Dispersive decay estimates for the magnetic Schr\"odinger equations

Abstract

In this paper, we present a proof of dispersive decay for both linear and nonlinear magnetic Schr\"odinger equations. To achieve this, we introduce the fractional distorted Fourier transforms with magnetic potentials and define the fractional differential operator $\arrowvert J_{A}(t)\arrowvert^{s}$. By leveraging the properties of the distorted Fourier transforms and the Strichartz estimates of $\arrowvert J_{A}\arrowvert^{s}u$, we establish the dispersive bounds with the decay rate $t^{-\frac{n}{2}}$. This decay rate provides valuable insights into the spreading properties and long-term dynamics of the solutions to the magnetic Schr\"odinger equations.