Laplacian and spectral gap in regular Hilbert geometries

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.

21 pages, 3 figures

References in corpus (3)

• On the Hilbert Geometry of Convex Polytopes
• A natural Finsler--Laplace operator
• Eigenvalues control for a Finsler--Laplace operator