Can One Determine the Shape of a Quantum Billiard Table through the Eigenenergies and Resonances?

Yuichiro Okada, Akira Shudo, Takahisa Harayama, Shuichi Tasaki

Research output: Contribution to journalArticle

Abstract

The quantum billiard problem, that is the Dirichlet problem for the Helmholtz equation, can be rewritten as a Fredholm integral equation of the second kind and the eigenenergies can be characterized as the zeros of the Fredholm determinant on the positive real axis. However the Fredholm determinant also has complex zeros corresponding to the resonances when the billiard table is regarded as a scatterer against the exterior wave function. That naturally leads us to a new question "Can one determine the shape of billiard table through the interior eigenenergies and exterior resonances, i.e., all zeros of the Fredholm determinant?" instead of the famous Kac's question "Can one hear the shape of a drum?", which was solved negatively.

Original languageEnglish
Pages (from-to)397-400
Number of pages4
JournalProgress of Theoretical Physics Supplement
Volume150
DOIs
Publication statusPublished - 2003 Jan 1
Externally publishedYes

ASJC Scopus subject areas

  • Physics and Astronomy (miscellaneous)

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