Zero Energy Scattering for One-Dimensional Schr\"odinger Operators and Applications to Dispersive Estimates

TL;DR

This paper proves that derivatives of the scattering matrix for 1D Schrödinger operators with certain potentials belong to the Wiener algebra, leading to improved dispersive estimates in resonant cases.

Contribution

It establishes the membership of scattering matrix derivatives in the Wiener algebra for potentials with integrable moments, enhancing dispersive estimate results.

Findings

Derivatives of the scattering matrix are in the Wiener algebra.

Improved dispersive decay estimates for the 1D Schrödinger equation in resonant cases.

Enhanced understanding of scattering theory for potentials with integrable moments.

Abstract

We show that for a one-dimensional Schr\"odinger 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 Schr\"odinger equation in the resonant case.