$L^p$-$L^q$ estimates for Electromagnetic Helmholtz equation. Singular potentials

Abstract

In space dimension $n\geq3$, we consider the electromagnetic Schr\"odinger Hamiltonian $H=(\nabla-iA(x))^2+V$ and the corresponding Helmholtz equation (\nabla-iA(x))^2u+u+V(x)u=f\quad \text{in}\quad \mathbb{R}^n, where the magnetic and electric potentials are allowed to have singularities at the origin and decay at infinity. We extend the well known $L^p$-$L^q$ estimates for the solution of the free Helmholtz equation to the case when the electromagnetic hamiltonian $H$ is considered. This work extends the results that appear in \cite{G}.