Decay estimates for one Aharonov-Bohm solenoid in a uniform magnetic field I: Schr\"odinger equation
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 FFFP1 and GYZZ22, FZZ22, and the second one is based on the Schulman-Sunada formula in sprit of 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.
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