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

Abstract

In space dimension $n\ge 3$, we consider the electromagnetic Schrödinger 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}.

arXiv comment25 pages, 4 figures. arXiv admin note: substantial text overlap with arXiv:1011.0915; and text overlap with arXiv:1104.4237 by other authors