Spectrum of generalized Hodge-Laplace operators on flat tori and round spheres

TL;DR

This paper explicitly computes the spectra of generalized Hodge-Laplace operators on flat tori and round spheres, analyzing their isospectral properties and how these depend on geometric parameters.

Contribution

It provides explicit spectral calculations for a family of generalized Hodge-Laplace operators on specific manifolds, and studies conditions for isospectrality.

Findings

Explicit spectra for operators on flat tori and spheres

Conditions for isospectrality between different geometries

Dependence of spectra on radii and parameters

Abstract

We consider generalized Hodge-Laplace operators $\alpha d \delta + \beta \delta d$ for $\alpha, \beta > 0$ on $p$-forms on compact Riemannian manifolds. In the case of flat tori and round spheres of different radii, we explicitly calculate the spectrum of these operators. Furthermore, we investigate under which circumstances they are isospectral.