 by the authors under Creative Commons Attribution 3.0 License (CC BY 3.0)
Article References
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  • DOI 10.1090/bproc/19
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  • 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}, volume={2}, number={4}, date={2015}, pages={51-59}, issn={2330-1511}, review={3450570}, doi={10.1090/bproc/19}, }

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