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 $η$-pseudo-Hermiticity (Mostafazadeh, quant-ph/0107001) of a Hamiltonian, provided the Hermitian linear automorphism, $η$, is introduced as $e^{-θp}$ which affects an imaginary shift of the co-ordinate : $e^{-θp} x e^{θp} = x+iθ$.