Density of proper delay times in chaotic and integrable quantum billiards

We calculate the density P(\tau) of the eigenvalues of the Wigner-Smith time delay matrix for two-dimensional rectangular and circular billiards with one opening. For long times, the density of these so-called “proper delay times” decays algebraically, in contradistinction to chaotic quantum billiards for which P(\tau) exhibits a long-time cut-off.