Schr\"odinger dispersive estimates for a scaling-critical class of potentials

TL;DR

This paper establishes dispersive estimates for three-dimensional Schrödinger operators with a class of scaling-critical potentials, using a novel Wiener's L^1 inversion theorem approach.

Contribution

It introduces a new dispersive estimate for Schrödinger operators with potentials in a critical class, under specific spectral conditions, expanding understanding of quantum evolution.

Findings

Dispersive estimates hold for potentials in the closure of bounded compactly-supported functions under the global Kato norm.

Spectral conditions exclude resonances and eigenfunctions on the positive half-line.

A new version of Wiener's L^1 inversion theorem is applied in the proof.

Abstract

We prove a dispersive estimate for the evolution of Schroedinger operators H = -\Delta + V(x) in three dimensions. The potential should belong to the closure of bounded compactly-supported functions with respect to the golbal Kato norm. Some additional spectral conditions are imposed, namely that no resonances or eigenfunctions of H exist anywhere on the positive half-line. The proof is an application of a new version of Wiener's L^1 inversion theorem.