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arXiv · 0909.1155

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

Hishamuddin Zainuddin·Abubaker Ahmed M. Siddig
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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 $Γ$($2)$-automorphicity for the triply punctured sphere and $Γ^{\prime}$-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.

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Hishamuddin Zainuddin, Abubaker Ahmed M. Siddig. 2009-09-17. Maass Cusp Forms on Singly Punctured Two-Torus and Triply Punctured Two-Sphere. https://arxiv.org/abs/0909.1155

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