Semiclassical Fredholm determinant for strongly chaotic billiards

Takahisa Harayama, Akira Shudo, Shuichi Tasaki

Research output: Contribution to journalArticle

9 Citations (Scopus)

Abstract

We investigate the "semiclassical Fredholm determinant" for strongly chaotic billiards derived from the semiclassical limit of the Fredholm determinant of the boundary element method. We show that it is the same as a cycle-expanded Gutzwiller-Voros zeta function when the bounce number of the periodic orbit with the billiard boundary corresponds to the length of the symbolic sequence of its symbolic dynamical expression. A numerical experiment on a "concave triangle billiard" shows that the series defining the semiclassical Fredholm determinant does not converge absolutely in spite of the absolute convergence of the series defining the Fredholm determinant. However, the series behaves like an asymptotic series, and the finite sum obtained by optimal truncation of the series defining the semiclassical Fredholm determinant gives the semiclassical eigenenergies precisely enough such that the error of the semiclassical approximation is much smaller than the mean spacing of the exact eigenenergies.

Original languageEnglish
Pages (from-to)1113-1149
Number of pages37
JournalNonlinearity
Volume12
Issue number4
DOIs
Publication statusPublished - 1999 Jul
Externally publishedYes

Fingerprint

Fredholm Determinant
Billiards
Boundary element method
determinants
Orbits
Series
Experiments
asymptotic series
Absolute convergence
Semiclassical Approximation
Asymptotic series
Bounce
Semiclassical Limit
boundary element method
approximation
Riemann zeta function
triangles
Truncation
Periodic Orbits
Boundary Elements

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics
  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

Semiclassical Fredholm determinant for strongly chaotic billiards. / Harayama, Takahisa; Shudo, Akira; Tasaki, Shuichi.

In: Nonlinearity, Vol. 12, No. 4, 07.1999, p. 1113-1149.

Research output: Contribution to journalArticle

Harayama, Takahisa ; Shudo, Akira ; Tasaki, Shuichi. / Semiclassical Fredholm determinant for strongly chaotic billiards. In: Nonlinearity. 1999 ; Vol. 12, No. 4. pp. 1113-1149.
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