Semiempirical theory of level spacing distribution beyond the Berry–Robnik regime: modeling the localization and the tunneling effects

In this work we study the level spacing distribution in the classically mixed-type quantum systems (which are generic), exhibiting regular motion on invariant tori for some initial conditions and chaotic motion for the complementary initial conditions. In the asymptotic regime of the sufficiently deep semiclassical limit (sufficiently small effective Planck constant) the Berry and Robnik (1984 J. Phys. A: Math. Gen. 17 2413) picture applies, which is very well established. We present a new quasi-universal semiempirical theory of the level spacing distribution in a regime away from the Berry–Robnik regime (the near semiclassical limit), by describing both the dynamical localization effects of chaotic eigenstates, and the tunneling effects which couple regular and chaotic eigenstates. The theory works extremely well in the 2D mixed-type billiard system introduced by Robnik (1983 J. Phys. A: Math. Gen. 16 3971) and is also tested in other systems (mushroom billiard and Prosen billiard).