3 A similarity transformation toy: isospectral- ity and mass signature 9 tion by θ = ± π/ 2 of the full real x -axis and would, effectively, just map a point z$_{1}$ = x into a point z$_{2}$ = ± iy on the imaginary y -axis of the complex z - plane. In the search for a more technical metric operators’ language, one may very well use Ben-Aryeh’s and Barak’s [5] similarity transformation with a metric operator Which transforms a power series into Where G ( x ) is a non-Hermitian PT -symmetric function and satisfies the simi- larity transformation relation η − $^{1}$G ( x ) η = F ( x ) ∈ R . To reflect such a result onto the transformation toy in the above section we choose β = ± π/ 2. This immediately mandates that a non-Hermitian PT -symmetric Hamiltonian H$_{PT}$ can be mapped into its partner Hermitian Hamiltonian H ( but not necessarily isospectral neither necessarily PT -symmetric) through a similarity transforma- tion η = exp ( − iβ x∂$_{x}$ ) ; β, x ∈ R . (21) F ( x ) = ∞ ∑ n =0 A$_{n}$ x n ∈ R , (22) G ( x ) = η F ( x ) η − 1 = ∞ ∑ n =0 A$_{n}$ ( e − $^{iβ}$x ) n ∈ C . (23) η − $^{1}$H$_{PT}$ η = H ⇐⇒ H$_{PT}$ = ηHη − $^{1}$. (24)