Non-trapping magnetic fields and Morrey-Campanato estimates for Schroedinger operators

TL;DR

This paper establishes uniform a priori estimates for solutions to Schrödinger equations with magnetic fields, using Morrey-Campanato norms, to demonstrate the absence of zero-resonances under specific conditions on the electromagnetic fields.

Contribution

It introduces new Morrey-Campanato estimates for Schrödinger operators with non-trapping magnetic fields, aiding in the analysis of zero-resonances and electromagnetic Hamiltonians.

Findings

Uniform a priori estimates in Morrey-Campanato norms

Conditions for absence of zero-resonances

Analysis of non-trapping magnetic fields

Abstract

We prove some uniform in $\epsilon$ a priori estimates for solutions of the equation $$(\nabla-iA)^2u-V(x)u+(\lambda\pm i\epsilon)u=f, \lambda\geq0, \epsilon\neq0.$$ The estimates are obtained in terms of Morrey-Campanato norms, and can be used to prove absence of zero-resonances, in a suitable sense, for electromagnetic Hamiltonians. Precise conditions on the size of the \textit{trapping component} of the magnetic field and the non repulsive component of the electric field are given.