A spectral analogue of the Meinardus theorem on asymptotics of the number of partitions

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Abstract

We discuss asymptotics of the number of states of Boson gas whose Hamiltonian is given by a positive elliptic pseudo-differential operator of order one on a compact manifold. We obtain an asymptotic formula for the average of the number of states. Furthermore, when the operator has integer eigenvalues and the periodic orbits of period less than 2π of the classical mechanics form clean submanifolds of lower dimensions, we give an asymptotic formula for the number of states itself. This is regarded as an analogue of the Meinardus theorem on asymptotics of the number of partitions of a positive integer. We use the Meinardus saddle point method of obtaining the asymptotics of the number of partitions, combined with a theorem due to Duistermaat-Guillemin and other authors on the singularities of the trace of the wave operators.

Original languageEnglish
Pages (from-to)101-123
Number of pages23
JournalAsymptotic Analysis
Volume67
Issue number1-2
DOIs
Publication statusPublished - 2010
Externally publishedYes

Keywords

  • Meinardus suddle point method
  • Number of partitions
  • Number of states of Boson gas
  • Singularities of traces of wave operators

ASJC Scopus subject areas

  • Mathematics(all)

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