Decay estimates for one Aharonov-Bohm solenoid in a uniform magnetic field I: Schr\"odinger equation

TL;DR

This paper establishes local dispersive and Strichartz estimates for the Schrödinger equation with a single Aharonov-Bohm solenoid in a uniform magnetic field, introducing two methods for constructing the propagator.

Contribution

It provides the first dispersive and Strichartz estimates for this quantum model, with novel propagator construction methods combining existing strategies and the Schulman-Sunada formula.

Findings

Proved local-in-time dispersive estimates.

Established Strichartz estimates for the model.

Developed two methods for Schrödinger propagator construction.

Abstract

This is the first 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 this first starting paper, we prove the local-in-time dispersive estimates and Strichartz estimates for Schr\"odinger equation with one Aharonov-Bohm solenoid in a uniform magnetic field. The key ingredient is the construction of Schr\"odinger propagator, we provide two methods to construct the propagator. The first one is combined the strategies of \cite{FFFP1} and \cite{GYZZ22, FZZ22}, and the second one is based on the Schulman-Sunada formula in sprit of \cite{stov, stov1} in which the heat kernel has been studied. In future papers, we will continue investigating this quantum model for wave with one or multiple Aharonov-Bohm solenoids in a uniform magnetic field.