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

Autor: Jesús Montejo-Gámez, José L. López
Rok vydání: 2011
Předmět:
Zdroj: Journal of Mathematical Analysis and Applications. 383:365-378
ISSN: 0022-247X
Popis: In this paper we make it mathematically rigorous the formulation of the following quantum Schrodinger–Langevin nonlinear operator for the wavefunction A QSL = i ℏ ∂ t + ℏ 2 2 m Δ x − λ ( S ψ − 〈 S ψ 〉 ) − Θ ℏ [ n ψ , J ψ ] in bounded domains via its mild interpretation. The a priori ambiguity caused by the presence of the multi-valued potential λ S ψ , proportional to the argument of the complex-valued wavefunction ψ = | ψ | exp { i ℏ S ψ } , is circumvented by subtracting its positional expectation value, 〈 S ψ 〉 ( t ) : = ∫ Ω S ψ ( t , x ) n ψ ( t , x ) d x , as motivated in the original derivation (Kostin, 1972 [45] ). The problem to be solved in order to find S ψ 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, n ψ = | ψ | 2 is the local probability density, J ψ = ℏ m Im ( ψ ¯ ∇ x ψ ) denotes the electric current density, and Θ ℏ is a general operator (eventually nonlinear) that only depends upon the macroscopic observables n ψ and J ψ . 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 Schrodinger–Langevin equation with Poisson coupling and enthalpy dependence (Jungel et al., 2002 [41] ).
Databáze: OpenAIRE
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