Distribution and fluctuation properties of transition probabilities in a system between integrability and chaos

The authors study the statistical properties of transition probabilities (or generalized intensities, equal to the squared matrix elements of operators having a classical limit). They 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. They 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). Their approach is illustrated numerically by analysing the dipole transition probabilities in a family of billiards between integrability and chaos. The average values of the intensities as a function of frequency are excellently described by the generalized Feingold-Peres formula.