  We discuss the Hamiltonian H = p^2/2 - (ix)^{2n+1} and the mixed Hamiltonian
H = (p^2 + x^2)/2 - g(ix)^{2n+1}, which are crypto-Hermitian in a sense that,
in spite of apparent complexity of the potential, a quantum spectral problem
can be formulated such that the spectrum is real. We note first that the
corresponding classical Hamiltonian system can be treated as a gauge system,
with imaginary part of the Hamiltonian playing the role of the first class
constraint. We observe then that, on the basis of this classical problem,
several different nontrivial quantum problems can be formulated. We formulate
and solve some such problems. We find in particular that the spectrum of the
mixed Hamiltonian undergoes in certain cases rather amazing transformation when
the coupling g is sent to zero. There is an infinite set of phase transitions
in g when a couple of eigenstates of H coalesce and disappear from the
spectrum. When quantization is done in the most natural way such that gauge
constraints are imposed on quantum states, the spectrum should not be positive
definite, but must involve the negative energy states (ghosts). We speculate
that, in spite of the appearance of ghost states, unitarity might still be
preserved.
