Comment on Energy Level Statistics in the Mixed Regime

Abstract

We comment on the recent paper by Abul-Magd (J.Phys.A: Math.Gen. 29 (1996) 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 $ρ_1$ and $ρ_2=q$ is not justified, since among other things, it relies on the type of Berry's ergodic assumption, which however is right only in a homogeneous ensemble of ergodic systems, but 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 $\hbar$ is not yet small enough. Thus we refer to the derivation, arguments and conclusions in our paper (Prosen and Robnik, J.Phys.A: Math.Gen. 26 (1994) 8059), and explain again the behaviour in this double transition region.