Decay and Strichartz estimates in critical electromagnetic fields

TL;DR

This paper establishes decay and Strichartz estimates for dispersive equations influenced by critical electromagnetic fields, specifically in Aharonov-Bohm magnetic settings, advancing understanding of wave behavior in such quantum scenarios.

Contribution

It introduces explicit constructions of spectral measures and heat kernels for Schrödinger operators with Aharonov-Bohm potentials, a novel approach in this context.

Findings

Constructed Schwartz kernels for spectral measure and heat propagator.

Proved Gaussian boundedness of the heat kernel in critical electromagnetic fields.

Established decay and Strichartz estimates for dispersive equations in these fields.

Abstract

We study the $L^1\to L^\infty$-decay estimates for dispersive equations in the Aharonov-Bohm magnetic fields, and further prove Strichartz estimates for the Klein-Gordon equation with critical electromagnetic potentials. The novel ingredients are the construction of Schwartz kernels of the spectral measure and heat propagator for the Schr\"odinger operator in Aharonov-Bohm magnetic fields. In particular, we explicitly construct the representation of the spectral measure and resolvent of the Schr\"odinger operator with Aharonov-Bohm potentials, and show that the heat kernel in critical electromagnetic fields satisfies Gaussian boundedness. In future papers, this result on the spectral measure will be used to (i) study the uniform resolvent estimates, and (ii) prove the $L^p$-regularity property of wave propagation in the same setting.