Calculation of the hidden symmetry operator in   -symmetric quantum mechanics

In a recent paper it was shown that if a Hamiltonian H has an unbroken PT symmetry, then it also possesses a hidden symmetry represented by the linear operator C. The operator C commutes with both H and PT. The inner product with respect to CPT is associated with a positive norm and the quantum theory built on the associated Hilbert space is unitary. In this paper it is shown how to construct the operator C for the non-Hermitian PT-symmetric Hamiltonian H=12p²+12x² +i x³ using perturbative techniques. It is also shown how to construct the operator C for H=12p²+12x²- x⁴ using nonperturbative methods.