Position-dependent mass quantum Hamiltonians: General approach and duality

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.