Density of proper delay times in chaotic and integrable quantum billiards

We calculate the density $P\left(\tau \right)$ 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 $P\left(\tau \right)$ exhibits a long-time cutoff.