Abstract
Compact billiards in phase space, or action billiards, are constructed by truncating the classical Hamiltonian in the action variables. The corresponding quantum mechanical system has a finite Hamiltonian matrix. In previous papers we defined the compact analog of common billiards, i.e., straight motion in phase space followed by specular reflections at the boundaries. Computation of their quantum energy spectra establishes that their properties are exactly those of common billiards: the short-range statistics follow the known universality classes depending on the regular or chaotic nature of the motion, while the long-range fluctuations are determined by the periodic orbits. In this work we show that the eigenfunctions also follow qualitatively the general characteristics of common billiards. In particular, we show that the low-lying levels can be classified according to their nodal lines as usual and that the high excited states present scars of several short periodic orbits. Moreover, since all the eigenstates of action billiards can be computed with great accuracy, Bogomolny's semiclassical formula for the scars can also be tested successfully.
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References
A. M. Ozorio de Almeida and M. A. M. de Aguiar,Chaos Solitons Fractals 2:377 (1992).
A. M. Ozorio de Almeida and M. A. M. de Aguiar, InQuantum Chaos, B. Chirikov and G. Casati, Eds. (Cambridge University Press, Cambridge, 1994).
M. A. M. de Aguiar and A. M. Ozorio de Almeida,Nonlinearity 5:523 (1992).
M. A. M. de Aguiar,Phys. Lett. A,164:284 (1992).
E. Bogomolny,Physica D 31:169 (1988).
A. M. Ozorio de Almeida, InHamiltonian Systems: Chaos and Quantization (Cambridge University Press, Cambridge, 1988).
D. Provost and M. Baranger,Phys. Rev. Lett. 71:662 (1993).
A. M. Ozorio de Almeida,Proc. R. Soc. Lond. A 439:139 (1992).
M. Baranger and M. A. M. de Aguiar, In preparation.
E. J. Heller,Phys. Rev. Lett. 53:1515 (1984).
K. Furuya, M. A. M. de Aguiar, C. H. Lewenkopf, and M. C. Nemes,Ann. Phys. (N.Y.)216:313 (1992).
G. M. Zaslavsky et al. InWeak Chaos and Quasi-Regular Patterns (Cambridge University Press, Cambridge, 1991), Section 7.4.
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Ortiz, J.S.E., de Aguiar, M.A.M. & Ozorio de Almeida, A.M. Scars of periodic orbits in the stadium action billiard. J Stat Phys 83, 275–287 (1996). https://doi.org/10.1007/BF02183650
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DOI: https://doi.org/10.1007/BF02183650