TY - JOUR

T1 - Can One Determine the Shape of a Quantum Billiard Table through the Eigenenergies and Resonances?

AU - Okada, Yuichiro

AU - Shudo, Akira

AU - Harayama, Takahisa

AU - Tasaki, Shuichi

N1 - Copyright:
Copyright 2018 Elsevier B.V., All rights reserved.

PY - 2003

Y1 - 2003

N2 - 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.

AB - 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.

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U2 - 10.1143/PTPS.150.397

DO - 10.1143/PTPS.150.397

M3 - Article

AN - SCOPUS:0344118081

VL - 150

SP - 397

EP - 400

JO - Progress of Theoretical Physics

JF - Progress of Theoretical Physics

SN - 0033-068X

ER -