Discontinuity, Nonlinearity, and Complexity
On the Well-posedness of the Magnetic, Semi-relativistic Schrödinger-Poisson System
Discontinuity, Nonlinearity, and Complexity 7(3) (2018) 233--241 | DOI:10.5890/DNC.2018.09.002
Vitali Vougalter
University of Toronto, Department of Mathematics,Toronto, Ontario, M5S 2E4, Canada
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Abstract
We prove global existence and uniqueness of strong solutions for the Schrödinger-Poisson system in the repulsive Coulomb case with relativistic, magnetic kinetic energy.
Acknowledgments
V.V. is grateful to F. Gesztesy for the stimulating discussions and to I.M. Sigal for the support.
References
-
[1]  | Barbaroux, J.M. and Vougalter, V. (2016), Existence and nonlinear stability of stationary states for the magnetic Schrödinger-Poisson system, J.Math. Sci. (N.Y.), 219(6), Problems in mathematical analysis, 87, (Russian), 874-898. |
-
[2]  | Anapolitanos, I. (2011), Rate of convergence towards the Hartree-von Neumann limit in the mean-field regime, Lett. Math. Phys., 98(1), 1-31. |
-
[3]  | Anapolitanos, I. and Sigal, I.M., The Hartree-von Neumann limit of many body dynamics, Preprint[http://arxiv.org/abs/0904.4514]. |
-
[4]  | Brezzi, F. and Markowich, P.A. (1991), The three-dimensional Wigner-Poisson problem: existence, uniqueness and approximation,Math. Methods Appl. Sci., 14(1), 35-61. |
-
[5]  | Markowich, P.A., Rein, G., and Wolansky, G. (2002), Existence and nonlinear stability of stationary states of the Schrödinger-Poisson system, J. Statist. Phys., 106(5-6), 1221-1239. |
-
[6]  | Abou Salem, W., Chen, T., and Vougalter, V. (2012), On the well-posedness of the semi-relativistic Schrödinger- Poisson system, Dyn. Partial Differ. Equ., 9(2), 121-132. |
-
[7]  | Abou Salem, W., Chen, T., and Vougalter, V. (2014), Existence and nonlinear stability of stationary states for the semi-relativistic Schrödinger-Poisson system, Ann. Henri Poincare, 15(6), 1171-1196. |
-
[8]  | Barbaroux, J.M. and Vougalter, V. (2017), On the well-posedness of the magnetic Schrödinger-Poisson system in R3, Math. Model. Nat. Phenom., 12(1), 15-22. |
-
[9]  | Steinrück, H. (1991), The one-dimensionalWigner-Poisson problem and its relation to the Schrödinger-Poisson problem, SIAM J. Math. Anal., 22(4), 957C972. |
-
[10]  | Aki, G.L.,Markowich, P.A., and Sparber, C. (2008), Classical limit for semirelativistic Hartree systems, J. Math. Phys., 49(10), 102110, 10pp. |
-
[11]  | Lenzmann, E. (2007),Well-posedness for semi-relativistic Hartree equations of critical type, Math. Phys. Anal. Geom., 10(1), 43-64. |
-
[12]  | Lieb, E.H., Loss, M., and Siedentop, H. (1996), Stability of relativistic matter via Thomas-Fermi theory, Helv. Phys. Acta, 69(5-6), 974-984. |
-
[13]  | Pazy, A. (1983), Semigroups of linear operators and applications to partial differential equations, Springer: Berlin. |
-
[14]  | Reed, M. and Simon, B. (1975), Methods of modern mathematical physics. II. Fourier analysis, self-adjointness, Academic Press, New York, 361pp. |
-
[15]  | Lieb, E.H. and Loss, M. (1997), Analysis, Graduate Studies in Mathematics, 14, American Mathematical Society, Providence, RI, 278pp. |