Cryptogauge symmetry and cryptoghosts for crypto-Hermitian Hamiltonians

We discuss the Hamiltonian H = p(2)/2 - (ix)(2n+1) and the mixed Hamiltonian H-mixed = (p(2) + x(2))/2 - g(ix)(2n+1). The Hamiltonians H and in some cases also H-mixed are crypto-Hermitian in a sense that, in spite of their apparent non-Hermiticity, a quantum spectral problem can be formulated such that the spectrum is real. We note that the corresponding classical Hamiltonian system can be treated as a gauge system, with the imaginary part of the Hamiltonian playing the role of the first class constraint. Several different nontrivial quantum problems can be formulated on the basis of this classical problem. We formulate and solve some such problems. We consider then the mixed Hamiltonian and find that its spectrum undergoes in certain cases a rather amazing transformation when the coupling g is sent to zero. There is an infinite set of exceptional points g(*)((j)) where a couple of eigenstates of H coalesce and their eigenvalues cease to be real. 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.