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The character of motion (regular or chaotic) on the quantum level manifestsitself in the statistics of the set of the system's energy levels. Thecompletely regular case generates the sequence of the levels with exponential(Poisson) level spacing distribution; the completely chaotic one -- with Wignerdistribution. The most interesting case is the co-existence of the regular andchaotic motion domains in the phase space of the system under consideration.The assumption of independent generation of two level sequencies by one chaoticdomain and by all regular domains leads to Berry--Robnik distribution. However,the presence of the chaotic motion domain in the phase space affects the levelsproduced by the regular domains via the so-called chaos-assisted tunneling(CAT) that leads to Podolskiy--Narimanov distribution of the level spacings.This distribution needs the mean amplitude of the tunnel transition and therelative contribution of the regular domains to the mean level density as theparameters. Using their estimations for the transverse motion of the highenergy positrons channeling near [100] direction in the silicon crystal wefound that Podolskiy--Narimanov distribution demonstrates the best agreementwith the level spacing distribution.

Mixed (Regular and Chaotic) Dynamics under the Channeling of the High  Energy Positrons in [100] Direction of the Silicon Crystal and  Podolskiy-Narimanov Distribution