Calculation of the operator C in PT-symmetric quantum mechanics

Abstract:
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 [1]. 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 [2].

References:

1. C. M. Bender, D. C. Brody and H. F. Jones, Phys. Rev. Lett. V.89, 270402 (2002).
2. C. M. Bender, P. N. Meisinger and Q. Wang, J. Phys. A: Math. Gen. V.36, 1973-1983 (2003).