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Prog. Theor. Phys. Vol. 102 No. 3 (1999) pp. 551-598

[ Full Text PDF : FREE ACCESS (1019K) ]

Symmetry Breaking and Bifurcations in the Periodic Orbit Theory. I

— Elliptic Billiard —

Alexander G. Magner,1,2,3 Sergey N. Fedotkin,1,2 Ken-ichiro Arita,4 Toshiyuki Misu,5 Kenichi Matsuyanagi,3 Thomas Schachner2 and Matthias Brack2

1Institute for Nuclear Research, 252028 Prospekt Nauki 47, Kiev–28, Ukraine
2Institute for Theoretical Physics, University of Regensburg
D-93040 Regensburg, Germany
3Department of Physics, Graduate School of Science, Kyoto University
Kyoto 606-8502, Japan
4Department of Physics, Nagoya Institute of Technology
Nagoya 466-8555, Japan
5Cyclotron Radio-isotope Center, Tohoku University, Sendai 980-8578, Japan

(Received May 27, 1999)

Abstract:

We derive an analytical trace formula for the level density of two-dimensional elliptic billiards using an improved stationary phase method. The result is a continuous function of the deformation parameter (eccentricity) through all bifurcation points of the short diameter orbit and its repetitions, and possesses the correct limit of circular billiard at zero eccentricity. Away from the circular limit and the bifurcations, it reduces to the usual (extended) Gutzwiller trace formula, which for the leading-order families of periodic orbits is identical to the result of Berry and Tabor. We show that the circular disk limit of the diameter-orbit contribution is also reached through contributions from closed (periodic and non-periodic) orbits of the hyperbolic type with an even number of reflections from the boundary. We obtain the Maslov indices depending on deformation and energy in terms of the phases of the complex error and Airy functions. We find enhancement of the amplitudes near the common bifurcation points of short-diameter and hyperbolic orbits. The calculated semiclassical level densities and shell energies are in good agreement with the quantum mechanical ones.


URL : http://ptp.ipap.jp/link?PTP/102/551/
DOI : 10.1143/PTP.102.551

[ Full Text PDF : FREE ACCESS (1019K) ] Citation:

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Citing Article(s) :

  1. Progress of Theoretical Physics Vol. 108 No. 5 (2002) pp. 853-901 :
    Symmetry Breaking and Bifurcations in the Periodic Orbit Theory. II
    Alexander G. Magner, Ken-ichiro Arita, Sergey N. Fedotkin and Kenichi Matsuyanagi
  2. Progress of Theoretical Physics Vol. 115 No. 3 (2006) pp. 523-546 :
    Semiclassical Approach for Bifurcations in a Smooth Finite-Depth Potential
    Alexander G. Magner, Ken-ichiro Arita and Sergey N. Fedotkin

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