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just happen to have their partners that are strictly equivalent to real potentials after being exposed to some supersymmetric quantum mechanical treatment [11] or integral, Fourier-like transformation [12]. Jones and Mateo [4] have, moreover, used a Darboux-type similarity transformation and have shown that for the Bender’s and Boettcher’s [1] non-Hermitian PT -symmetric Hamiltonian H = p 2 − g ( ix ) N ; N = 4, there exists an equivalent Hermitian Hamiltonian h = σ − $^{1}$Hσ ; σ = exp ( Q/ 2), where σ is Hermitian and positive definite. Similar proposal was carried out by Bender et al. [3]. For more details the reader is advised to refer to [3,4]. In our current methodical proposal, we try to have our input in this direction and fill this gap partially, at least. 

Through the forthcoming proposition (in section 2) or through a similarity transformation (in section 3) with a metric operator η (defined in (21) below) we report that for every non-Hermitian complex PT -symmetric Hamiltonian (with positive mass m = m$_{+}$ = + | m | ) there exists a Hermitian partner Hamil- tonian (with negative mass m = m$_{−}$ = − | m | ) in Hilbert space L $^{2}$( R ) = H . In section 3, we also discuss isospectrality and orthonormalization conditions as- sociated with both the Hermitian partner (not necessarily PT -symmetric) and the non-Hermitian PT -symmetric Hamiltonians. An obvious correspondence is constructed, therein. This has not been discussed elsewhere, to the best of our knowledge. We give our concluding remarks in section 4. 

## 2 A transformation toy: x −→ ± iy ; x, y ∈ R 

In connection with an over simplified transformation toy x −→ ± iy ∈ i R ; x, y ∈ R ( x −→ ± iy to be understood as x −→ + iy and/or x −→ − iy ), t’ Hooft and Nobbenhuis [44] have used a complex space-time symmetry transformation 

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