The spectrum on $p$-forms of a lens space

TL;DR

This paper explicitly describes the spectrum of the Hodge--Laplace operator on p-forms of lens spaces, providing rational function formulas and characterizing p-isospectral lens spaces across intervals.

Contribution

It offers a comprehensive spectral description for all p-forms on lens spaces and characterizes p-isospectrality for ranges of p, extending previous spectral analysis results.

Findings

Explicit spectrum formulas as rational functions

Characterization of p-isospectral lens spaces

Spectral invariants across p-forms

Abstract

We give an explicit description of the spectrum of the Hodge--Laplace operator on $p$-forms of an arbitrary lens space for any $p$. We write the two generating functions encoding the $p$-spectrum as rational functions. As a consequence, we prove a geometric characterization of lens spaces that are $p$-isospectral for every $p$ in an interval of the form $[0,p_0]$.