The eigenvalues of the Laplacian on domains with small slits

TL;DR

This paper investigates how introducing a small slit into a planar domain affects the Laplacian's eigenvalues, showing that as the slit shrinks, eigenvalues approach those of the original domain, with implications for spectral simplicity.

Contribution

It demonstrates the limiting behavior of Laplacian eigenvalues under small slit perturbations and applies this to prove generic simplicity of spectra in multiply connected polygons.

Findings

Eigenvalues tend to those of the original domain as slit length approaches zero.

Small slits do not cause eigenvalue divergence, maintaining spectral stability.

Generic multiply connected polygons have simple spectra.

Abstract

We introduce a small slit into a planar domain and study the resulting effect upon the eigenvalues of the Laplacian. In particular, we show that as the length of the slit tends to zero, each real-analytic eigenvalue branch tends to an eigenvalue of the original domain. By combining this with our earlier work (arXiv:math/0703616), we obtain the following application: The generic multiply connected polygon has simple spectrum.