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

详细信息    查看全文

• 作者：José ; Luis L&#xf3 ; pez ; ^a ; ^{jllopez@ugr.es"" rel=""nofollow} ; J. Montejo-Gá ; mez^a ; ^{jmontejo@ugr.es"" rel=""nofollow}
• 关键词：Quantum open system ; Dissipative quantum mechanics ; Madelung decomposition of the wavefunction ; Schr&#xf6 ; dinger– ; Langevin operator ; Kostin equation ; Quantum Fokker– ; Planck equation ; Caldeira– ; Leggett master equation
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2011
• 出版时间：15 November 2011
• 年：2011
• 卷：383
• 期：2
• 页码：365-378
• 全文大小：227 K

文摘

In this paper we make it mathematically rigorous the formulation of the following quantum Schrödinger–Langevin nonlinear operator for the wavefunction in bounded domains via its mild interpretation. The a priori ambiguity caused by the presence of the multi-valued potential λS_ψ, proportional to the argument of the complex-valued wavefunction is circumvented by subtracting its positional expectation value, 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 is the (reduced) Planck constant, m is the particle mass, λ is a friction coefficient, n_ψ=ψ² is the local probability density, denotes the electric current density, and Θ 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 Schrödinger–Langevin operator 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]).