arXiv:quant-ph/0211166 · Journal of Physics A Mathematical and General

Calculation of the hidden symmetry operator in -symmetric quantum mechanics

Carl M. Bender, Peter N. Meisinger, Qinghai Wang

Published Feb 5, 2003• 93 citations

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 = 1/2 p 2 + 1/2 x 2 + i x 3 using perturbative techniques. It is also shown how to construct the operator 𝒞 for H = 1/2 p 2 + 1/2 x 2 − x 4 using nonperturbative methods.

DOI: 10.1088/0305-4470/36/7/312

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