  We analyze a general family of position-dependent mass quantum Hamiltonians
which are not self-adjoint and include, as particular cases, some Hamiltonians
obtained in phenomenological approaches to condensed matter physics. We build a
general family of self-adjoint Hamiltonians which are quantum mechanically
equivalent to the non self-adjoint proposed ones. Inspired in the probability
density of the problem, we construct an ansatz for the solutions of the family
of self-adjoint Hamiltonians. We use this ansatz to map the solutions of the
time independent Schrodinger equations generated by the non self-adjoint
Hamiltonians into the Hilbert space of the solutions of the respective dual
self-adjoint Hamiltonians. This mapping depends on both the position-dependent
mass and on a function of position satisfying a condition that assures the
existence of a consistent continuity equation. We identify the non self-adjoint
Hamiltonians here studied to a very general family of Hamiltonians proposed in
a seminal article of Harrison [1] to describe varying band structures in
different types of metals. Therefore, we have self-adjoint Hamiltonians that
correspond to the non self-adjoint ones found in Harrison's article. We analyze
three typical cases by choosing a physical position-dependent mass and a
deformed harmonic oscillator potential . We completely solve the Schrodinger
equations for the three cases; we also find and compare their respective energy
levels.
