Small eigenvalues of pseudo-Laplacians

TL;DR

This paper extends bounds on small eigenvalues from Laplacians to pseudo-Laplacians on hyperbolic surfaces, broadening spectral theory applications to surfaces with multiple cusps.

Contribution

It generalizes the Otal-Rosas bound and Colin de Verdière's work to include pseudo-Laplacians on hyperbolic surfaces with multiple cusps.

Findings

Extended bounds on small eigenvalues for pseudo-Laplacians.

Generalized spectral theory to hyperbolic surfaces with multiple cusps.

Broadened understanding of eigenvalue distribution in geometric analysis.

Abstract

We extend the Otal-Rosas bound on the number of small eigenvalues of the Laplacian on a hyperbolic surface to the small eigenvalues of pseudo-Laplacians. In the process, we extend the work of Colin de Verdi\`ere on the spectral theory of pseudo-Laplacians to hyperbolic surfaces with more than one cusp.