Linear vs. nonlinear effects for nonlinear Schrödinger equations with potential

Communications in Contemporary Mathematics

Volumn 7, Issue 4, 2005, Pages 483-508

  Carles, Rémi ^a,b  

^a UNIVERSITÉ DE RENNES 1   (France)

^b UNIVERSITÉ BORDEAUX 1   (France)

Author keywords

Finite time blow up; Linear potential; Nonlinear Schr dinger equation; Scattering; Semiclassical analysis

Indexed keywords

EID: 24144475004     PISSN: 02191997     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0219199705001829     Document Type: Article

References (58)

1. J. E. Avron and I. W. Herbst, Spectral and scattering theory of Schrödinger operators related to the Stark effect, Commun. Math. Phys. 52(3) (1977) 239-254.
2. C. C. Bradley, C. A. Sackett, J. J. Tollett and R. G. Hulet, Evidence of Bose-Einstein condensation in an atomic gas with attractive interactions, Phys. Rev. Lett. 75 (1995) 1687-1690.
3. J. C. Bronski and R. L. Jerrard, Soliton dynamics in a potential, Math. Res. Lett. 7(2-3) (2000) 329-342.
4. d, Math. Res. Lett. 9(2-3) (2002) 323-335.
5. N. Burq, P. Gérard and N. Tzvetkov, Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds, Amer. J. Math. 126(3) (2004) 569-605.
6. N. Burq, P. Gérard and N. Tzvetkov, On nonlinear Schrödinger equations in exterior domains, Ann. Inst. H. Poincaré Anal. Non Linéaire 21(3) (2004) 295-318.
7. R. Carles, Remarks on nonlinear Schrödinger equations with harmonic potential, Ann. Henri Poincaré 3(4) (2002) 757-772.
8. R. Carles, Critical nonlinear Schrödinger equations with and without harmonic potential, Math. Models Methods Appl. Sci. 12(10) (2002) 1513-1523.
9. R. Carles, Semi-classical Schrödinger equations with harmonic potential and nonlinear perturbation, Ann. Inst. H. Poincaré Anal. Non Linéaire 20(3) (2003) 501-542.
10. R. Carles, Nonlinear Schrödinger equations with repulsive harmonic potential and applications, SIAM J. Math. Anal. 35(4) (2003) 823-843.
11. R. Carles, Global existence results for nonlinear Schrödinger equations with quadratic potentials, Discrete Contin. Dyn. Syst. 13(2) (2005) 385-398.
12. R. Carles, N. J. Mauser and H. P. Stimming, (Semi)classical limit of the Hartree equation with harmonic potential, to appear in SIAM J. Appl. Math.
13. R. Carles and L. Miller, Semiclassical nonlinear Schrödinger equations with potential and focusing initial data, Osaka J. Math. 41(3) (2004) 693-725.
14. R. Carles and Y. Nakamura, Nonlinear Schrödinger equations with Stark potential, Hokkaido Math. J. 33(3) (2004) 719-729.
15. T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, Vol. 10 (New York University Courant Institute of Mathematical Sciences, New York, 2003).
16. T. Cazenave and F. Weissler, Rapidly decaying solutions of the nonlinear Schrödinger equation, Commun. Math. Phys. 147 (1992) 75-100.
17. W. Craig, T. Kappeler and W. Strauss, Microlocal dispersive smoothing for the Schrödinger equation, Commun. Pure Appl. Math. 48(8) (1995) 769-860.
18. B. Deconinck, B. Frigyik and J. N. Kutz, Dynamics and stability of Bose-Einstein condensates: the nonlinear Schrödinger equation with periodic potential, J. Nonlinear Sci. 12(3) (2002) 169-205.
19. S.-I. Doi, Smoothing effects for Schrödinger evolution equation and global behavior of geodesic flow, Math. Ann. 318 (2000) 355-389.
20. J. J. Duistermaat, Oscillatory integrals, Lagrange immersions and unfolding of singularities, Commun. Pure Appl. Math. 27 (1974) 207-281.
21. N. Dunford and J. T. Schwartz, Linear Operators. Part II: Spectral Theory. Self Adjoint Operators in Hilbert Space, with the assistance of William G. Bade and Robert G. Bartle (Interscience Publishers John Wiley & Sons, New York, London, 1963).
22. R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, International Series in Pure and Applied Physics (McGraw-Hill Publishing Company, Maidenhead, Berksh, 1965).
23. D. Fujiwara, A construction of the fundamental solution for the Schrödinger equation, J. Anal. Math. 35 (1979) 41-96.
24. D. Fujiwara, Remarks on the convergence of the Feynman path integrals, Duke Math. J. 47(3) (1980) 559-600.
25. J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations. I The Cauchy problem, general case, J. Funct. Anal. 32 (1979) 1-32.
26. J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations. II Scattering theory, general case, J. Funct. Anal. 32 (1979) 33-71.
27. J. Ginibre and G. Velo, The global Cauchy problem for the nonlinear Schrödinger equation revisited, Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985) 309-327.
28. R. T. Glassey, On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations, J. Math. Phys. 18 (1977) 1794-1797.
29. N. Hayashi and Y. Tsutsumi, Remarks on the scattering problem for nonlinear Schrödinger equations, in Differential Equations and Mathematical Physics (Birmingham, Ala., 1986), Lectures Notes in Mathematics, Vol. 1285 (Springer, Berlin, 1987), pp. 162-168.
30. L. Hörmander, Symplectic classification of quadratic forms, and general Mehler formulas, Math. Z. 219(3) (1995) 413-449.
31. A. Jensen and T. Kato, Spectral properties of Schrödinger operators and time decay of the wave functions, Duke Math. J. 46(3) (1979) 583-611.
32. R. Johnson and X. Pan, On an elliptic equation related to the blow-up phenomenon in the nonlinear Schrödinger equation, Proc. Roy. Soc. Edinburgh Sect. A123(4) (1993) 763-782.
33. L. Kapitanski and I. Rodnianski, Regulated smoothing for Schrödinger evolution, Internat. Math. Res. Notices (2) (1996) 41-54.
34. O. Kavian and F. Weissler, Self-similar solutions of the pseudo-conformally invariant nonlinear Schrödinger equation, Michigan Math. J. 41(1) (1994) 151-173.
35. M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math. 120(5) (1998) 955-980.
36. S. Keraani, Semiclassical limit for a class of nonlinear Schrödinger equations with potential, Commun. Partial Differential Equations 27(3-4) (2002) 693-704.
37. S. Klainerman, Uniform decay estimates and the Lorentz invariance of the classical wave equation, Commun. Pure Appl. Math. 38(3) (1985) 321-332.
38. E. B. Kolomeisky, T. J. Newman, J. P. Straley and X. Qi, Low-dimensional Bose liquids: Beyond the Gross-Pitaevskii approximation, Phys. Rev. Lett. 85(6) (2000) 1146-1149.
39. F. Merle, Construction of solutions with exactly k blow-up points for the Schrödinger equation with critical nonlinearity, Commun. Math. Phys. 129(2) (1990) 223-240.
40. F. Merle and P. Raphaël, Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation, Geom. Funct. Anal. 13(3) (2003) 591-642.
41. K. Nakanishi and T. Ozawa, Remarks on scattering for nonlinear Schrödinger equations, NoDEA Nonlinear Differential Equations Appl. 9(1) (2002) 45-68.
42. U. Niederer, The maximal kinematical invariance groups of Schrödinger equations with arbitrary potentials, Helv. Phys. Acta 47 (1974) 167-172.
43. F. Nier, A semi-classical picture of quantum scattering, Ann. Sci. Ecole Norm. Sup. (4) 29(2) (1996) 149-183.
44. F. Nier and A. Soffer, Dispersion and Strichartz estimates for some finite rank perturbations of the Laplace operator, J. Funct. Anal. 198(2) (2003) 511-535.
45. Y.-G. Oh, Cauchy problem and Ehrenfest's law of nonlinear Schrödinger equations with potentials, J. Differential Equations 81(2) (1989) 255-274.
46. L. Pitaevskii and S. Stringari, Bose-Einstein Condensation, International Series of Monographs on Physics, Vol. 116 (The Clarendon Press Oxford University Press, Oxford, 2003).
47. M. Reed and B. Simon, Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness (Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1975).
48. D. Robert, Autour de l'Approximation Semi-Classique, Progress in Mathematics, Vol. 68 (Birkhäuser Boston Inc., Boston, MA, 1987).
49. I. Rodnianski and W. Schlag, Time decay for solutions of Schrödinger equations with rough and time-dependent potentials, Invent. Math. 155(3) (2004) 451-513.
50. A. V. Rybin, G. G. Varzugin, M. Lindberg, J. Timonen and R. K. Bullough, Similarity solutions and collapse in the attractive Gross-Pitaevskii equation, Phys. Rev. E (3) 62(5A) (2000) 6224-6228.
51. G. Staffilani and D. Tataru, Strichartz estimates for a Schrödinger operator with nonsmooth coefficients, Commun. Partial Differential Equations 27(7-8) (2002) 1337-1372.
52. R. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44(3) (1977) 705-714.
53. W. Thirring, A Course in Mathematical Physics, Vol. 3 (Springer-Verlag, New York, 1981). Quantum Mechanics of Atoms and Molecules, Translated from the German by Evans M. Harrell, Lecture Notes in Physics, Vol. 141.
54. 2-solutions for nonlinear Schrodinger equations and nonlinear groups, Funkcial. Ekvac. 30(1) (1987) 115-125.
55. M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Commun. Math. Phys. 87(4) (1982/83) 567-576.
56. K. Yajima, Existence of solutions for Schrödinger evolution equations, Commun. Math. Phys. 110 (1987) 415-426.
57. K. Yajima, Smoothness and non-smoothness of the fundamental solution of time dependent Schrödinger equations, Commun. Math. Phys. 181(3) (1996) 605-629.
58. K. Yajima and G. Zhang, Smoothing property for Schrödinger equations with potential superquadratic at infinity, Commun. Math. Phys. 221(3) (2001) 573-590.