Non-Hermitian PT Symmetric Hamiltonian with Position-Dependent Masses: Associated Schrödinger Equation and Finite-Norm Solutions

Abstract

A one-dimensional non-Hermitian PT symmetric Hamiltonian, characterized by position-dependent masses, defines a Schrödinger equation in terms of a field Ψ(x, t). Based on an exact classical field theory, the necessity of an extra field Φ(x, t) (which satisfies a conjugate equation and in general different is from Ψ∗(x, t)) is shown. Simple applications are investigated by solving analytically both equations and it is shown that the effective masses proposed lead to a probability density characterized by a finite norm, typical of the physical situation that occurs with the concentration of electrons in some semiconductor heterojunctions. An extension to a three-dimensional space is also presented.