Decay estimates for one Aharonov-Bohm solenoid in a uniform magnetic field II: wave equation

TL;DR

This paper investigates decay estimates for the wave equation influenced by an Aharonov-Bohm solenoid in a uniform magnetic field, addressing mathematical challenges and establishing bounds for related heat kernels and inequalities.

Contribution

It introduces new decay estimates for the wave equation with Aharonov-Bohm solenoids, and develops methods to bound the heat kernel and establish inequalities in this magnetic setting.

Findings

Proved Gaussian upper bounds for the heat kernel.

Established Davies-Gaffney inequality in this context.

Demonstrated Bernstein and square function inequalities for the Schrödinger operator.

Abstract

This is the second of a series of papers in which we investigate the decay estimates for dispersive equations with Aharonov-Bohm solenoids in a uniform magnetic field. In our first starting paper \cite{WZZ}, we have studied the Strichartz estimates for Schr\"odinger equation with one Aharonov-Bohm solenoid in a uniform magnetic field. The wave equation in this setting becomes more delicate since a difficulty is raised from the square root of the eigenvalue of the Schr\"odinger operator $H_{\alpha, B_0}$ so that we cannot directly construct the half-wave propagator. An independent interesting result concerning the Gaussian upper bounds of the heat kernel is proved by using two different methods. The first one is based on establishing Davies-Gaffney inequality in this setting and the second one is straightforward to construct the heat kernel (which efficiently captures the magnetic effects) based on the Schulman-Sunada formula. As byproducts, we prove optimal bounds for the heat kernel and show the Bernstein inequality and the square function inequality for Schr\"odinger operator with one Aharonov-Bohm solenoid in a uniform magnetic field.