Decay estimates for Schrödinger's equation with magnetic potentials in three dimensions

Abstract

In this paper we prove that Schrödinger'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 Schrödinger 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.

34 pages