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Distribution and fluctuation properties of transition probabilities in a system between integrability and chaos
Prosen, Tomaž, 1970- ; Robnik, Marko, 1954-
We study the statistical properties of transition probabilities (or generalized intensities, equal to the squared matrix elements of operators having a classical limit). We first generalize the ...Shnirelman theorem to include semiclassically integrable states and use this to generalize the Feingold-Peres formula for the average value of generalized intensities. We perform an unfolding procedure to separate the smooth mean part of the intensities (as a function of frequency) from its fluctuating part by applying the generalized Feingold-Peres formula. (This formula relates the mean value of squared matrix elements to the power spectrum of the given observable over classical trajectories.) Our approach is illustrated numerically by analysing the dipole transition probabilities in a family between integrability and chaos as introduced by Robnik. The average values of the intensities as a function of frequency are excellently described by the generalized Feingold-Peres formula, especially in the classically ergodic case where the agreement is excellent. In the ergodic case the fluctuations of intensities are perfectly well described by the Porter-Thomas distribution, whilst in the predominantly regular regime (almost integrable KAM) we find a great abundance of approximate selection rules, some apparent systematics of line series and some strongly enhanced transition probabilities which we believe is typical for such a regime. Our approach is expected to be very useful and practical in the context of nuclear, atomic and molecular physics
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