Abstract
We numerically study quantum mechanical features of the Bunimovich stadium billiard and the rational billiards which approach the former as the number of their sides increases. The statistics of energy levels and eigenfunctions of the rational billiards becomes indistinguishable from that of the Bunimovich stadium billiard below a certain energy. This fact contradicts the classical picture in which the Bunimovich stadium billiard is chaotic, but the rational billiard is pseudointegrable. It is numerically confirmed that the wave functions do not detect the fine structure, which is much smaller than the wavelength.
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References
M. V. Berry, InSemiclassical Mechanics Regular and Irregular Motion (Les Houches, Session XXXVI, 1981), G. Iooss, R. H. G. Helleman, and R. Stora, eds. (North-Holland, Amsterdam, 1983), p. 171.
A. N. Zemlyalov and A. B. Katoh,Math Notes 18:760 (1976);20:1051 (1976).
P. J. Richens and M. V. Berry,Physica D 2:495 (1981).
B. Eckhardt, J. Ford, and F. Vivaldi,Physica D 13:329 (1984).
J. L. Vega, T. Uzer and J. Ford,Phys. Rev. E 48:3414 (1993).
T. Cheon and T. D. Cohen,Phys. Rev. Lett. 62:1769 (1989).
A. Shudo and Y. Shimizu,Phys. Rev. E 47:54 (1993).
E. Heller, inWavepacket Dynamics and Quantum Chaology (Les Houches Summer School. Session LII, 1989), M.-J. Giannoni, A. Voros and J. Zinn-Justin, eds. (North-Holland, Amsterdam, 1991), p. 547.
T. Takami,Phys. Rev. Lett. 68:3371 (1992).
H. P. Baltes, and E. R. Hilf,Spectra of Finite Systems (BI Wissenschaftsverlag, Mannheim, 1976).
H.-D. Gröf, H. L. Harney, H. Lengeler, C. H. Lewenkopf C. Rangacharyulu, A. Richter, P. Schardt, and H. A. Weidenmüller,Phys. Rev. Lett. 69:1296 (1992).
M. Sieber, U. Smilansky, S. C. Creagh, and R. G. Littlejohn,J. Phys. A: Math. Gen. 26:6217 (1993).
E. B. Bogomolny,Physica D 31:169 (1988).
S. W. McDonald and A. N. Kaufman,Phys. Rev. A 37:3067 (1988).
Y. Shimizu and A. Shudo,Prog. Theor. Phys. Suppl. 116:267 (1994).
P. Seba,Phys. Rev. Lett. 64:1855 (1990).
T. Shigehara,Phys. Rev. E 50:4357 (1994).
A. Shudo and Y. Shimizu, Statistical properties of eigen-functions for quantum billiards with and without positive exponent, to be published.
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Tomiya, M., Yoshinaga, N. Scars in nonintegrable and rational billiards. J Stat Phys 83, 215–242 (1996). https://doi.org/10.1007/BF02183647
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DOI: https://doi.org/10.1007/BF02183647