Global solutions of the mean–field, very high temperature Caldeira–Leggett master equation
Authors:
José Luis López and Juanjo Nieto
Journal:
Quart. Appl. Math. 64 (2006), 189-199
MSC (2000):
Primary 35Q40, 35S10, 81Q99, 81V99
DOI:
https://doi.org/10.1090/S0033-569X-06-01005-3
Published electronically:
January 24, 2006
MathSciNet review:
2211384
Full-text PDF Free Access
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Additional Information
Abstract: In this paper, global well–posedness as well as regularity of very high temperature Caldeira–Leggett models with repulsive Poisson coupling are proved by using Green function techniques and Fokker–Planck smoothing arguments along with kinetic energy and elliptic estimates.
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Ar Arnold, A.: Self–consistent relaxation–time models in quantum mechanics. Comm. Part. Diff. Equ. 21 (1996), 473–506.
ALMS Arnold, A., López, J.L., Markowich, P.A. and Soler, J.: An analysis of Wigner–Fokker–Planck models: a Wigner function approach, Rev. Mat. Iberoamericana 20 (2004), 771–814.
ADM Arnold, A., Dhamo, E. and Mancini, C.: The Wigner–Poisson–Fokker–Planck system: global–in–time solutions and dispersive effects, submitted.
CL Caldeira, A.O. and Leggett, A.J.: Path integral approach to quantum Brownian motion. Physica A 121 (1983), 587–616.
CLN Cañizo, J.A., López, J.L. and Nieto, J.: Global $L^1$ theory and regularity for the 3D nonlinear Wigner–Poisson–Fokker–Planck system, J. Diff. Equ. 198 (2004), 356–373.
Dav Davies, E.B.: Quantum theory of open systems, Academic Press, New York, 1976.
Dekker Dekker, H.: Classical and quantum mechanics of the damped harmonic oscillator, Phys. Reports 80 (1981), 1–112.
Diosi Diósi, L.: On high–temperature Markovian equation for quantum Brownian motion. Europhys. Lett. 22 (1993), 1–3.
GJT Gualdani, M.P., Jüngel, A. and Toscani, G.: Exponential decay in time of solutions of the viscous quantum hydrodynamic equations. Appl. Math. Lett. 16 (2003), 1273–1278.
Lin Lindblad, G.: On the generators of quantum dynamical semigroups. Comm. Math. Phys. 48 (1976), 119–130.
Pa Pazy, A.: Semigroups of linear operators and applications to partial differential equations. Applied Mathematical Sciences, Vol. 44 Springer-Verlag, 1983.
S Stein, E.M.: Singular integrals and differentiability properties of functions. Princeton Mathematical Series, No. 30, Princeton University Press, New Jersey, 1970.
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Additional Information
José Luis López
Affiliation:
Departamento de Matemática Aplicada, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain
ORCID:
0000-0002-6050-9015
Email:
jllopez@ugr.es
Juanjo Nieto
Affiliation:
Departamento de Matemática Aplicada, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain
Email:
jjmnieto@ugr.es
Keywords:
Open quantum system,
quantum dissipation,
Fokker–Planck operator,
Wigner equation,
Caldeira–Leggett equation,
Lindblad class,
global solution
Received by editor(s):
August 16, 2005
Published electronically:
January 24, 2006
Article copyright:
© Copyright 2006
Brown University
