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.
@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://archive.numdam.org/articles/10.1016/S0294-1449(02)00027-6/}
}
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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://archive.numdam.org/articles/10.1016/S0294-1449(02)00027-6/

[1] Bahouri H., Gérard P., Concentration effects in critical nonlinear wave equation, in: Colombini F., Lerner N. (Eds.), Geometrical Optics and Related Topics, Birkäuser, 1997, pp. 17-30. | MR | Zbl

[2] Bahouri H., Gérard P., 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] Bahouri H., Gérard P., High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math. 121 (1) (1999) 131-175. | MR | Zbl

[4] Carles R., Focusing on a line for nonlinear Schrödinger equations in R2, Asymptotic Anal. 24 (3-4) (2000) 255-276. | MR | Zbl

[5] Carles R., Geometric optics with caustic crossing for some nonlinear Schrödinger equations, Indiana Univ. Math. J. 49 (2) (2000) 475-551. | MR | Zbl

[6] Carles R., 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] Carles R., Geometric optics and long range scattering for one-dimensional nonlinear Schrödinger equations, Comm. Math. Phys. 220 (1) (2001) 41-67. | MR | Zbl

[8] Cazenave T., An Introduction to Nonlinear Schrödinger Equations, Text. Met. Mat., 26, Univ. Fed. Rio de Janeiro, 1993.

[9] Cazenave T., Weissler F., 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] Duistermaat J.J., Oscillatory integrals, Lagrange immersions and unfolding of singularities, Comm. Pure Appl. Math. 27 (1974) 207-281. | MR | Zbl

[12] Feynman R.P., Hibbs A.R., Quantum Mechanics and Path Integrals, International Series in Pure and Applied Physics, McGraw-Hill, Maidenhead, 1965, p. 365. | Zbl

[13] Folland G., Harmonic Analysis in Phase Space, Annals of Mathematics Studies, 122, Princeton University Press, Princeton, NJ, 1989. | MR | Zbl

[14] Fujiwara D., Remarks on the convergence of the Feynman path integrals, Duke Math. J. 47 (3) (1980) 559-600. | MR | Zbl

[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. | MR | Zbl

[16] Ginibre J., Introduction aux équations de Schrödinger non linéaires, Cours de DEA, Onze Édition, Paris, 1995.

[17] Ginibre J., An introduction to nonlinear Schrödinger equations, in: Agemi R., Giga Y., Ozawa T. (Eds.), Nonlinear Waves (Sapporo, 1995), GAKUTO International Series, Math. Sciences and Appl., Gakkōtosho, Tokyo, 1997, pp. 85-133. | MR | Zbl

[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. | MR | Zbl

[19] Guillemin V., Sternberg S., Symplectic Techniques in Physics, Cambridge University Press, Cambridge, 1984. | MR | Zbl

[20] Hayashi N., Tsutsumi Y., Remarks on the scattering problem for nonlinear Schrödinger equations, in: Lectures Notes in Math., 1285, Springer, Berlin, 1987, pp. 162-168. | MR | Zbl

[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. | MR | Zbl

[22] Kato T., Nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor. 46 (1987) 113-129. | Numdam | MR | Zbl

[23] Keel M., Tao T., Endpoint Strichartz estimates, Amer. J. Math. 120 (5) (1998) 955-980. | MR | Zbl

[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. | MR | Zbl

[25] Nier F., 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] 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. | MR | Zbl

[28] Yajima K., Existence of solutions for Schrödinger evolution equations, Comm. Math. Phys. 110 (1987) 415-426. | MR | Zbl

[29] Yajima K., Smoothness and non-smoothness of the fundamental solution of time dependent Schrödinger equations, Comm. Math. Phys. 181 (3) (1996) 605-629. | MR | Zbl

[30] Zelditch S., Reconstruction of singularities for solutions of Schrödinger's equation, Comm. Math. Phys. 90 (1) (1983) 1-26. | MR | Zbl

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