Multiple eigenvalues and the width

TL;DR

This paper investigates the simplicity and comparison of the first few Neumann eigenvalues in convex thin domains and manifolds, revealing how geometric properties influence eigenvalue multiplicity.

Contribution

It establishes the simplicity of the first Neumann eigenvalue in convex thin domains and extends results to the first k eigenvalues in 2D, linking eigenvalue properties to geometric ratios.

Findings

First Neumann eigenvalue is simple in convex thin domains.

Simplicity of the first k eigenvalues depends on width-to-diameter ratio in 2D.

Eigenvalue comparison with collapsing segments in higher dimensions.

Abstract

We obtain the simplicity of the first Neumann eigenvalue of convex thin domain with boundary in $R^n$ and compact thin manifolds with non-negative Ricci curvature. For convex thin domain in $R^2$, we get the simplicity of the first k Neumann eigenvalues. The number k depends on the ratio of the corresponding width over the diameter of the domain. For convex thin domain in $R^n$, we obtain the eigenvalue comparison with collapsing segment.