Calculation of the Hidden Symmetry Operator in PT-Symmetric Quantum Mechanics

Abstract

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 $H=\frac{1}{2}{p}^2+\frac{1}{2}{x}^2+i\epsilon x^3$ using perturbative techniques. It is also shown how to construct the operator C for $H=\frac{1}{2}{p}^2+\frac{1}{2}{x}^2-\epsilon x^4$ using nonperturbative methods.