Random matrices associated with general barrier billiards

Abstract

The paper is devoted to the derivation of random unitary matrices whose spectral statistics is the same as statistics of quantum eigenvalues of certain deterministic two-dimensional barrier billiards. These random matrices are extracted from the exact billiard quantisation condition by applying a random phase approximation for high-excited states. An important ingredient of the method is the calculation of S-matrix for the scattering in the slab with a half-plane inside by the Wiener–Hopf method. It appears that these random matrices have the form similar to the one obtained by the author in (2022 J. Phys. A: Math. Theor. 55 024001) for a particular case of symmetric barrier billiards but with different choices of parameters. The local correlation functions of the resulting random matrices are well approximated by the semi-Poisson distribution which is a characteristic feature of various models with intermediate statistics. Consequently, local spectral statistics of the considered barrier billiards is (i) universal for almost all values of parameters and (ii) well described by the semi-Poisson statistics.