Time-delay matrix, midgap spectral peak, and thermopower of an Andreev billiard

We derive the statistics of the time-delay matrix (energy derivative of the scattering matrix) in an ensemble of superconducting quantum dots with chaotic scattering (Andreev billiards), coupled ballistically to $M$ conducting modes (electron-hole modes in a normal metal or Majorana edge modes in a superconductor). As a first application we calculate the density of states ${\rho }_0$ at the Fermi level. The ensemble average $⟨{\rho }_0⟩={\delta }_0^{-1}M[\max (0,M+2\alpha \mathrm{/}\beta ){]}^{-1}$ deviates from the bulk value $1\mathrm{/}{\delta }_0$ by an amount depending on the Altland-Zirnbauer symmetry indices $\alpha ,\beta$. The divergent average for $M=1,2$ in symmetry class D ($\alpha =-1$, $\beta =1$) originates from the mid-gap spectral peak of a closed quantum dot, but now no longer depends on the presence or absence of a Majorana zero-mode. As a second application we calculate the probability distribution of the thermopower, contrasting the difference for paired and unpaired Majorana edge modes.