Dispersive estimates for two-particles Schr\"odinger and wave equations in the Aharonov-Bohm field

TL;DR

This paper establishes dispersive and Strichartz estimates for two-particle Schr"odinger and wave equations in the Aharonov-Bohm field, and applies these results to develop a scattering theory for nonlinear equations with such potentials.

Contribution

It constructs the spectral measure of the Schr"odinger operator with Aharonov-Bohm potential in four dimensions and proves dispersive estimates in this setting.

Findings

Proved Strichartz estimates for Schr"odinger and wave equations in Aharonov-Bohm field

Constructed spectral measure for Schr"odinger operator with Aharonov-Bohm potential

Established scattering theory for nonlinear two-particle Schr"odinger equation with Aharonov-Bohm potential

Abstract

We study the dispersive behaviors of two-particles Schr\"odinger and wave equations in the Aharonov-Bohm field. In particular, we prove the Strichartz estimates for Schr\"odinger and wave equations in this setting. The key point is to construct spectral measure of Schr\"odinger operator with an Aharonov-Bohm type potential in $\R^4$. As applications, we finally prove a scattering theory for the nonlinear defocusing subcritical two-particles Schr\"odinger equation with Aharonov-Bohm potential.