×

zbMATH — the first resource for mathematics

Semi-classical limit of Schrödinger–Poisson equations in space dimension \(n\geqslant 3\). (English) Zbl 1107.35018
Summary: We prove the existence of solutions to the Schrödinger-Poisson system on a time interval independent of the Planck constant, when the doping profile does not necessarily decrease at infinity, in the presence of a subquadratic external potential. The lack of integrability of the doping profile is resolved by working in Zhidkov spaces, in space dimension at least three. We infer that the main quadratic quantities (position density and modified momentum density) converge strongly as the Planck constant goes to zero. When the doping profile is integrable, we prove pointwise convergence.

MSC:
35B40 Asymptotic behavior of solutions to PDEs
35C20 Asymptotic expansions of solutions to PDEs
35Q40 PDEs in connection with quantum mechanics
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
82D37 Statistical mechanical studies of semiconductors
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Bohun, S.; Illner, R.; Lange, H.; Zweifel, P.F., Error estimates for Galerkin approximations to the periodic schrödinger – poisson system, Z. angew. math. mech., 76, 1, 7-13, (1996) · Zbl 0864.65062
[2] Brezzi, F.; Markowich, P.A., The three-dimensional wigner – poisson problem: existence, uniqueness and approximation, Math. methods appl. sci., 14, 1, 35-61, (1991) · Zbl 0739.35080
[3] Carles, R., Linear vs. nonlinear effects for nonlinear Schrödinger equations with potential, Commun. contemp. math., 7, 4, 483-508, (2005) · Zbl 1095.35044
[4] R. Carles, Geometric optics and instability for semi-classical Schrödinger equations, Arch. Ration. Mech. Anal. (2006), doi: 10.1007/s00205-006-0017-5, in press
[5] R. Carles, WKB analysis for nonlinear Schrödinger equations with potential, Comm. Math. Phys. (2006), doi: 10.1007/s00220-006-0077-2, in press
[6] Castella, F., \(L^2\) solutions to the schrödinger – poisson system: existence, uniqueness, time behaviour, and smoothing effects, Math. models methods appl. sci., 7, 8, 1051-1083, (1997) · Zbl 0892.35141
[7] Chemin, J.-Y., Perfect incompressible fluids, Oxford lecture ser. math. appl., vol. 14, (1998), Oxford Univ. Press New York, translated from the 1995 French original by I. Gallagher, D. Iftimie
[8] Cycon, H.L.; Froese, R.G.; Kirsch, W.; Simon, B., Schrödinger operators with application to quantum mechanics and global geometry, Texts monogr. phys., (1987), Springer-Verlag Berlin · Zbl 0619.47005
[9] Dereziński, J.; Gérard, C., Scattering theory of quantum and classical N-particle systems, Texts monogr. phys., (1997), Springer-Verlag Berlin · Zbl 0899.47007
[10] Fujiwara, D., Remarks on the convergence of the Feynman path integrals, Duke math. J., 47, 3, 559-600, (1980) · Zbl 0457.35026
[11] Gallo, C., Schrödinger group on zhidkov spaces, Adv. differential equations, 9, 5-6, 509-538, (2004) · Zbl 1103.35093
[12] Gérard, P., Remarques sur l’analyse semi-classique de l’équation de Schrödinger non linéaire, (), pp. Exp. No. XIII, 13 · Zbl 0874.35111
[13] Gérard, P., The Cauchy problem for the gross – pitaevskii equation, Ann. inst. H. Poincaré anal. non linéaire, 23, 5, 765-779, (2006) · Zbl 1122.35133
[14] Gosse, L.; Mauser, N.J., Multiphase semiclassical approximation of an electron in a one-dimensional crystalline lattice. III. from ab initio models to WKB for schrödinger – poisson, J. comput. phys., 211, 1, 326-346, (2006) · Zbl 1081.81041
[15] Grenier, E., Semiclassical limit of the nonlinear Schrödinger equation in small time, Proc. amer. math. soc., 126, 2, 523-530, (1998) · Zbl 0910.35115
[16] Hörmander, L., The analysis of linear partial differential operators. I, (1990), Springer-Verlag Berlin, Distribution theory and Fourier analysis
[17] Lannes, D., Sharp estimates for pseudo-differential operators with symbols of limited smoothness and commutators, J. funct. anal., 232, 2, 495-539, (2006) · Zbl 1099.35191
[18] Lin, F.; Zhang, P., Semiclassical limit of the gross – pitaevskii equation in an exterior domain, Arch. ration. mech. anal., 179, 1, 79-107, (2005) · Zbl 1079.76016
[19] Majda, A., Compressible fluid flow and systems of conservation laws in several space variables, Appl. math. sci., vol. 53, (1984), Springer-Verlag New York · Zbl 0537.76001
[20] Markowich, P.A.; Ringhofer, C.A.; Schmeiser, C., Semiconductors equations, (1990), Springer-Verlag Wien
[21] Sulem, C.; Sulem, P.-L., The nonlinear Schrödinger equation, self-focusing and wave collapse, (1999), Springer-Verlag New York · Zbl 0928.35157
[22] Taylor, M., Partial differential equations. III, Appl. math. sci., vol. 117, (1997), Springer-Verlag New York, Nonlinear equations
[23] Zhang, P., Wigner measure and the semiclassical limit of schrödinger – poisson equations, SIAM J. math. anal., 34, 3, 700-718, (2002) · Zbl 1032.35132
[24] Zhang, P.; Zheng, Y.; Mauser, N.J., The limit from the schrödinger – poisson to the vlasov – poisson equations with general data in one dimension, Comm. pure appl. math., 55, 5, 582-632, (2002) · Zbl 1032.81011
[25] P.E. Zhidkov, The Cauchy problem for a nonlinear Schrödinger equation, JINR Commun., P5-87-373, Dubna, 1987 (in Russian)
[26] Zhidkov, P.E., Korteweg – de Vries and nonlinear Schrödinger equations: qualitative theory, Lecture notes in math., vol. 1756, (2001), Springer-Verlag Berlin · Zbl 0987.35001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.