References
- Coffey WT, Kalmykov YP, Titov SV, Mulligan BP. Wigner function approach to the quantum Brownian motion of a particle in a potential. Phys. Chem. Chem. Phys. 2007;9:3361–3382.
- Elk M, Lambropoulos P. Connection between approximate Fokker--Planck equations and the Wigner function applied to the micromaser. Quantum Semiclass. Opt. 1996;8:23–37.
- Donarini A, Novotn\’{y} T, Jauho AP. Simple models suffice for the single-dot quantum shuttle. New J. Phys. 2005;7:237–262.
- Castella F. The Vlasov--Poisson--Fokker--Planck system with infinite kinetic energy. Indiana Univ. Math. J. 1998;47:939–964.
- Carrillo JA, Soler J, Vázquez JL. Asymptotic behaviour and self-similarity for the three dimensional Vlasov--Poisson--Fokker--Planck system. J. Funct. Anal. 1996;141:99–132.
- Arnold A, Dhamo E, Manzini C. The Wigner--Poisson--Fokker--Planck system: global-in-time solution and dispersive effects. Ann. Inst. Henri Poincaré. 2007;24:645–676.
- Bosi R. Classical limit for linear and nonlinear quantum Fokker--Planck systems. Commun. Pure Appl. Anal. 2009;8:845–870.
- Castella F, Erdös L, Frommlet F, et al. Fokker--Planck equations as scaling limit of reversible quantum systems. J. Stat. Phys. 2000;100:543–601.
- Sparber C, Carrillo JA, Dolbeault J, et al. On the long time behavior of the quantum Fokker--Planck equation. Monatsh. Math. 2004;141:237–257.
- Cañizo JL, López JL, Nieto J. Global L1 theory and regularity for the 3D nonlinear Wigner--Poisson--Fokker--Planck system. J. Differ. Equ. 2004;198:356–373.
- Arnold A, Carrillo JA, Dhamo E. On the periodic Wigner--Poisson--Fokker--Planck system. J. Math. Anal. Appl. 2002;275:263–276.
- Arnold A, Lopez JL, Markowich PA, et al. An analysis of quantum Fokker--Planck models: a Wigner function approach. Rev. Mat. Iberoam. 2004;20:771–814.
- Arnold A, Sparber C. Quantum dynamical semigroups for diffusion models with Hartree interaction. Commun. Math. Phys. 2004;251:179–207.
- Arnold A, Dhamo E, Manzini C. Dispersive effects in quantum kinetic equations. Indiana Univ. Math. J. 2007;56:1299–1332.
- Arnold A, Gamba IM, Gualdani MP, et al. The Wigner--Fokker--Planck equation: stationary states and large time behavior. Math. Models Methods Appl. Sci. 2012;22:1250034–1250065.
- Bellazzini J, Bonanno C. Nonlinear Schrödinger equations with strongly singular potentials. Proc. R. Soc. Edinburgh, Sect. A Math. 2010;140:707–721.
- Burq N, Planchon F, Stalker JG, et al. Strichartz estimates for the wave and Schrödinger equations with the inverse-square potential. J. Funct. Anal. 2003;203:519–549.
- Deng Y, Jin L, Peng S. Solutions of Schrödinger equations with inverse square potential and critical nonlinearity. J. Differ. Equ. 2012;253:1376–1398.
- Jiang YS, Zhou HS. Multiple solutions for a Schrödinger--Poisson--Slater equation with external Coulomb potential. Sci. China Math. 2014;57:1163–1174.
- Jiang YS, Zhou HS. Schrödinger--Poisson equations with singular potentials in R3. 2012. Available from: http://arxiv.org/pdf/1204.1825.pdf.
- Jiang YS, Zhou HS, Wiwatanapataphee B, et al. Nonexistence results for the Schrödinger--Poisson equations with spherical and cylindrical potentials in R3. Abstract Appl. Anal. 2013;2013:1–6.
- Wigner E. On the quantum correction for the thermodynamic equilibrium. Phys. Rev. 1932;40:749–759.
- Markowich PA. On the equivalence of the Schröinger and the quantum Liouville equations. Math. Methods Appl. Sci. 1989;11:459–469.
- Illner R, Lange H, Zweifel P. Global existence, uniqueness, and asymptotic behaviour of solutions of the Wigner--Poisson and Schrödinger systems. Math. Methods Appl. Sci. 1994;17:349–376.
- Manzini C. The three dimensional Wigner--Poisson problem with inflow boundary conditions. J. Math. Anal. Appl. 2006;313:184–196.
- Manzini C. Quantum kinetic models of open quantum systems in semiconductor theory [PhD thesis]. Münster: Westfälischen Wilhelms-Universität Münster; 2005.
- Durante T, Rhandi A. On the essential self-adjointness of Ornstein--Uhlenbeck operators perturbed by inverse-square potentials. Discrete Cont. Dyn. Syst. S. 2013;6:649–655.
- Goldstein GR, Goldstein JA, Rhandi A. Weighted Hardy’s inequality and the Kolmogorov equation perturbed by an inverse-square potential. Appl. Anal. 2011;91:2057–2071.
- Arnold A, Markowich P, Toscani G, et al. On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker--Planck type equations. Commun. Partial Differ. Equ. 2001;26:43–100.
- Pazy A. Semigroups of linear operators and applications to partial differential equations. Berlin: Springer; 1983.
- Reed M, Simon B. Methods of modern mathematical physics I, functional analysis. New York: Academic Press; 1980.
- Barletti L. A mathematical introduction to the Wigner formulation of quantum mechanics. B. Unione Mat. Ital. 2003;6B:693–716.