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.
Abstract A classical field theory for a Schrödinger equation with a non-Hermitian Hamiltonian describing a particle with position-dependent mass has been recently advanced by Nobre and Rego–Monteiro (NR) [Phys. Rev. A 88 (2013) 032105 ]. This field theory is based on a variational principle involving the wavefunction Ψ(x,t) and an auxiliary field ϕ(x,t). It is here shown that the relation between the dynamics of the auxiliary field ϕ(x,i) and that of the original wavefunction Ψ(x,t) is deeper than suggested by the NR approach. Indeed, we formulate a variational principle for the aforementioned Schrödinger equation which is based solely on the wavefunction Ψ(x,t). A continuity equation for an appropriately defined probability density, and the concomitant preservation of the norm, follows from this variational principle via Noether's theorem. Moreover, the norm-conservation law obtained by NR is reinterpreted as the preservation of the inner product between pairs of solutions of the variable mass Schrödinger equation.