Computing Quantum Bound States on Triply Punctured Two-Sphere Surface

Abstract

We are interested in a quantum mechanical system on a triply punctured two-sphere surface with hyperbolic metric. The bound states on this system are described by the Maass cusp forms (MCFs) which are smooth square integrable eigenfunctions of the hyperbolic Laplacian. Their discrete eigenvalues and the MCF are not known analytically. We solve numerically using a modified Hejhal and Then algorithm, which is suitable to compute eigenvalues for a surface with more than one cusp. We report on the computational results of some lower-lying eigenvalues for the triply punctured surface as well as providing plots of the MCF using GridMathematica.

Universal Quantum Computing and Three-Manifolds

“…Σ(2, 3, 5) and Σ(2, 3, 7)], to the toroidal Seifert fibered Σ ′ , to Akbulut's manifold Σ Y and to a maximum symmetry hyperbolic manifold Σ 120e slightly breaking the icosahedral symmetry. It is expected that our work will have importance for new ways of implementing quantum computing and for the understanding of the link between quantum information and cosmology [47,48,49]. A subsequent paper of ours develops the field of 3-manifold based UQC with its relationship to Bianchigroups [50].…”

“…The conjugacy class of subgroups of index 3 in G is represented as 7,4,47,19,66,42,484, · · · } corresponding to the manifold otet06 00003 , alias s961.…”

Universal Quantum Computing and Three-Manifolds

“…Finally, in Section 4, Dehn fillings on T 1 were used to explore the connection of quantum computing to important exotic 3-manifolds (i.e., Σ(2, 3, 5) and Σ (2, 3, 7)), to the toroidal Seifert fibered Σ , to Akbulut's manifold Σ Y and to a maximum symmetry hyperbolic manifold Σ 120e slightly breaking the icosahedral symmetry. It is expected that our work will have importance for new ways of implementing quantum computing and for the understanding of the link between quantum information and cosmology [45][46][47]…”

The Desymmetrized PSL(2, Z) Group; Its ‘Square-Box’ One-Cusp Congruence Subgroups

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