Classical field theory for a non-Hermitian Schrödinger equation with position-dependent masses

Abstract

A linear one-dimensional Schr\"odinger equation, defined by means of a non-Hermitian Hamiltonian characterized by position-dependent masses, was proposed lately. Herein we present an exact classical field theory for this equation, showing the need for an extra field $\ensuremath{\Phi}(x,t)$, in addition to the usual one, $\ensuremath{\Psi}(x,t)$, similar to what was done recently in the analysis of a class of nonlinear quantum equations. These generalizations of the Schr\"odinger equation depend on an index $q$, in such a way that the standard case is recovered in the limit $q\ensuremath{\rightarrow}1$. Particularly, the field $\ensuremath{\Phi}(x,t)$ becomes ${\ensuremath{\Psi}}^{*}(x,t)$ only when $q\ensuremath{\rightarrow}1$ and satisfies a similar Schr\"odinger equation for the Hermitian conjugate of the Hamiltonian operator. In terms of these two fields one may define a probability density following a standard continuity equation, leading to the preservation of probability in Cartesian space. Simple applications are performed by solving the equations for the two fields.