Semi-classical Schrödinger equations with harmonic potential and nonlinear perturbation

Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire

Volumn 20, Issue 3, 2003, Pages 501-542

  Carles, Rémi ^a  

^a UNIVERSITÉ BORDEAUX 1   (France)

Author keywords

[No Author keywords available]

Indexed keywords

FOURIER TRANSFORMS; INITIAL VALUE PROBLEMS; MATHEMATICAL OPERATORS; NONLINEAR EQUATIONS; PERTURBATION TECHNIQUES; PROBLEM SOLVING;

CRITICAL NONLINEARITY; HARMONIC POTENTIAL; MASLOV INDEXES;

WAVE EQUATIONS;

EID: 0141814240     PISSN: 02941449     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0294-1449(02)00027-6     Document Type: Article

References (30)

1. Bahouri H., Gérard P. Concentration effects in critical nonlinear wave equation. Colombini F., Lerner N. Geometrical Optics and Related Topics. 1997;17-30 Birkäuser.
2. Bahouri H., Gérard P. Optique géométrique généralisée pour les ondes non linéaires critiques. Séminaire sur les Équations aux Dérivées Partielles, 1996-1997. 1997;École Polytechnique, Palaiseau. Exp. VIII, 17.
3. Bahouri H., Gérard P. High frequency approximation of solutions to critical nonlinear wave equations. Amer. J. Math. 121:(1):1999;131-175.
4. 2 . Asymptotic Anal. 24:(3-4):2000;255-276.
5. Carles R. Geometric optics with caustic crossing for some nonlinear Schrödinger equations. Indiana Univ. Math. J. 49:(2):2000;475-551.
6. Carles R. Équation de Schrödinger semi-classique avec potentiel harmonique et perturbation non-linéaire. Séminaire sur les Équations aux Dérivées Partielles, 2001-2002. 2001;École Polytechnique, Palaiseau. Exp. III, 12.
7. Carles R. Geometric optics and long range scattering for one-dimensional nonlinear Schrödinger equations. Comm. Math. Phys. 220:(1):2001;41-67.
8. Cazenave T. An Introduction to Nonlinear Schrödinger Equations. Text. Met. Mat. 26:1993;Univ. Fed. Rio de Janeiro.
9. Cazenave T., Weissler F. Rapidly decaying solutions of the nonlinear Schrödinger equation. Comm. Math. Phys. 147:1992;75-100.
10. C. Cohen-Tannoudji, Cours du collège de France, 1998-99, available at: www.lkb.ens.fr/̃laloe/PHYS/cours/college-de-france/.
11. Duistermaat J.J. Oscillatory integrals, Lagrange immersions and unfolding of singularities. Comm. Pure Appl. Math. 27:1974;207-281.
12. Feynman R.P., Hibbs A.R. Quantum Mechanics and Path Integrals. International Series in Pure and Applied Physics. 1965;McGraw-Hill, Maidenhead. p. 365.
13. Folland G. Harmonic Analysis in Phase Space. Annals of Mathematics Studies. 122:1989;Princeton University Press, Princeton, NJ.
14. Fujiwara D. Remarks on the convergence of the Feynman path integrals. Duke Math. J. 47:(3):1980;559-600.
15. Gallagher I., Gérard P. Profile decomposition for the wave equation outside a convex obstacle. J. Math. Pures Appl. (9). 80:(1):2001;1-49.
16. Ginibre J. Introduction aux équations de Schrödinger non linéaires, Cours de DEA. 1995;Onze Édition, Paris.
17. Ginibre J. An introduction to nonlinear Schrödinger equations. Agemi R., Giga Y., Ozawa T. Nonlinear Waves (Sapporo, 1995), GAKUTO International Series. Math. Sciences and Appl. 1997;85-133 Gakkōtosho, Tokyo.
18. Ginibre J., Velo G. Scattering theory in the energy space for a class of nonlinear Schrödinger equations. J. Math. Pures Appl. (9). 64:(4):1985;363-401.
19. Guillemin V., Sternberg S. Symplectic Techniques in Physics. 1984;Cambridge University Press, Cambridge.
20. Hayashi N., Tsutsumi Y. Remarks on the scattering problem for nonlinear Schrödinger equations. Lectures Notes in Math. 1285:1987;162-168 Springer, Berlin.
21. Kapitanski L., Rodnianski I., Yajima K. On the fundamental solution of a perturbed harmonic oscillator. Topol. Methods Nonlinear Anal. 9:(1):1997;77-106.
22. Kato T. Nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Phys. Théor. 46:1987;113-129.
23. Keel M., Tao T. Endpoint Strichartz estimates. Amer. J. Math. 120:(5):1998;955-980.
24. Merle F. Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power. Duke Math. J. 69:(2):1993;427-454.
25. Nier F. A semi-classical picture of quantum scattering. Ann. Sci. École Norm. Sup. (4). 29:(2):1996;149-183.
26. J. Rauch, Lectures on geometric optics, Available at: www.math.lsa.umich.edu/̃rauch, 1995.
27. Strichartz R.S. Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations. Duke Math. J. 44:(3):1977;705-714.
28. Yajima K. Existence of solutions for Schrödinger evolution equations. Comm. Math. Phys. 110:1987;415-426.
29. Yajima K. Smoothness and non-smoothness of the fundamental solution of time dependent Schrödinger equations. Comm. Math. Phys. 181:(3):1996;605-629.
30. Zelditch S. Reconstruction of singularities for solutions of Schrödinger's equation. Comm. Math. Phys. 90:(1):1983;1-26.