Decay estimates for Schr\"{o}dinger's equation with magnetic potentials in three dimensions

TL;DR

This paper demonstrates that Schr"{o}dinger's equation with magnetic and scalar potentials in three dimensions exhibits decay properties similar to the free case, including $L^1 o L^ty$ decay, despite large, short-range potentials.

Contribution

It establishes decay estimates for Schr"{o}dinger's equation with magnetic potentials, extending known results to cases with large, short-range potentials in three dimensions.

Findings

Proves $L^1 o L^ty$ decay for Schr"{o}dinger with magnetic potentials.

Shows decay estimates hold despite large, short-range potentials.

Extends dispersive estimates to magnetic Schr"{o}dinger operators in 3D.

Abstract

In this paper we prove that Schr\"{o}dinger's equation with a Hamiltonian of the form $H=-\Delta+i(A \nabla + \nabla A) + V$, which includes a magnetic potential $A$, has the same dispersive and solution decay properties as the free Schr\"{o}dinger equation. In particular, we prove $L^1 \to L^\infty$ decay and some related estimates for the wave equation. The potentials $A$ and $V$ are short-range and $A$ has four derivatives, but they can be arbitrarily large. All results hold in three space dimensions.