Semiclassical resonances under local magnetic fields

TL;DR

This paper investigates semiclassical magnetic Laplacian resonances in the plane with localized magnetic fields, revealing their emergence near Landau levels and other magnetic features using complex scaling and scattering theory.

Contribution

It introduces new results on the existence and origin of semiclassical resonances for magnetic Laplacians with various magnetic field configurations.

Findings

Resonances exist near Landau levels with exponentially small imaginary parts.

Resonances emerge from magnetic step discontinuities and wells.

Resonances occur near anharmonic Landau levels with isolated zeroes.

Abstract

We study resonances for the semiclassical magnetic Laplacian in the full plane with a compactly supported magnetic field in the framework of semiclassical complex scaling and black box scattering theory. Assuming that the magnetic field is locally constant, we prove the existence of semiclassical resonances near the Landau levels with exponentially small imaginary parts. We also prove that resonances emerge from a magnetic step discontinuity along a curved interface or a non-degenerate magnetic well, and in the vicinity of anharmonic Landau levels if the field has an isolated zero.