4 Concluding remarks Which in turn yields An obvious and immediate correspondence between the regular quantum me- chanical orthonormalization condition (20) and that associated with the non- Hermitian complex PT -symmetric Hamiltonians (22) is constructed, therefore. However, we could not find any example that may satisfy such a condition. The orthonormalizable set of wave functions satisfying this condition is an empty set. This should be anticipated since the normalizable wave functions of the Her- mitian Hamiltonians are not expected to be safely transformed (along with the associated well-defined boundary conditions in Hilbert space) into the complex space. In this work, we have introduced a simple transformation, x −→ ± iy ; x, y ∈ R , that allowed non-Hermitian PT -symmetric Hamiltonians to find their Hermi- tian ( but not necessarily isospectral neither necessarily PT -symmetric) partners in Hilbert space. We have also introduced a similarity transformation recipe (with a metric operator η in (21)) that proved to provide a more mathematical accessibility to the orthonormalization conditions associated with both the Her- mitian (not necessarily PT -symmetric) and the non-Hermitian PT -symmetric (not necessarily isospectral) Hamiltonians. Moreover, the parametrized-mass signature (an almost forgotten and usually deliberately dismissed for the sake of mathematical manipulation simplicity) is shown to play a significant role in the current methodical proposal. An analogous coupling parameter’s recipe is shown to play a similar role as that of 12 〈 Ψ$_{k}$ ( x ) ∣ ∓ iη − 1 [ η − $^{1}$Ψ$_{n}$ ( x ) ] 〉 = δ$_{kn}$ ⇐⇒ 〈 Ψ$_{k}$ ( x ) | Ψ$_{n}$ ( − x ) 〉 = ± iδ$_{kn}$. (31)