Dispersive decay estimates for the magnetic Schr\"odinger equations

TL;DR

This paper proves dispersive decay estimates for magnetic Schr"odinger equations, both linear and nonlinear, using fractional distorted Fourier transforms and Strichartz estimates to analyze solution spreading over time.

Contribution

It introduces fractional distorted Fourier transforms with magnetic potentials and establishes decay bounds, advancing understanding of magnetic Schr"odinger dynamics.

Findings

Dispersive decay rate of t^{-rac{n}{2}} for solutions.

Extension of decay estimates to nonlinear magnetic Schr"odinger equations.

Development of fractional distorted Fourier transforms for magnetic potentials.

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.