# Eigenvalue Asymptotics of Narrow Domains

## TL;DR

This paper investigates the asymptotic behavior of eigenvalues of the Dirichlet Laplacian on narrow, variable-width domains, providing detailed spectral analysis as the domain narrows.

## Contribution

It offers a comprehensive asymptotic analysis of eigenvalues for the Dirichlet Laplacian on domains with variable width, extending understanding of spectral properties in narrow geometries.

## Key findings

- Derived full eigenvalue asymptotics for the Dirichlet Laplacian
- Analyzed spectral behavior in domains with maximum width at a single point
- Provided detailed asymptotic formulas for eigenvalues

## Abstract

In this paper, we considered the spectrum of the Dirichlet Laplacian $\Delta_\epsilon$ on $\Omega_\epsilon=\{(x,y): -l_1<x<l_2, 0<y<\epsilon h(x)]\}$ where $ l_1,l_2>0$ and $h(x)$ is a positive analytic function having $0$ the only point where it achieves its global maximum $M$. In particular we studied in details about the full asymptotics of the eigenvalues.