Dispersive Estimates in R^3 with Threshold Resonances

TL;DR

This paper establishes optimal dispersive estimates for the Schrödinger equation in three dimensions, accounting for zero energy eigenfunctions and resonances, which are critical for understanding long-term wave behavior.

Contribution

It provides the first sharp dispersive estimates in R^3 with threshold resonances and zero energy eigenfunctions under optimal decay conditions on the potential.

Findings

Proves dispersive decay rates for Schrödinger evolution in R^3.

Handles zero energy eigenfunctions and resonances.

Establishes optimal decay conditions on the potential V.

Abstract

We prove dispersive estimates in R^3 for the Schroedinger evolution generated by the Hamiltonian H = -\Delta+V, under optimal decay conditions on V, in the presence of zero energy eigenfunctions and resonances.