  The barrier billiard is the simplest example of pseudo-integrable models with
interesting and intricate classical and quantum properties. Using the
Wiener-Hopf method it is demonstrated that quantum mechanics of a rectangular
billiard with a barrier in the centre can be reduced to the investigation of a
certain unitary matrix. Under heuristic assumptions this matrix is substituted
by a special low-complexity random unitary matrix of independent interest. The
main results of the paper are (i) spectral statistics of such billiards is
insensitive to the barrier height and (ii) it is well described by the
semi-Poisson distributions.
