A perturbative approach to the spectral zeta functions of strings, drums, and quantum billiards

Journal Article · Sat Dec 15 04:00:00 EST 2012 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.4769212· OSTI ID:22094015

Amore, Paolo ^[1]

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1. Facultad de Ciencias, CUICBAS, Universidad de Colima, Bernal Diaz del Castillo 340, Colima, Colima (Mexico)

We show that the spectral zeta functions of inhomogeneous strings and drums can be calculated using Rayleigh-Schroedinger perturbation theory. The inhomogeneities that can be treated with this method are small but otherwise arbitrary and include the previously studied case of a piecewise constant density. In two dimensions the method can be used to derive the spectral zeta function of a domain obtained from the small deformation of a square. We also obtain exact sum rules that are valid for arbitrary densities and that correspond to the values taken by the spectral zeta function at integer positive values; we have tested numerically these sum rules in specific examples. We show that the Dirichlet or Neumann Casimir energies of an inhomogeneous string, evaluated to first order in perturbation theory, contain in some cases an irremovable divergence, but that the combination of the two is always free of divergences. Finally, our calculation of the Casimir energies of a string with piecewise constant density and of two perfectly conducting concentric cylinders, of similar radius, reproduce the results previously published.

• OSTI ID: 22094015
• Journal Information: Journal of Mathematical Physics, Journal Name: Journal of Mathematical Physics Journal Issue: 12 Vol. 53; ISSN JMAPAQ; ISSN 0022-2488
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
CASIMIR EFFECT
DEFORMATION
DENSITY
DIRICHLET PROBLEM
PERTURBATION THEORY
PLATES
SCHROEDINGER EQUATION
SIMULATION
STRING MODELS
SUM RULES