Counting function of magnetic resonances for exterior problems

TL;DR

This paper analyzes the distribution of resonances near Landau levels for the exterior Dirichlet, Neumann, and Robin problems of a 3D Schrödinger operator with magnetic field, revealing accumulation and resonance-free sectors.

Contribution

It provides the first detailed asymptotic analysis of resonance distribution near Landau levels for exterior magnetic Schrödinger problems, including accumulation and resonance-free sectors.

Findings

Resonances accumulate at Landau levels.

Existence of resonance-free sectors near Landau levels.

Discreteness of embedded eigenvalues near Landau levels.

Abstract

We study the asymptotic distribution of the resonances near the Landau levels $\Lambda\_q =(2q+1)b$, $q \in \mathbb{N}$, of the Dirichlet (resp. Neumann, resp. Robin) realization in the exterior of a compact domain of $\mathbb{R}^3$ of the 3D Schr{\"o}dinger operator with constant magnetic field of scalar intensity $b\textgreater{}0$. We investigate the corresponding resonance counting function and obtain the main asymptotic term. In particular, we prove the accumulation of resonances at the Landau levels and the existence of resonance free sectors. In some cases, it provides the discreteness of the set of embedded eigenvalues near the Landau levels.