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

Abstract

We consider the Helmholtz equation −∆u + V u − λ u = f on R n where the potential V : R n → R is constant on each of the half-spaces R n−1 ×(−∞, 0) and R n−1 ×(0, ∞). We prove an L p − L q-Limiting Absorption Principle for frequencies λ > max V with the aid of Fourier Restriction Theory and derive the existence of nontrivial solutions of linear and nonlinear Helmholtz equations. As a main analytical tool we develop new L p − L q estimates for a singular Fourier multiplier supported in an annulus.