Magnetic Neumann Laplacian on a domain with hole

TL;DR

This paper investigates how small holes in a domain affect the spectrum of the magnetic Neumann Laplacian, showing that the spectrum remains stable under certain conditions and converges to that of the unperturbed domain.

Contribution

It provides a rigorous analysis of spectral stability for the magnetic Neumann Laplacian under domain perturbations involving small holes.

Findings

Spectrum converges in Hausdorff distance to the original spectrum.

Holes that are sufficiently small do not significantly alter the spectral properties.

The spectral stability depends on the size and placement of the holes.

Abstract

This article gives a domain with a small compact set of removed and the magnetic Neumann Laplacian on such set. The main theorem of this article shows the description of the holes which do not change the spectrum drastically. In this article we prove that the spectrum of the magnetic Neumann Laplacian converges in the Hausdorff distance sense to the spectrum of the original operator defined on the unperturbed domain.