MR2109646 Lévai, Géza . Supersymmetry without Hermiticity.
Czechoslovak J. Phys. 54 (2004), no. 10, 1121--1124.



The subject of this text belongs to the so called supersymmetric
quantum mechanics [where, typically, a Hamiltonian $K$ of a system
$S$ coincides with one of generators of a graded Lie algebra, say,
$sl(1|1)$ in the models in question] and to the so called
$PT-$symmetric quantum mechanics (where, typically, a one-particle
Hamiltonian $H= T + V$ in one dimension is merely
parity-pseudo-Hermitian, $H^\dagger = P H P \neq H$). In the
former context, a key to applications lies in a representation of
$K$ as a direct sum of two (traditionally, Hermitian) ``partner
Hamiltonians" $H^{(\pm)}$. Typically, we may ``start" from a
known, solvable potential $V^{(-)}(x)$ and {\em construct} a new,
``partner" potential $V^{(+)}(x)$ as well as all its bound state
solutions in closed form.
 \par
In the spirit of several independent proposals of a combination of
the two formalisms [I cannot resist self-citing M. Z. et al, Phys.
Lett. B 483 (2000) 284-9 and M. Z. Czech. J. Phys. 51 (2001) 420-8
at least] the author explains how the supersymmetric partnership
works in the parity-pseudo-Hermitian setting. Discussing the
explicit construction (via so called superpotentials $W$) he
emphasizes that the requirement of $PT-$symmetry of $V^{(-)}(x)$
may though need not be preserved for the partner $V^{(+)}(x)$.
These observations are illustrated using a few explicit examples
of the Rosen-Morse type. The study nicely complements the author's
review paper [3] (where the page update 10179 - 92 should be
added) as well as his recent Doctor-of-Science dissertation.