content_image/325962.jpg content_image/325964.jpg content_image/325965.jpg content_image/325966.jpg content_image/325963.jpg
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) 

<img src='content_image/325962.jpg'>

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 

<img src='content_image/325964.jpg'>

<img src='content_image/325965.jpg'>

<img src='content_image/325966.jpg'>

<img src='content_image/325963.jpg'>

