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

A linear one-dimensional Schrodinger 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 Phi(x, t), in addition to the usual one, Psi(x, t), similar to what was done recently in the analysis of a class of nonlinear quantum equations. These generalizations of the Schrodinger equation depend on an index q, in such a way that the standard case is recovered in the limit q -> 1. Particularly, the field Phi(x, t) becomes Psi*(x, t) only when q -> 1 and satisfies a similar Schrodinger 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.