On a rigorous interpretation of the quantum Schrodinger-Langevin operator in bounded domains with applications

In this paper we make it mathematically rigorous the formulation of the following Received 2 March 2011 quantum Schrodinger-Langevin nonlinear operator for the wavefunction A(QSL) = ih partial derivative(t) + h(2)/2m Delta(x) - lambda(S-psi - < S-psi >) -Theta(h)[n(psi), J(psi)] in bounded domains via its mild interpretation. The a priori ambiguity caused by the presence of the multi-valued potential lambda S-psi, proportional to the argument of the complex-valued wavefunction psi = vertical bar psi vertical bar exp{i/hS psi} as motivated in the original derivation (Kostin, 1972 [45]). The problem to be solved in order to find S-psi is mostly deduced from the modulus-argument decomposition of psi and dealt with much like in Guerrero et al. (2010)[37]. Here h is the (reduced) Planck constant, m is the particle mass, A is a friction coefficient, n(psi) =vertical bar psi vertical bar(2) is the local probability density, J(psi) = h/m Im((psi) over bar Delta(x)psi) denotes the electric current density, and Theta(h) is a general operator (eventually nonlinear) that only depends upon the macroscopic observables n(psi) and J(psi). In this framework, we show local well-posedness of the initial-boundary value problem associated with the Schrodinger-Langevin operator A(QSL) 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 Schrtidinger-Langevin equation with Poisson coupling and enthalpy dependence (Jungel et al., 2002 [41]). (C) 2011 Elsevier Inc. All rights reserved.