Discrete spectrum of the magnetic Laplacian on perturbed half-planes

TL;DR

This paper investigates the spectral properties of the magnetic Laplacian in perturbed half-planes, demonstrating the existence of bound states under certain geometric conditions related to boundary curvature.

Contribution

It establishes the presence of bound states for the magnetic Laplacian in almost flat and slightly curved half-planes with positive total boundary curvature, extending understanding of spectral behavior in unbounded domains.

Findings

Bound states exist in almost flat corners with positive boundary curvature.

Positive total curvature of the boundary guarantees bound states.

Results apply to slightly curved half-planes, broadening spectral analysis in unbounded domains.

Abstract

The existence of bound states for the magnetic Laplacian in unbounded domains can be quite challenging in the case of a homogeneous magnetic field. We provide an affirmative answer for almost flat corners and slightly curved half-planes when the total curvature of the boundary is positive.