The paper develops, in the context of the quantum models exhibiting
a position-dependent mass, the idea of studying (Hamiltonian)
operators which are similar to their conjugates. This idea (which
dates back, at least, to Dieudonne, cf. J. Dieudonne,
Quasi-Hermitian operators, in Proc. Int. Symp. Lin. Processes, 1960,
Jerusalem Academic Press) found recently applications in physics.
The authors consider the particular (viz., the first-order
differential) choice of the similarity intertwinner $\eta$ as
proposed a few years ago [cf. E. Caliceti, F. Cannata, M. Znojil and
A. Ventura, Construction of PT-asymmetric non-Hermitian Hamiltonians
with CPT- symmetry.  Phys. Lett. A 335 (2005) 26-30, and B. Bagchi,
A. Banerjee, E. Caliceti, F. Cannata, H. B. Geyer, C. Quesne and M.
Znojil, CPT-conserving Hamiltonians and their nonlinear
supersymmetrization using differential charge-operators C, Int. J.
Mod. Phys. A 20 (2005) 7107 - 7128]. Selected technical details are
illustrated using the Scarf II and Samsonov-Roy interaction
potentials.

MR2396775 Mustafa, Omar; Mazharimousavi, S. Habib
First-order intertwining operators with position dependent mass
and $\eta$-weak-pseudo-Hermiticity generators.
Internat. J. Theoret. Phys.  47  (2008), no. 2, 446--454. 81Q10