Spectral simplicity and asymptotic separation of variables

TL;DR

This paper introduces a method to compare eigenvalues of two quadratic form families, showing that spectral simplicity in a separable case implies simplicity in a more complex case, with applications to triangles in various geometries.

Contribution

It presents a novel approach to relate spectral properties of degenerate quadratic forms, extending results on eigenvalue simplicity to non-separable cases with geometric applications.

Findings

Generic triangles in Euclidean and constant curvature spaces have simple Laplacian spectra.

Most hyperbolic triangles with angles 0, t, t have simple spectra for all but countably many t.

Eigenbranches of non-separable families are generically one-dimensional under certain conditions.

Abstract

We describe a method for comparing the real analytic eigenbranches of two families of quadratic forms that degenerate as t tends to zero. One of the families is assumed to be amenable to `separation of variables' and the other one not. With certain additional assumptions, we show that if the families are asymptotic at first order as t tends to 0, then the generic spectral simplicity of the separable family implies that the eigenbranches of the second family are also generically one-dimensional. As an application, we prove that for the generic triangle (simplex) in Euclidean space (constant curvature space form) each eigenspace of the Laplacian is one-dimensional. We also show that for all but countably many t, the geodesic triangle in the hyperbolic plane with interior angles 0, t, and t, has simple spectrum.