  The character of motion (regular or chaotic) on the quantum level manifests
itself in the statistics of the set of the system's energy levels. The
completely regular case generates the sequence of the levels with exponential
(Poisson) level spacing distribution; the completely chaotic one -- with Wigner
distribution. The most interesting case is the co-existence of the regular and
chaotic motion domains in the phase space of the system under consideration.
The assumption of independent generation of two level sequencies by one chaotic
domain and by all regular domains leads to Berry--Robnik distribution. However,
the presence of the chaotic motion domain in the phase space affects the levels
produced 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 the
relative contribution of the regular domains to the mean level density as the
parameters. Using their estimations for the transverse motion of the high
energy positrons channeling near [100] direction in the silicon crystal we
found that Podolskiy--Narimanov distribution demonstrates the best agreement
with the level spacing distribution.
