# A localized boundary deformation which splits the spectrum of the Laplacian

## TL;DR

This paper presents a method to create tiny, localized boundary deformations in Lipschitz domains that split the Laplacian's spectrum into simple eigenvalues, using Hadamard's formula and spectral stability analysis.

## Contribution

It introduces a novel technique for spectrum splitting via localized boundary perturbations in Lipschitz domains, expanding spectral control methods.

## Key findings

- Spectrum can be split into simple eigenvalues with small boundary changes
- Uses Hadamard's formula for spectral analysis
- Perturbations are arbitrarily small and localized

## Abstract

For any Lipschitz domain we construct an arbitrarily small, localized perturbation which splits the spectrum of the Laplacian into simple eigenvalues. We use for this purpose a Hadamard's formula and spectral stability results.