Dispersion Estimates for One-Dimensional Schr\"odinger and Klein-Gordon Equations Revisited

TL;DR

This paper establishes that for certain one-dimensional Schrödinger operators, the scattering matrix belongs to the Wiener algebra, enabling improved dispersion estimates for Schrödinger and Klein-Gordon equations, especially in resonant cases.

Contribution

It demonstrates the inclusion of the scattering matrix in the Wiener algebra under minimal conditions and derives new dispersion estimates without extra decay assumptions.

Findings

Scattering matrix in Wiener algebra for integrable first moment potentials

Dispersion estimates valid even with resonances at spectrum edges

Removal of additional decay conditions in resonance cases

Abstract

We show that for a one-dimensional Schr\"odinger operator with a potential whose first moment is integrable the scattering matrix is in the unital Wiener algebra of functions with integrable Fourier transforms. Then we use this to derive dispersion estimates for solutions of the associated Schr\"odinger and Klein-Gordon equations. In particular, we remove the additional decay conditions in the case where a resonance is present at the edge of the continuous spectrum.