@article{AIHPC_2003__20_3_501_0, author = {Carles, R\'emi}, title = {Semi-classical {Schr\"odinger} equations with harmonic potential and nonlinear perturbation}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, pages = {501--542}, publisher = {Elsevier}, volume = {20}, number = {3}, year = {2003}, doi = {10.1016/S0294-1449(02)00027-6}, mrnumber = {1972872}, zbl = {1031.35119}, language = {en}, url = {http://www.numdam.org/articles/10.1016/S0294-1449(02)00027-6/} }
TY - JOUR AU - Carles, Rémi TI - Semi-classical Schrödinger equations with harmonic potential and nonlinear perturbation JO - Annales de l'I.H.P. Analyse non linéaire PY - 2003 SP - 501 EP - 542 VL - 20 IS - 3 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/S0294-1449(02)00027-6/ DO - 10.1016/S0294-1449(02)00027-6 LA - en ID - AIHPC_2003__20_3_501_0 ER -
%0 Journal Article %A Carles, Rémi %T Semi-classical Schrödinger equations with harmonic potential and nonlinear perturbation %J Annales de l'I.H.P. Analyse non linéaire %D 2003 %P 501-542 %V 20 %N 3 %I Elsevier %U http://www.numdam.org/articles/10.1016/S0294-1449(02)00027-6/ %R 10.1016/S0294-1449(02)00027-6 %G en %F AIHPC_2003__20_3_501_0
Carles, Rémi. Semi-classical Schrödinger equations with harmonic potential and nonlinear perturbation. Annales de l'I.H.P. Analyse non linéaire, Tome 20 (2003) no. 3, pp. 501-542. doi : 10.1016/S0294-1449(02)00027-6. http://www.numdam.org/articles/10.1016/S0294-1449(02)00027-6/
[1] Concentration effects in critical nonlinear wave equation, in: , (Eds.), Geometrical Optics and Related Topics, Birkäuser, 1997, pp. 17-30. | MR | Zbl
, ,[2] Optique géométrique généralisée pour les ondes non linéaires critiques, in: Séminaire sur les Équations aux Dérivées Partielles, 1996-1997, École Polytechnique, Palaiseau, 1997, Exp. VIII, 17. | MR | Zbl
, ,[3] High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math. 121 (1) (1999) 131-175. | MR | Zbl
, ,[4] Focusing on a line for nonlinear Schrödinger equations in R2, Asymptotic Anal. 24 (3-4) (2000) 255-276. | MR | Zbl
,[5] Geometric optics with caustic crossing for some nonlinear Schrödinger equations, Indiana Univ. Math. J. 49 (2) (2000) 475-551. | MR | Zbl
,[6] Équation de Schrödinger semi-classique avec potentiel harmonique et perturbation non-linéaire, in: Séminaire sur les Équations aux Dérivées Partielles, 2001-2002, École Polytechnique, Palaiseau, 2001, Exp. III, 12. | Numdam
,[7] Geometric optics and long range scattering for one-dimensional nonlinear Schrödinger equations, Comm. Math. Phys. 220 (1) (2001) 41-67. | MR | Zbl
,[8] An Introduction to Nonlinear Schrödinger Equations, Text. Met. Mat., 26, Univ. Fed. Rio de Janeiro, 1993.
,[9] Rapidly decaying solutions of the nonlinear Schrödinger equation, Comm. Math. Phys. 147 (1992) 75-100. | MR | Zbl
, ,[10] C. Cohen-Tannoudji, Cours du collège de France, 1998-99, available at: www.lkb.ens.fr/~laloe/PHYS/cours/college-de-france/.
[11] Oscillatory integrals, Lagrange immersions and unfolding of singularities, Comm. Pure Appl. Math. 27 (1974) 207-281. | MR | Zbl
,[12] Quantum Mechanics and Path Integrals, International Series in Pure and Applied Physics, McGraw-Hill, Maidenhead, 1965, p. 365. | Zbl
, ,[13] Harmonic Analysis in Phase Space, Annals of Mathematics Studies, 122, Princeton University Press, Princeton, NJ, 1989. | MR | Zbl
,[14] Remarks on the convergence of the Feynman path integrals, Duke Math. J. 47 (3) (1980) 559-600. | MR | Zbl
,[15] Profile decomposition for the wave equation outside a convex obstacle, J. Math. Pures Appl. (9) 80 (1) (2001) 1-49. | MR | Zbl
, ,[16] Introduction aux équations de Schrödinger non linéaires, Cours de DEA, Onze Édition, Paris, 1995.
,[17] An introduction to nonlinear Schrödinger equations, in: , , (Eds.), Nonlinear Waves (Sapporo, 1995), GAKUTO International Series, Math. Sciences and Appl., Gakkōtosho, Tokyo, 1997, pp. 85-133. | MR | Zbl
,[18] Scattering theory in the energy space for a class of nonlinear Schrödinger equations, J. Math. Pures Appl. (9) 64 (4) (1985) 363-401. | MR | Zbl
, ,[19] Symplectic Techniques in Physics, Cambridge University Press, Cambridge, 1984. | MR | Zbl
, ,[20] Remarks on the scattering problem for nonlinear Schrödinger equations, in: Lectures Notes in Math., 1285, Springer, Berlin, 1987, pp. 162-168. | MR | Zbl
, ,[21] On the fundamental solution of a perturbed harmonic oscillator, Topol. Methods Nonlinear Anal. 9 (1) (1997) 77-106. | MR | Zbl
, , ,[22] Nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor. 46 (1987) 113-129. | Numdam | MR | Zbl
,[23] Endpoint Strichartz estimates, Amer. J. Math. 120 (5) (1998) 955-980. | MR | Zbl
, ,[24] Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power, Duke Math. J. 69 (2) (1993) 427-454. | MR | Zbl
,[25] A semi-classical picture of quantum scattering, Ann. Sci. École Norm. Sup. (4) 29 (2) (1996) 149-183. | Numdam | MR | Zbl
,[26] J. Rauch, Lectures on geometric optics, Available at: www.math.lsa.umich.edu/~rauch.
[27] Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 (3) (1977) 705-714. | MR | Zbl
,[28] Existence of solutions for Schrödinger evolution equations, Comm. Math. Phys. 110 (1987) 415-426. | MR | Zbl
,[29] Smoothness and non-smoothness of the fundamental solution of time dependent Schrödinger equations, Comm. Math. Phys. 181 (3) (1996) 605-629. | MR | Zbl
,[30] Reconstruction of singularities for solutions of Schrödinger's equation, Comm. Math. Phys. 90 (1) (1983) 1-26. | MR | Zbl
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