We deal with classical and “semiclassical limits” , i.e. vanishing Planck constant , eventually combined with a homogenization limit of a crystal lattice, of a class of “weakly nonlinear” NLS. The Schrödinger-Poisson (S-P) system for the wave functions is transformed to the Wigner-Poisson (W-P) equation for a “phase space function” , the Wigner function. The weak limit of , as tends to , is called the “Wigner measure” (also called “semiclassical measure” by P. Gérard). The mathematically rigorous classical limit from S-P to the Vlasov-Poisson (V-P) system has been solved first by P.L. Lions and T. Paul (1993) and, independently, by P.A. Markowich and N.J. Mauser (1993). There the case of the so called “completely mixed state”, i.e. , was considered where strong additional assumptions can be posed on the initial data. For the so called “pure state” case where only one (or a finite number) of wave functions is considered, recently P. Zhang, Y. Zheng and N.J. Mauser (2002) have given the limit from S-P to V-P in one space dimension for a very weak class of measure valued solutions of V-P that are not unique. For the setting in a crystal, as it occurs in semiconductor modeling, we consider Schrödinger equations with an additional periodic potential. This allows for the use of the concept of “energy bands”, Bloch decomposition of etc. On the level of the Wigner transform the Wigner function is replaced by the “Wigner series” , where the “kinetic variable” lives on the torus (“Brioullin zone”) instead of the whole space. Recently P. Bechouche, N.J. Mauser and F. Poupaud (2001) have given the rigorous “semiclassical” limit from S-P in a crystal to the “semiclassical equations”, i.e. the “semiconductor V-P system”, with the assumption of the initial data to be concentrated in isolated bands.
@article{JEDP_2002____A11_0,
author = {Mauser, Norbert J.},
title = {Semi-classical limits of Schr\"odinger-Poisson systems via Wigner transforms},
journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
year = {2002},
pages = {1-12},
doi = {10.5802/jedp.609},
mrnumber = {1968207},
language = {en},
url = {http://dml.mathdoc.fr/item/JEDP_2002____A11_0}
}
Mauser, Norbert J. Semi-classical limits of Schrödinger-Poisson systems via Wigner transforms. Journées équations aux dérivées partielles, (2002), pp. 1-12. doi : 10.5802/jedp.609. http://gdmltest.u-ga.fr/item/JEDP_2002____A11_0/
[1] and Solid State Physics, Holt, Rinehart, 1976
[2] , , , and , "Derivation of the Schrödinger-Poisson equation from the quantum N-particle Coulomb problem", C. R. Acad. Sci., t 334 (6), Serie I Math. (2002) 515-520 | MR 1890644 | Zbl 1018.81009
[3] , and "Weak coupling limit of the N -particle Schrödinger equation", Math. Analysis and Applications 7 (2) (2000) 275-293 | MR 1869286 | Zbl 1003.81027
[4] , and and , "Mean field dynamics of fermions and the time-dependent Hartree-Fock equation", to appear in J.d.Mathematiques Pures et Appl. (2002) | MR 1996777 | Zbl 1029.82022
[5] , and and , "Derivation of the timedependent Hartree-Fock equation : the Coulomb interaction case", manuscript
[6] , and , "(Semi)-nonrelativistic limits of the Dirac equation with external time-dependent electromagnetic field", Comm. Math. Phys. 197 (1998) 405-425 | MR 1652738 | Zbl 0926.35124
[7] , and , "Semiclassical Limit for the Schrödinger-Poisson Equation in a Crystal", Comm. Pure and Appl. Math. 54 (2001) 1-40 | MR 1823422 | Zbl 1031.81024
[8] , and , "Nonrelativistic limit of KleinGordon Maxwell to Schrödinger-Poisson", submitted to American J. of Math. (2002) | MR 2033563 | Zbl 02058032
[9] , , , "Nonrelativistic limit of Dirac Maxwell to Schrödinger-Poisson", manuscript (2002)
[10] and , "Local approximation for the Hartree-Fock exchange potential: a deformation approach", Math.Meth. and Mod.in the Appl.Sci. 9 (6) (1999) 941-961 XI-10 | MR 1702877 | Zbl 0956.81097
[11] , and , "Rigorous derivation of the "X- alpha" exchange potential: a deformation approach", in "Density Functional and Ab Initio Theories Applied to Atoms, Molecules and Solids" J. Mol. Struct. (Theochem), Vol. 501-502 (2000) 47-58
[12] and , "The three dimensional Wigner-Poisson problem: existence, uniqueness and approximation", Math. Meth. Appl. Sci., 14 (1991) 35-62. | MR 1087449 | Zbl 0739.35080
[13] , " L 2 solutions to the Schrödinger-Poisson system: existence, uniqueness, time behavior and smoothing effects", Math. Meth. Mod. Appl. Sci., 7 (1997), 1051-1083. | MR 1487521 | Zbl 0892.35141
[14] and , Introduction aux problèmes d'évolution semilinéaires, Mathématiques et Applications, Ellipses, 1990. | MR 1299976 | Zbl 0786.35070
[15] and , "Solution globales d'equations du type VlasovPoisson", C. R. Acad. Sci. Paris, 307 (1988), 655-658. | MR 967806 | Zbl 0682.35022
[16] and , " Derivation of the nonlinear Schrödinger equation with Coulomb potential" Preprint (2001) | MR 1926667
[17] , , and , Homogenization limits and Wigner transforms, Comm. Pure Appl. Math., 50 (1997) 323-379. | MR 1438151 | Zbl 0881.35099
[18] and , On the global Cauchy problem for some nonlinear Schrödinger equations, Ann. Inst. H. Poincaré, Analyse non linéaire, 1 (1984), 309-323. | Numdam | MR 778977 | Zbl 0569.35070
[19] , Mathematical Topics in Fluid Mechanics, Vol. 1, Incompressible Models, Oxford lecture series in Mathematics and its Applications, Oxford University Press, New York, 1996. | MR 1422251 | Zbl 0866.76002
[20] and , Propagation of moments and regularity for 3- dimensional Vlasov-Poisson system, Invent. Math., 105 (1991), 415-430 | MR 1115549 | Zbl 0741.35061
[21] and , Sur les measures de Wigner, Rev. Mat. Iberoamericana, 9 (1993), 553-618 | MR 1251718 | Zbl 0801.35117
[22] , and , "Concentrations in the one-dimensional Vlasov-Poisson equations, I : Temporal development and non-unique weak solutions in the single component case", Physica D, 74 (1994) 268-300 | MR 1286201 | Zbl 0813.35091
[23] and , The classical limit of a self-consistent quantum-Vlasov equation in 3-D, Math. Meth. Mod. Appl. Sci., 3 (1993), 109- 124 | MR 1203274 | Zbl 0772.35061
[24] , and , "A Wignerfunction Approach to (Semi)classical Limits : Electrons in a Periodic Potential", J. of Math. Phys. 35 (1994) 1066-1094 XI-11 | MR 1262733 | Zbl 0805.35106
[25] , and , Semiconductor Equations, Springer (1990) | MR 1063852 | Zbl 0765.35001
[26] and , "The selfconsistent Pauli equation", Math. Monatshefte 132 (2001) 19-24 | MR 1825716 | Zbl 0973.35160
[27] , , private communication (2001)
[28] , "The Schrödinger-Poisson-X model", Appl. Math. Lett. 14 (2001) 759-763 | MR 1836081 | Zbl 0990.81024
[29] , "Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data", J. Diff. Eq., 95 (1992) 281-303 | MR 1165424 | Zbl 0810.35089
[Ro] , "Unicité de la solution faible à support compact de l'équation de Vlasov Poisson", C.R.A.S., Sér. I, 324 (8) (1997) 873-877 | MR 1450441 | Zbl 0886.35118
[30] , "Global existence of smooth solutions to the Vlasov-Poisson system in three dimensions", Comm. Partial Differential Equations, 16 (1991), 1313- 1335 | MR 1132787 | Zbl 0746.35050
[31] , "On the quantum correction for thermodynamic equilibrium", Phys. Rev., 40 (1932) 742-759. | JFM 58.0948.07 | Zbl 0004.38201
[32] and , Analysis in Classes of Discontinuous Functions and Equations of Mathematical Physics (1985) | MR 785938 | Zbl 0564.46025
[33] , and , "The limit from the Schrödinger-Poisson to Vlasov-Poisson equation with general data in one dimension", Comm. Pure and Appl. Math. 55 (5) (2002) 582-632 | MR 1880644 | Zbl 1032.81011
[34] and , "Existence of global weak solutions to one-component Vlasov-Poisson and Fokker-Planck-Poisson systems in one space dimension with measures as initial data", Comm. Pure Appl. Math., 47 (1994), pp. 1365-1401 | MR 1295933 | Zbl 0809.35088
[35] , "Vlasov-Poisson systems with measures as initial data", Proc of the ICIAM 95. (1995)