content_image/325927.jpg content_image/325928.jpg content_image/325924.jpg content_image/325926.jpg content_image/325925.jpg
parametrization recipe does the trick, therefore. 

Such observations would unavoidably manifest the following proposition. 

Proposition 1 For every non-Hermitian complex PT -symmetric Hamiltonian with positive mass (i.e., m = m$_{+}$ = + | m | ) there exists a Hermitian (but not necessarily isospectral neither necessarily PT -symmetric ) partner Hamiltonian with negative mass (i.e., m = m$_{−}$ = − | m | ) in Hilbert space L $^{2}$( R ) = H . 

Proof. Let 

be a non-Hermitian complex PT -symmetric Hamiltonian (with m = + | m | ) with a corresponding PT -symmetric eigenfunctions Ψ ( x ; m$_{+}$ ) such that 

Then a mapping of the sort 

would imply 

<img src='content_image/325927.jpg'>

where the substitution m$_{+}$ = − m$_{−}$ is used and 

<img src='content_image/325928.jpg'>

which is Hermitian ( but not necessarily isospectral with H$_{x}$$_{;}$$_{m}$$_{+}$ of (12) neither 

<img src='content_image/325924.jpg'>

<img src='content_image/325926.jpg'>

<img src='content_image/325925.jpg'>

