Calculation of the hidden symmetry operator in -symmetric quantum mechanics

Abstract

In a recent paper it was shown that if a Hamiltonian H has an unbroken PT\nsymmetry, then it also possesses a hidden symmetry represented by the linear\noperator C. The operator C commutes with both H and PT. The inner product with\nrespect to CPT is associated with a positive norm and the quantum theory built\non the associated Hilbert space is unitary. In this paper it is shown how to\nconstruct the operator C for the non-Hermitian PT-symmetric Hamiltonian\n$H={1\\over2}p^2+{1\\over2}x^2 +i\\epsilon x^3$ using perturbative techniques. It\nis also shown how to construct the operator C for\n$H={1\\over2}p^2+{1\\over2}x^2-\\epsilon x^4$ using nonperturbative methods.\n