Generic spectrum of warped products and G-manifolds

TL;DR

This paper investigates the spectral properties of Laplacians on warped products and G-manifolds, establishing generic simplicity results and addressing a longstanding question about eigenvalue multiplicities on principal bundles.

Contribution

It introduces a splitting theorem for eigenvalues on warped products and proves that generically, the Laplacian spectrum on certain G-manifolds is simple, partially answering Zelditch's 1990 question.

Findings

Eigenvalues of specific operators on warped product bases can be split.

A density theorem for warping functions makes the Laplacian spectrum warped-simple.

The generic eigenvalue multiplicity on certain G-manifolds is one.

Abstract

In this paper, we establish a kind of splitting theorem for the eigenvalues of a specific family of operators on the base of a warped product. As a consequence, we prove a density theorem for a set of warping functions that makes the spectrum of the Laplacian a warped-simple spectrum. This is then used to study the generic situation of the eigenvalues of the Laplacian on a class of compact G-manifolds. In particular, we give a partial answer to a question posed in 1990 by Steven Zelditch about the generic situation of multiplicity of the eigenvalues of the Laplacian on principal bundles.