$L^p$-$L^q$ estimates for Electromagnetic Helmholtz equation

TL;DR

This paper extends classical $L^p$-$L^q$ estimates for the Helmholtz equation to electromagnetic Hamiltonians in space dimensions three and higher, providing new bounds for solutions involving electromagnetic potentials.

Contribution

It introduces novel $L^p$-$L^q$ estimates for the electromagnetic Helmholtz equation, broadening the understanding of solutions with electromagnetic potentials in higher dimensions.

Findings

Extended $L^p$-$L^q$ estimates to electromagnetic Hamiltonians

Applicable for space dimensions $n \\geq 3$

Provides bounds for solutions involving electromagnetic 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 $(\nabla-iA(x))^2u+u-V(x)u=f \in \mathbb{R}^n$. 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.