(Semi) classical limit of the Hartree equation with harmonic potential

SIAM Journal on Applied Mathematics

Volumn 66, Issue 1, 2006, Pages 29-56

  Carles, Rémi ^a   Mauser, Norbert J ^b   Stimming, Hans Peter ^c,d  

^a UNIVERSITÉ BORDEAUX 1   (France)

^b UNIVERSITY OF VIENNA   (Austria)

^c Vienna University   (Austria)

^d UNIVERSITÉ DE LYON   (France)

Author keywords

Harmonic potential; Hartree equation; Schr dinger Poisson; Semiclassical limit

Indexed keywords

ASYMPTOTIC STABILITY; HARMONIC GENERATION; MATHEMATICAL MODELS; POISSON DISTRIBUTION; QUANTUM THEORY;

HARMONIC POTENTIAL; HARTREE EQUATION; SCHRÖDINGER-POISSON; SEMICLASSICAL LIMIT;

NONLINEAR EQUATIONS;

EID: 33644603110     PISSN: 00361399     EISSN: None     Source Type: Journal    
DOI: 10.1137/040609732     Document Type: Article

References (31)

1. W. BAO, N. J. MAUSER, AND H. P. STIMMING, Effective one particle quantum dynamics of electrons: A numerical study of the Schrödinger-Poisson-Xa model, Commun. Math. Sci., 1 (2003), pp. 809-828.
2. P. BECHOUCHE, N. MAUSER, AND F. POUPAUD, Semiclassical limit for the Schrödinger equation in a crystal with Coulomb interaction, Comm. Pure Appl. Math., 54 (2001), pp. 851-890.
3. R. CARLES, Geometric optics with caustic crossing for some nonlinear Schrödinger equations, Indiana Univ. Math. J., 49 (2000), pp. 475-551.
4. R. CARLES, Geometric optics and long range scattering for one-dimensional nonlinear Schrö dinger equations, Comm. Math. Phys., 220 (2001), pp. 41-67.
5. R. CARLES, Remarques sur les mesures de Wigner, C. R. Acad. Sci. Paris Sér. I Math., 332 (2001), pp. 981-984.
6. R. CARLES, Semi-classical Schrödinger equations with harmonic potential and nonlinear perturbation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 20 (2003), pp. 501-542.
7. T. CAZENAVE, Semilinear Schrödinger Equations, Courant Lect. Notes Math. 10, New York University Courant Institute of Mathematical Sciences, New York, 2003.
8. J. DEREZIŃSKI AND C. GÉRARD, Scattering Theory of Quantum and Classical N-Particle Systems, Texts Monogr. Phys., Springer-Verlag, Berlin, Heidelberg, 1997.
9. J. J. DUISTERMAAT, Oscillatory integrals, Lagrange immersions and unfolding of singularities, Comm. Pure Appl. Math., 27 (1974), pp. 207-281.
10. R. P. FEYNMAN AND A. HIBBS, Quantum Mechanics and Path Integrals, International Series in Pure and Applied Physics, McGraw-Hill, Maidenhead, Berkshire, UK, 1965.
11. P. GÉRARD, P. A. MARKOWICH, N. J. MAUSER, AND F. POUPAUD, Homogenization limits and Wigner transforms, Comm. Pure Appl. Math., 50 (1997), pp. 323-379.
12. J. GINIBRE, An Introduction to Nonlinear Schrödinger Equations, in Nonlinear Waves (Sapporo, 1995), GAKUTO Internat. Ser. Math. Sci. Appl., R. Agemi, Y. Giga, and T. Ozawa, eds., Gakkōtosho, Tokyo, 1997, pp. 85-133.
13. J. GINIBRE AND G. VELO, On a class of nonlinear Schrödinger equations with nonlocal interaction, Math. Z., 170 (1980), pp. 109-136.
14. J. GINIBRE AND G. VELO, Long range scattering and modified wave operators for some Hartree type equations. II, Ann. Henri Poincaré, 1 (2000), pp. 753-800.
15. J. GINIBRE AND G. VELO, Long range scattering and modified wave operators for some Hartree type equations. III. Gevrey spaces and low dimensions, J. Differential Equations, 175 (2001), pp. 415-501.
16. N. HAYASHI AND P. NAUMKIN, Asymptotics for large time of solutions to the nonlinear Schrö dinger and Hartree equations, Amer. J. Math., 120 (1998), pp. 369-389.
17. N. HAYASHI AND Y. TSUTSUMI, Scattering theory for Hartree type equations, Ann. Inst. H. Poincaré Phys. Théor., 46 (1987), pp. 187-213.
18. L. HÖRMANDER, Symplectic classification of quadratic forms, and general Mehler formulas, Math. Z., 219 (1995), pp. 413-449.
19. J. HUNTER AND J. KELLER, Caustics of nonlinear waves, Wave Motion, 9 (1987), pp. 429-443.
20. S. JIN, D. LEVERMORE, AND D. MCLAUGHLIN, The semiclassical limit of the defocusing NLS hierarchy, Comm. Pure Appl. Math., 52 (1999), pp. 613-654.
21. J.-L. JOLY, G. MÉTIVIER, AND J. RAUCH, Caustics for dissipative semilinear oscillations, Mem. Amer. Math. Soc., 144 (2000), 72 pp.
22. S. KAMVISSIS, K. MCLAUGHLIN, AND P. MILLER, Semiclassical Soliton Ensembles for the Focusing Nonlinear Schrödinger Equation, Ann. of Math. Stud. 154, Princeton University Press, Princeton, NJ, 2003.
23. P.-L. LIONS AND T. PAUL, Sur les mesures de Wigner, Rev. Mat. Iberoamericana, 9 (1993), pp. 553-618.
24. P. A. MARKOWICH AND N. J. MAUSER, The classical limit of a self-consistent quantum-Vlasov equation in 3D, Math. Models Methods Appl. Sci., 3 (1993), pp. 109-124.
25. N. J. MAUSER, The Schrödinger-Poisson-Xa equation, Appl. Math. Lett., 14 (2001), pp. 759-763.
26. N. J. MAUSER, (Semi)classical limits of Schrödinger-Poisson systems via Wigner transforms, in Journées "Équations aux Dérivées Partielles" (Forges-les-Eaux, 2002), Univ. Nantes, Nantes, France, 2002, pp. Exp. No. XI, 12.
27. Y.-G. OH, Cauchy problem and Ehrenfest's law of nonlinear Schrödinger equations with potentials, J. Differential Equations, 81 (1989), pp. 255-274.
28. J. RAUCH AND M. KEEL, Lectures on geometric optics, in Hyperbolic Equations and Frequency Interactions (Park City, UT, 1995), LAS/Park City Math. Ser. 5, AMS, Providence, RI, 1999, pp. 383-466.
29. C. SPARBER, P. A. MARKOWICH, AND N. J. MAUSER, Wigner functions versus WKB-methods in multivalued geometrical optics, Asymptot. Anal., 33 (2003), pp. 153-187.
30. W. THIRRING, A Course in Mathematical Physics. Vol. 3. Quantum Mechanics of Atoms and Molecules, Lecture Notes in Phys. 141, Springer-Verlag, New York, Vienna, 1981.
31. P. ZHANG, Y. ZHENG, AND N. J. MAUSER, The limit from the Schrödinger-Poisson to the Vlasov-Poisson equations with general data in one dimension, Comm. Pure Appl. Math., 55 (2002), pp. 582-632.