Comment on energy level statistics in the mixed regime

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Abstract
We comment on the recent paper by Abul-Magd (1996 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 assumption on the additive property of the level-repulsion function r(S) (conditional probability density) in the sense of dividing it linearly into the regular and chaotic part in proportion to the classical fractional phase space volumes and is not justified, since among other things, it relies on the type of Berry's ergodic assumption, which, however, is only correct in a homogeneous ensemble of ergodic systems, not in the neighbourhood of an integrable system. Thus, his resulting distribution cannot be regarded as a theoretically well-founded object. We point out that the semiclassical limiting energy level spacing distribution must be of Berry - Robnik (1984) type, and explain what transitional behaviour of the Brody-type (with fractional power-law energy level repulsion) we observe in the near semiclassical regime where effective is not yet small enough. Thus we refer to the derivation, arguments and conclusions in our previous paper (Prosen and Robnik 1994 26 8059), and explain again the behaviour in this double transition region.
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