parametrization recipe does the trick, therefore. Such observations would unavoidably manifest the following proposition. Proposition 1 For every non-Hermitian complex PT -symmetric Hamiltonian with positive mass (i.e., m = m$_{+}$ = + | m | ) there exists a Hermitian (but not necessarily isospectral neither necessarily PT -symmetric ) partner Hamiltonian with negative mass (i.e., m = m$_{−}$ = − | m | ) in Hilbert space L $^{2}$( R ) = H . Proof. Let be a non-Hermitian complex PT -symmetric Hamiltonian (with m = + | m | ) with a corresponding PT -symmetric eigenfunctions Ψ ( x ; m$_{+}$ ) such that Then a mapping of the sort would imply where the substitution m$_{+}$ = − m$_{−}$ is used and which is Hermitian ( but not necessarily isospectral with H$_{x}$$_{;}$$_{m}$$_{+}$ of (12) neither 6 H$_{x}$$_{;}$$_{m}$$_{+}$ = p 2 x 2 m$_{+}$ + V ( x ; m$_{+}$ ) ; V ( x ; m$_{+}$ ) = V $^{∗}$( − x ; m$_{+}$ ) ∈ C , (12) H$_{x}$$_{;}$$_{m}$$_{+}$ Ψ ( x ; m$_{+}$ ) = E$_{m}$$_{+}$ Ψ ( x ; m$_{+}$ ) . x ∈ R −→ ± iy ∈ i R ⇐⇒ p$_{x}$ −→ ∓ ip$_{y}$ ∈ i R ; x, y ∈ R , (13) H$_{x}$$_{;}$$_{m}$$_{+}$ Ψ ( x ; m$_{+}$ ) = E$_{m}$$_{+}$ Ψ ( x ; m$_{+}$ ) ⇐⇒ H$_{y}$$_{;}$$_{m}$$_{−}$ Φ ( y ; m$_{−}$ ) = E$_{m}$$_{−}$ Φ ( y ; m$_{−}$ ) , (14) H$_{y}$$_{;}$$_{m}$$_{−}$ = p 2 y 2 m$_{−}$ + V ( y ; m$_{−}$ ) ∈ L $^{2}$( R ) ; V ( y ; m$_{−}$ ) = V $^{∗}$( y ; m$_{−}$ ) ∈ R . (15)