between de-Sitter and anti-de-Sitter space to identify vacuum solutions with zero cosmological constant (used later on by Assis and Fring [45] to provide a simple proof of the reality of the spectrum of p 2 + z $^{2}$( iz ) 2 m $^{+1}$). However, in their instructive harmonic oscillator [44] example with the annihilation and creation operators they have shown (using x −→ iy, p$_{x}$ → − ip$_{y}$ ) that the corresponding Hamilto- nian reads with Under such settings, H$_{x}$ −→ − H$_{y}$ and whilst the eigenvalues of H$_{x}$ are [ ω ( n + 1 / 2)] those of H$_{y}$ read [ − ω ( n + 1 / 2)]. Consequently, the ground state ∼ exp ( − mωx $^{2}$/ 2 ) in the x -space is normalizable, whereas the ground state ∼ exp ( + mωy $^{2}$/ 2 ) in the y -space is non-normalizable. Within similar spiritual lines, Tanaka [46] has shown that a transformation of the form would map a non-Hermitian PT -symmetric potential V ( x ) ∈ C (or any non- Hermitian PT -symmetric function f ( x ) ∈ C in general, so to speak) into a Hermitian (but not necessarily PT -symmetric) potential V ( y ) ∈ R . The proof of which is straightforward. Using equation (1), one would write (with z = − iy 4 H$_{x}$ = p 2 x 2 m + 1 2 mω $^{2}$x2 = ω ( a † $_{x}$a$_{x}$ + 1 2 ) , (2) a$_{x}$ = √ 1 2 mω ( mωx + ip$_{x}$ ) , a † x = √ 1 2 mω ( mωx − ip$_{x}$ ) , (3) H$_{y}$ = ω ( a † $_{y}$a$_{y}$ − 1 2 ) , (4) a$_{y}$ = √ 1 2 mω ( mωy + ip$_{y}$ ) , a † y = √ 1 2 mω ( − mωy + ip$_{y}$ ) . (5) x ∈ R −→ − iy ∈ i R , p$_{x}$ −→ ip$_{y}$ ∈ i R (6)