Can one hear the shape of a Majorana Billiard
Majorana billiards are finite sized superconductors or semiconductors in proximity to a superconductor, featuring ground state fermion parity switches. We study their "spectra", i.e. the set of external parameters at which said parity switches occur, and find formulae for the asymptotic mean value of the density of crossings and its oscillations that are analogous to the Weyl expansion and the Gutzwiller trace formula, respectively. Moreover, the statistics of the spacings of parity-crossings are universal and are described by a random matrix ensemble choice of which depends on the antiunitary symmetries of the system in its normal state. We thus demonstrate that "one can hear (information about) the shape of a Majorana billiard" by investigating its spectrum.