  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.
