Limiting absorption principle for the Magnetic Dirichlet Laplacian in a half-plane

TL;DR

This paper establishes a limiting absorption principle for the magnetic Dirichlet Laplacian in a half-plane, analyzing spectral properties and decay behaviors at thresholds related to Landau levels, with broader applicability to fibered magnetic operators.

Contribution

It proves the limiting absorption principle at thresholds for the magnetic Laplacian in a half-plane and discusses its generalization to other fibered magnetic Laplacians.

Findings

Established limiting absorption principle at spectral thresholds.

Analyzed decay and analytic properties of absorption space functions.

Demonstrated the generality of the approach for fibered magnetic Laplacians.

Abstract

We consider the Dirichlet Laplacian in the half-plane with constant magnetic field. Due to the translational invariance this operator admits a fiber decomposition and a family of dis- persion curves, that are real analytic functions. Each of them is simple and monotically decreasing from positive infinity to a finite value, which is the corresponding Landau level. These finite limits are thresholds in the purely absolutely continuous spectrum of the magnetic Laplacian. We prove a limiting absorption principle for this operator both outside and at the thresholds. Finally, we establish analytic and decay properties for functions lying in the absorption spaces. We point out that the analysis carried out in this paper is rather general and can be adapted to a wide class of fibered magnetic Laplacians with thresholds in their spectrum that are finite limits of their band functions.