  We introduce and study a class of non-Hermitian Hamiltonians which have
velocity dependent potentials. Since stability can not be advocated directly
from the classical potential, we show that the energy spectra are real and
bounded from below which proves the stability of the spectra of all members in
the class. We find that the introduced class of non-Hermitian Hamiltonians do
have a corresponding superpartner class of non-Hermitian Hamiltonians. We were
able to introduce supercharges which in conjunction with the corresponding
super Hamiltonians constitute a closed super algebra. Among the introduced
Hamiltonians, we show that non-$\mathcal{PT }$-symmetric Hamiltonians can be
transformed into their corresponding superpartner Hamiltonians via a specific
canonical transformation while the $\mathcal{PT }$-symmetric ones failed to be
mapped to their corresponding superpartner Hamiltonians via the same canonical
transformation. Since canonical transformations preserve the spectrum, we
conclude that non-$\mathcal{PT }$-symmetric Hamiltonians out of the introduced
class of Hamiltonians have the same spectrum as the corresponding superpartner
Hamiltonians and thus Susy is broken for such Hamiltonians. This kind of
intertwining of $\mathcal{PT }$-symmetry and Supersymmetry is new as all the so
far discussed cases concentrate on Hamiltonians of broken $\mathcal{PT }%
$-symmetry that have broken Supersymmetry too while we showed that Susy can be
also broken for non-$\mathcal{PT }$-symmetric and non-Hermitian Hamiltonians .

Quantum Physics