necessarily PT -symmetric). QED. Illustrative examples are ample. In the complex ”shifted by an imaginary constant” PT -symmetric oscillator Hamiltonian (cf., e.g., Mustafa and Znojil [18]) a companied by a properly regularized attractive/repulsive core (with the mass term kept intact) would, under the transformation x −→ ± iy and with imply Which is not only real valued but also PT -symmetric (with parity performing reflection about y = ± c rather than y = 0). In such a case, where Obviously, H$_{y}$$_{;}$$_{m}$$_{−}$ is not only Hermitian but also PT -symmetric and shares the 7 H$_{x}$$_{;}$$_{m}$$_{+}$ = p 2 x 2 m$_{+}$ + V ( x ; m$_{+}$ ) = p 2 x 2 m$_{+}$ + m$_{+}$ω 2 2 ( x − ic ) 2 + G ( m$_{+}$, α ) ( x − ic ) 2 , (16) G ( m$_{+}$, α ) = ℏ 2 ( α 2 − 1 / 4 ) 2 m$_{+}$ , V ( y ; m$_{+}$ ) = − m$_{+}$ω 2 2 ( y ∓ c ) 2 − ℏ 2 2 m$_{+}$ ( α 2 − 1 / 4 ) ( y ∓ c ) 2 ∈ R . (17) H$_{x}$$_{;}$$_{m}$$_{+}$ −→ H$_{y}$$_{;}$$_{m}$$_{−}$ = − H$_{y}$$_{;}$$_{m}$$_{+}$ ∈ L $^{2}$( R ) = H H$_{y}$$_{;}$$_{m}$$_{−}$ = P 2 y 2 m$_{−}$ + V ( y ; m$_{−}$ ) = P 2 y 2 m$_{−}$ + m$_{−}$ω 2 2 ( y ∓ c ) 2 + ℏ 2 2 m$_{−}$ ( α 2 − 1 / 4 ) ( y ∓ c ) 2 . (18)