Uniform resolvent estimates for critical magnetic Schr\"odinger operators in 2D

TL;DR

This paper establishes uniform resolvent estimates for 2D magnetic Schrödinger operators with critical magnetic fields, including the Aharonov-Bohm model, and applies these to eigenvalue localization for perturbed operators.

Contribution

It provides the first uniform resolvent estimates in the critical magnetic field setting and applies them to eigenvalue localization for non self-adjoint perturbations.

Findings

Established $L^p-L^q$ resolvent estimates for 2D magnetic Schrödinger operators.

Proved eigenvalue localization results for non self-adjoint perturbations.

Analyzed the Aharonov-Bohm model as a key example.

Abstract

We study the $L^p-L^q$-type uniform resolvent estimates for 2D-Schr\"odinger operators in scaling-critical magnetic fields, involving the Aharonov-Bohm model as a main example. As an application, we prove localization estimates for the eigenvalue of some non self-adjoint zero-order perturbations of the magnetic Hamiltonian.