Dispersive and Strichartz Estimates for Schrödinger Equation with One Aharonov-Bohm Solenoid in a Uniform Magnetic Field

We obtain dispersive and Strichartz estimates for solutions to the Schr & ouml;dinger equation with one Aharonov-Bohm solenoid in the uniform magnetic field. The main step of the proof is the construction of the Schr & ouml;dinger kernel, and the main obstacle is to obtain the explicit representation of the kernel, which requires a large set of careful calculations. To overcome this obstacle, we plan to construct the Schr & ouml;dinger kernel by two different strategies. The first one is to use the Poisson summation formula as Fanelli et al. (Adv Math 400:108333, 2022), while the second one relies on the Schulman-Sunada formula in & Scaron;t'ov & iacute;& ccaron;ek (Ann Phys 376:254-282, 2017), which reveals the intrinsic connections of the heat kernels on manifolds with group actions.