COMMENT ON ENERGY-LEVEL STATISTICS IN THE MIXED REGIME

We comment on the recent paper by Abul-Magd (1996 J. Phys. A: Math. Ge n. 29 1) concerning the energy level statistics in the mixed regime, i .e. such having the mixed classical dynamics where regular and chaotic regions coexist in the phase space. We point out that his basic assum ption on the additive property of the level-repulsion function r(S) (c onditional probability density) in the sense of dividing it linearly i nto the regular and chaotic part in proportion to the classical fracti onal phase space volumes rho(1) and rho(2) = q is not justified, since among other things, it relies on the type of Berry's ergodic assumpti on, which, however, is only correct in a homogeneous ensemble of ergod ic systems, not in the neighbourhood of an integrable system. Thus, hi s resulting distribution cannot be regarded as a theoretically well fo unded object. We point out that the semiclassical limiting energy leve l spacing distribution must be of Berry-Robnik (1984) type, and explai n what transitional behaviour of the Brody-type (with fractional power -law energy level repulsion) we observe in the near semiclassical regi me where effective A is not yet Small enough. Thus we refer to the der ivation, arguments and conclusions in our previous paper (Prosen and R obnik 1994 J. Phys. Ac Math. Cen. 26 8059), and explain again the beha viour in this double transition region.