Calculation of the operator C in PT-symmetric quantum mechanics
It has been 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.
We will give a formal derivation of the operator C. Then we will construct the operator C for two different types of non-Hermitian PT-symmetric Hamiltonians. The first is H = 1/2p2+1/2x2+iex3. We use perturbative techniques to calculate C to the third order in e for this theory. The second is H = 1/2p2+1/2x2-ex4. For this theory nonperturbative methods must be used .