Quantizing a classically ergodic system: Sinai's billiard and the KKR method |
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Authors: | MV Berry |
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Institution: | H. H. Wills Physics Laboratory, Tyndall Avenue, Bristol BS8 1TL, U.K. |
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Abstract: | Sinai's “billiards on a torus,” i.e., free motion of a particle in a plane amongst reflecting discs of radius R centred on points of the unit square lattice, is a classically ergodic system with two freedoms, parametrized by R. Quantal energy levels En are given by the vanishing of the Korringa-Kohn-Rostoker (KKR) determinant of solid state theory. This gives a rapid computational scheme for computing En as functions of R. Except for the integrable case R = 0, no degeneracies are found, illustrating the theorem that two parameters, not one, are required to make levels cross in a generic system. The same theorem leads to the prediction that the probability distribution of the spacings S of neighbouring levels is (S) as S → 0, in good agreement with computation. The KKR determinant is transformed analytically to give the level density d(E) semiclassically (i.e., as ? → 0) as the sum of a steady contribution and an oscillatory contribution dosc(E). is (??2) and is given by the Weyl “area” formula plus “edge,” “corner” and “curvature” corrections, in excellent agreement with computation. dosc is given by a sum over classical closed orbits (all unstable). Nonisolated closed orbits (not hitting discs) contribute terms with to dosc, while isolated closed orbits (bouncing between discs) contribute terms with (??1) to dosc. The isolated orbits are vastly more numerous than the nonisolated orbits and their contributions cannot be neglected. As a means of calculating the individual En (rather than the smoothed spectrum), the KKR method is much more efficient than the classical path sum. |
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