Pseudo-Hermiticity of Hamiltonians under imaginary shift of the co-ordinate : real spectrum of complex potentials

Abstract

We propose that the real spectrum and the orthogonality of the states for several known complex potentials of both types, PT-symmetric and non-PT-symmetric can be understood in terms of currently proposed $\eta$-pseudo-Hermiticity (Mostafazadeh, quant-ph/0107001) of a Hamiltonian, provided the Hermitian linear automorphism, $\eta$, is introduced as $e^{-\theta p}$ which affects an imaginary shift of the co-ordinate : $e^{-\theta p} x e^{\theta p} = x+i\theta$.