Nonunique CC operator in PTPT quantum mechanics

The three simultaneous algebraic equations, C2=1C2=1, [C,PT]=0[C,PT]=0, [C,H]=0[C,H]=0, which determine the CC operator for a non-Hermitian PTPT-symmetric Hamiltonian H  , are shown to have a nonunique solution. Specifically, the CC operator for the Hamiltonian H=12p2+12μ2q2+iϵq3 is determined perturbatively to first order in ϵ and it is demonstrated that the CC operator contains an infinite number of arbitrary parameters. For each different CC operator, the corresponding equivalent isospectral Dirac Hermitian Hamiltonian h is calculated.