content_image/316603.jpg content_image/316604.jpg content_image/316602.jpg content_image/316605.jpg
Where 

and 

<img src='content_image/316603.jpg'>

H denotes the Hermitian partner Hamiltonian in Hilbert space with real eigen- values, therefore. 

Under such settings, one can easily show that ηxη − 1 = ± ix (i.e., x → ± ix , which practically imitates our original transformation toy above) and con- sequently a non-Hermitian PT -symmetric potential V$_{PT}$ ( x ) would be trans- formed into its real-valued (by the virtue of equation (2)) partner potential V ( ± ix ; m$_{+}$ ) ∈ R through the relation 

<img src='content_image/316604.jpg'>

On the other hand, the proof of the related isospectrality between H$_{PT}$$_{,m}$$_{+}$ in (25) and its Hermitian partner Hamiltonian H$_{m}$$_{−}$ in (26) seems to be a straight- forward one. Let E$_{n,m}$$_{+}$ and Ψ$_{n}$ ( x ; m$_{+}$ ) be the eigenvalues and eigenfunctions of the complex PT -symmetric Hamiltonian H$_{PT}$$_{,m}$$_{+}$ , respectively, then 

<img src='content_image/316602.jpg'>

<img src='content_image/316605.jpg'>

