An annulus multiplier and applications to the Limiting absorption principle for Helmholtz equations with a step potential

Abstract

We consider the Helmholtz equation $-\Delta u+V \, u - \lambda \, u = f $ on $\mathbb{R}^n$ where the potential $V:\mathbb{R}^n\to\mathbb{R}$ is constant on each of the half-spaces $\mathbb{R}^{n-1}\times (-\infty,0)$ and $\mathbb{R}^{n-1}\times (0,\infty)$. We prove an $L^p-L^q$-Limiting Absorption Principle for frequencies $\lambda>\max \, V$ with the aid of Fourier Restriction Theory and derive the existence of nontrivial solutions of linear and nonlinear Helmholtz equations.