Laplacian and spectral gap in regular Hilbert geometries

TL;DR

This paper investigates the spectral properties of the Finsler-Laplace operator in regular Hilbert geometries, establishing bounds on the spectral gap and constructing examples with small eigenvalues.

Contribution

It provides the first bounds on the spectral gap for regular Hilbert geometries and constructs convex sets with arbitrarily small eigenvalues.

Findings

Spectral gap is bounded above by (n-1)^2/4.

The bound is the infimum of the essential spectrum.

Examples of convex sets with arbitrarily small eigenvalues are constructed.

Abstract

We study the spectrum of the Finsler--Laplace operator for regular Hilbert geometries, defined by convex sets with $C^2$ boundaries. We show that for an $n$-dimensional geometry, the spectral gap is bounded above by $(n-1)^2/4$, which we prove to be the infimum of the essential spectrum. We also construct examples of convex sets with arbitrarily small eigenvalues.