Exponential local energy decay of solutions to the wave equation with $L^\infty$ electric and magnetic potentials

TL;DR

This paper establishes sharp resolvent estimates for the magnetic Schrödinger operator with $L^ obreak{}^ ext{infty}$ potentials and demonstrates exponential local energy decay for wave equations with such potentials, under certain conditions.

Contribution

It provides new resolvent estimates for Schrödinger operators with $L^ obreak{}^ ext{infty}$ potentials and proves exponential decay of local energy for wave equations in various dimensions.

Findings

Sharp resolvent estimates for magnetic Schrödinger operators with $L^ obreak{}^ ext{infty}$ potentials.

Exponential decay of local energy for wave equations with these potentials.

No low-frequency cutoff needed in odd dimensions if zero is not an eigenvalue or resonance.

Abstract

In this paper we prove sharp resolvent estimates for the magnetic Schr\"odinger operator in $\mathbb{R}^d$, $d\ge 3$, with $L^\infty$ short-range electric and magnetic potentials. We also show that these resolvent estimates still hold for the Dirichlet self-adjoint realization of the Schr\"odinger operator in the exterior of a non-trapping obstacle in $\mathbb{R}^d$, $d\ge 2$, provided the magnetic potential is supposed identically zero. As an application of the resolvent estimates, we obtain an exponential decay of the local energy of solutions to the wave equation with $L^\infty$ electric and magnetic potentials which decay exponentially at infinity, in all odd and even dimensions, provided the low frequencies are cut off in a suitable way. We also show that in odd dimensions there is no need to cut off the low frequencies in order to get an exponential local energy decay, provided we assume that zero is neither an eigenvalue nor a resonance.