Maass Cusp Forms on Singly Punctured Two-Torus and Triply Punctured Two-Sphere

TL;DR

This paper investigates Maass cusp forms on hyperbolic surfaces of a punctured two-torus and a triply punctured sphere, introducing a numerical algorithm to compute eigenfunctions and reporting preliminary eigenvalue results.

Contribution

It presents a new numerical method for computing Maass cusp forms on these punctured hyperbolic surfaces, expanding understanding of their spectral properties.

Findings

Developed an algorithm for numerical computation of Maass cusp forms.

Computed lower-lying eigenvalues for odd and even Maass cusp forms.

Reported preliminary results on eigenvalues for both surfaces.

Abstract

In this paper we study two quantum mechanical systems on punctured surfaces modeled by hyperbolic spaces, namely the cases of the singly punctured two-torus and triply punctured two-sphere. We study the systems using their Maass waveforms in connection with the eigenfunctions of the Laplacian. The energy eigenfunctions on such surfaces are precisely the eigenfunctions of the hyperbolic Laplacian satisfying $\Gamma $($2)$-automorphicity for the triply punctured sphere and $\Gamma ^{\prime}$-automorphicity for singly punctured torus. We introduce the algorithm of numerically computing the Maass cusp forms on these two surfaces and report on the (preliminary) computational results of the lower-lying eigenvalues for each odd and even Maass cusp forms on both surfaces.