Maass cusp form on asymmetric hyperbolic torus

The quantum system describing a free particle moving on a cusped hyperbolic surface is represented using the eigenfunction of the hyperbolic Laplace-Beltrami operator. The eigenspectra contained both continuous and discrete spectra, but the focus here is only on the discrete part. The eigenfunctions have to be computed numerically and they are known as Maass cusp form (MCF). The hyperbolic surface of interest here is the singly punctured two-torus. Past research has shown that the case of the symmetric torus has degenerate eigenvalues. The purpose of this research is to find the eigenvalues for asymmetric torus, deformed from symmetric torus by moving the vertices of its fundamental domain at the real axis, as well as to investigate the degeneracy behavior of its eigenvalues. There are three models that are being explored, namely F1 with vertices at -1, 1/2 , 1, and ∞, F2 with vertices at -3, 0, 2 and ∞ and the last one F3 with vertices at -2, 0, 1 and ∞. Despite having different cusp widths, all models are ensured to have the same area. Since the domain of the torus in the hyperbolic plane needs an equivalent fundamental domain where the cusp is represented by the point of imaginary infinity for a convenient computation, a cusp reduction method is constructed including the equations for the generators in order to act as the side identification. Consider that the asymmetric torus has no parity symmetry, an algorithm of MCF with exponential expansion is developed using Mathematica. The computation of MCF is an adapted algorithm of Hejhal and Then, i.e. based on implicit automorphy and finite Fourier series. There are 37 eigenvalues found for asymmetric torus F1 and 24 eigenvalues for asymmetric torus F2 between range [0, 15]. Both domains have non-degenerate eigenvalues. Remarkably, all eigenvalues of F2 are also eigenvalues for F1, suggesting that the unique MCF for F1 are newforms while those of F2 are oldforms. In the same range, the computed algorithm for asymmetric torus F3 gives out 36 eigenvalues and surprisingly these eigenvalues are doubly degenerate. It is believed that the equivalent fundamental domain for F3 has extra symmetry compared to F1 and F2. Apparently, equivalent fundamental domain for F3 has symmetry at each vertices, meanwhile the other two does not have. All the candidate eigenvalues given by the algorithm went through checking procedure stated in the literature so that only authentic eigenvalues have been chosen. Those procedures are y-independent solution, automorphy condition, Hecke relation and Ramanujan- Petersson conjecture. Later, the eigenstates of selected eigenvalues from each surface are visualized using contour plot and density plot in the Mathematica.

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