On a rigorous interpretation of the quantum Schrödinger–Langevin operator in bounded domains with applications

In this paper we make it mathematically rigorous the formulation of the following quantum Schrödinger–Langevin nonlinear operator for the wavefunctionAQSL=iℏ∂t+ℏ22mΔx−λ(Sψ−〈Sψ〉)−Θℏ[nψ,Jψ] in bounded domains via its mild interpretation. The a priori   ambiguity caused by the presence of the multi-valued potential λSψλSψ, proportional to the argument of the complex-valued wavefunctionψ=|ψ|exp{iℏSψ}, is circumvented by subtracting its positional expectation value,〈Sψ〉(t):=∫ΩSψ(t,x)nψ(t,x)dx, as motivated in the original derivation (Kostin, 1972 [45]). The problem to be solved in order to find SψSψ is mostly deduced from the modulus-argument decomposition of ψ and dealt with much like in Guerrero et al. (2010) [37]. Here ℏ is the (reduced) Planck constant, m is the particle mass, λ   is a friction coefficient, nψ=|ψ|2nψ=|ψ|2 is the local probability density, Jψ=ℏmIm(ψ¯∇xψ) denotes the electric current density, and ΘℏΘℏ is a general operator (eventually nonlinear) that only depends upon the macroscopic observables nψnψ and JψJψ. In this framework, we show local well-posedness of the initial-boundary value problem associated with the Schrödinger–Langevin operator AQSLAQSL in bounded domains. In particular, all of our results apply to the analysis of the well-known Kostin equation derived in Kostin (1972) [45] and of the Schrödinger–Langevin equation with Poisson coupling and enthalpy dependence (Jüngel et al., 2002 [41]).