Numerical Study of Length Spectra and Low-lying Eigenvalue Spectra of Compact Hyperbolic 3-manifolds

Details Numerical Study of Length Spectra and Low-lying Eigenvalue Spectra of Compact Hyperbolic 3-manifolds Numerical Study of Length Spectra and Low-lying Eigenvalue Spectra of Compact Hyperbolic 3-manifolds

2000-11-09

arxiv.org/abs/math-ph/0011012v2

Class.Quant.Grav. 18 No.4 629-652 (2001)

Physics

Mathematical Physics

19 pages, 18 EPS figures and 2 GIF figures (fig.10) Description of cusped manifolds in section 2 is corrected

Scientific paper

10.1088/0264-9381/18/4/305

In this paper, we numerically investigate the length spectra and the low-lying eigenvalue spectra of the Laplace-Beltrami operator for a large number of small compact(closed) hyperbolic (CH) 3-manifolds. The first non-zero eigenvalues have been successfully computed using the periodic orbit sum method, which are compared with various geometric quantities such as volume, diameter and length of the shortest periodic geodesic of the manifolds. The deviation of low-lying eigenvalue spectra of manifolds converging to a cusped hyperbolic manifold from the asymptotic distribution has been measured by $\zeta-$ function and spectral distance.

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