Where and 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 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 H$_{PT}$ = p 2 x 2 m$_{+}$ + V ( x ; m$_{+}$ ) ; V ( x ; m$_{+}$ ) = [ V ( − x ; m$_{+}$ )] ∗ ∈ C , (25) H = p 2 x 2 m$_{−}$ + V ( ix ; m$_{−}$ ) ; V ( ix ; m$_{−}$ ) = V ( x ; m$_{−}$ ) = V $^{∗}$( x ; m$_{−}$ ) ∈ R . (26) η − $^{1}$V$_{PT}$ ( x ) η = η − $^{1}$V ( x ; m$_{+}$ ) η = V ( ± ix ; m$_{+}$ ) = [ V ( ± ix ; m$_{+}$ )] ∗ ∈ R . (27) H$_{PT}$$_{,m}$$_{+}$ Ψ$_{n}$ ( x ; m$_{+}$ ) = E$_{n,m}$$_{+}$ Ψ$_{n}$ ( x ; m$_{+}$ ) = ⇒ η − $^{1}$ηH$_{m}$$_{+}$ [ η − $^{1}$Ψ$_{n}$ ( x ; m$_{+}$ ) ] = E$_{n,m}$$_{+}$ [ η − $^{1}$Ψ$_{n}$ ( x ; m$_{+}$ ) ] = ⇒ H$_{m}$$_{+}$ Φ$_{n}$ ( x ; m$_{+}$ ) = E$_{n,m}$$_{+}$ Φ$_{n}$ ( x ; m$_{+}$ ) = ⇒ H$_{m}$$_{−}$ Φ$_{n}$ ( x ; m$_{−}$ ) = E$_{n,m}$$_{−}$ Φ$_{n}$ ( x ; m$_{−}$ ) , (28) 10