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

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.