Limiting absorption principle for the electromagnetic Helmholtz equation with singular potentials

TL;DR

This paper establishes the limiting absorption principle for the electromagnetic Helmholtz equation with singular potentials, proving existence, uniqueness, and a priori estimates of solutions using Morawetz-type multipliers.

Contribution

It extends the limiting absorption principle to Helmholtz equations with singular magnetic and electric potentials, providing new a priori estimates and solution existence results.

Findings

Existence and uniqueness of solutions under singular potentials

A priori estimates for solutions with decay at infinity

Application of Morawetz-type multiplier techniques

Abstract

We study the following Helmholtz equation $$ (\nabla +iA(x))^{2} u+ V_{1}(x) u + V_{2}(x) u + \lambda u = f(x) $$ in $\mathbb{R}^d$ with magnetic and electric potentials that are singular at the origin and decay at infinity. We prove the existence of a unique solution satisfying a suitable Sommerfeld radiation condition, together with some a priori estimates. We use the limiting absorption method and a multiplier technique of Morawetz type.