MR2116805 Nanayakkara, Asiri . Comparison of quantal and classical
behavior of $PT$-symmetric systems at avoided crossings. Phys.
Lett. A 334 (2005), no. 2-3, 144--153.


Barbanis' (= two-dimensional non-separable $\sqrt{g}xy^2$ plus
harmonic) potential is numerically studied in the two contexts
(viz., of classical and quantum theory) and in two regimes (viz.,
the real and PT-symmetric one, with $g>0$ and $g < 0$,
respectively).

In the former context, several samples of the Poincare surfaces of
sections confirm the expected transition, with the growth of the
energy, from the regular domain via mixed domain to the almost
complete chaos at both the positive $g$ (mind the misprint) and
negative $g$. This opens the terrain for the Percival's prediction
(applied usually to Hermitian Hamiltonians) that at the lowest
energies many non-integrable systems possess almost solely regular
trajectories in classical phase space (which means that these
trajectories are confined to an N-dimensional torus where N is the
number of degrees of freedom) while their quantized spectrum
remains ``regular". In contrast, the classical trajectories become
chaotic [filling all the (2N-1)-dimensional fixed-energy subspace
(called ``shell")] at the high energies while, in parallel, the
quantized spectrum becomes ``irregular", exhibiting, typically,
the large second derivatives (with respect to $g$) and level
repulsion.

The main attention of the paper is paid to the quantum and
PT-symmetric context. Empirically (using matrix diagonalization
and Rayleigh-Schroedinger perturbation methods), the Percival's
correspondence is claimed confirmed. In this frame, my doubts
concern one of the key author's claims that the levels cannot
cross in the PT-symmetric regime. In fact, even if they did (as
they really do in many solvable PT-symmetric models, say, of ref.
[7]), the phenomenon could not have been detected by the
diagonalization method once the operator itself ceases to be
diagonalizable at the points of the crossing.