Spectral Asymptotics Revisited

TL;DR

This paper introduces heuristic concepts related to the spectrum of Laplacians across manifolds, graphs, and fractals, proposing new ideas like spectral mass and an asymptotic Schur's lemma, supported by theorems and conjectures.

Contribution

It presents novel heuristics for analyzing Laplacian spectra, including spectral mass and spectral symmetry proportions, with formulations applicable to various geometric and algebraic structures.

Findings

Spectral mass can substitute eigenvalue counting in non-discrete spectra

Asymptotic Schur's lemma describes spectral proportions under symmetry groups

Results include reformulations of known theorems and new conjectures

Abstract

We introduce two new heuristic ideas concerning the spectrum of a Laplacian, and we give theorems and conjectures from the realms of manifolds, graphs and fractals that validate these heuristics. The first heuristic concerns Laplacians that do not have discrete spectra: here we introduce a notion of "spectral mass", an average of the diagonal of the kernel of the spectral projection operator, and show that this can serve as a substitute for the eigenvalue counting function. The second heuristic is an "asymptotic Schur's lemma" that describes the proportions of the spectrum that transforms according to the irreducible representations of a finite group that acts as a symmetric group of the Laplacian. For this to be valid we require the existence of a fundamental domain with relatively small boundary. We also give a version in the case that the symmetry groups is a compact Lie group. Many of our results are reformulations of known results, and some are merely conjectures, but there is something to be gained by looking at them together with a new perspective.