Nonlinear Coherent States and Ehrenfest Time for Schrödinger Equation*

Communications in Mathematical Physics

Volumn 301, Issue 2, 2011, Pages 443-472

  Carles, Rémi ^a   Fermanian Kammerer, Clotilde ^b  

^a CNRS   (France)

^b UNIVERSITÉ PARIS EST CRÉTEIL   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 78651515692     PISSN: 00103616     EISSN: 14320916     Source Type: Journal    
DOI: 10.1007/s00220-010-1154-0     Document Type: Article

References (42)

1. Ablowitz, M. J., Clarkson, P. A.: Solitons, nonlinear evolution equations and inverse scattering. London Mathematical Society Lecture Note Series, Vol. 149, Cambridge: Cambridge University Press, 1991.
2. Bahouri H., Gérard P.: High frequency approximation of solutions to critical nonlinear wave equations. Amer. J. Math. 121(1), 131-175 (1999).
3. Bambusi D., Graffi S., Paul T.: Long time semiclassical approximation of quantum flows: a proof of the Ehrenfest time. Asymptot. Anal. 21(2), 149-160 (1999).
4. Bily, J. M., Robert, D.: The semi-classical van Vleck formula. Application to the Aharonov-Bohm effect. In: Long time behaviour of classical and quantum systems (Bologna, 1999), Ser. Concr. Appl. Math., Vol. 1, River Edge, NJ: World Sci. Publ., 2001, pp. 89-106.
5. Bronski J. C., Jerrard R. L.: Soliton dynamics in a potential. Math. Res. Lett. 7(2-3), 329-342 (2000).
6. Carles R.: Semi-classical analysis for nonlinear Schrödinger equations. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2008).
7. Carles, R.: Nonlinear Schrödinger equation with time dependent potential. http://arXiv. org/abs/0910. 4893v2 [math. AP], 2010.
8. Carles R., Dumas E., Sparber C.: Multiphase weakly nonlinear geometric optics for Schrödinger equations. SIAM J. Math. Anal. 42(1), 489-518 (2010).
9. Carles R., Fermanian C., Gallagher I.: On the role of quadratic oscillations in nonlinear Schrödinger equations. J. Funct. Anal. 203(2), 453-493 (2003).
10. Carles R., Miller L.: Semiclassical nonlinear Schrödinger equations with potential and focusing initial data. Osaka J. Math. 41(3), 693-725 (2004).
11. Combescure M., Robert D.: Semiclassical spreading of quantum wave packets and applications near unstable fixed points of the classical flow. Asymptot. Anal. 14(4), 377-404 (1997).
12. Combescure M., Robert D.: Quadratic quantum Hamiltonians revisited. Cubo 8(1), 61-86 (2006).
13. Combescure M., Robert D.: A phase-space study of the quantum Loschmidt echo in the semiclassical limit. Ann. Henri Poincaré 8(1), 91-108 (2007).
14. Dalfovo F., Giorgini S., Pitaevskii L. P., Stringari S.: Theory of Bose-Einstein condensation in trapped gases. Rev. Mod. Phys. 71(3), 463-512 (1999).
15. Fröhlich J., Gustafson S., Jonsson B. L. G., Sigal I. M.: Solitary wave dynamics in an external potential. Commun. Math. Phys. 250(3), 613-642 (2004).
16. Fujiwara D.: A construction of the fundamental solution for the Schrödinger equation. J. Analyse Math. 35, 41-96 (1979).
17. Fujiwara D.: Remarks on the convergence of the Feynman path integrals. Duke Math. J. 47(3), 559-600 (1980).
18. Gang Z., Sigal I. M.: Relaxation of solitons in nonlinear Schrödinger equations with potential. Adv. Math. 216(2), 443-490 (2007).
19. Gérard P.: Oscillations and concentration effects in semilinear dispersive wave equations. J. Funct. Anal. 141(1), 60-98 (1996).
20. Gérard P., Markowich P. A., Mauser N. J., Poupaud F.: Homogenization limits and Wigner transforms. Commun. Pure Appl. Math. 50(4), 323-379 (1997).
21. Ginibre J., Velo G.: Scattering theory in the energy space for a class of nonlinear Schrödinger equations. J. Math. Pures Appl. (9) 64(4), 363-401 (1985).
22. Hagedorn G. A.: Semiclassical quantum mechanics. I. The 0 limit for coherent states. Commun. Math. Phys. 71(1), 77-93 (1980).
23. Hagedorn G. A., Joye A.: Exponentially accurate semiclassical dynamics: propagation, localization, Ehrenfest times, scattering, and more general states. Ann. Henri Poincaré 1(5), 837-883 (2000).
24. Hagedorn G. A., Joye A.: A time-dependent Born-Oppenheimer approximation with exponentially small error estimates. Commun. Math. Phys. 223(3), 583-626 (2001).
25. Holmer J., Zworski M.: Slow soliton interaction with delta impurities. J. Mod. Dyn. 1(4), 689-718 (2007).
26. Jonsson B. L. G., Fröhlich J., Gustafson S., Sigal I. M.: Long time motion of NLS solitary waves in a confining potential. Ann. Henri Poincaré 7(4), 621-660 (2006).
27. Josserand C., Pomeau Y.: Nonlinear aspects of the theory of Bose-Einstein condensates. Nonlinearity 14(5), R25-R62 (2001).
28. Keel M., Tao T.: Endpoint Strichartz estimates. Amer. J. Math. 120(5), 955-980 (1998).
29. Keraani S.: Semiclassical limit for a class of nonlinear Schrödinger equations with potential. Commun. Part. Diff. Eq. 27(3-4), 693-704 (2002).
30. Keraani S.: Semiclassical limit for nonlinear Schrödinger equation with potential. II. Asymptot. Anal. 47(3-4), 171-186 (2006).
31. Al Khawaja U.: Soliton localization in Bose-Einstein condensates with time-dependent harmonic potential and scattering length. J. Phys. A: Math. Theor. 42, 265206 (2009).
32. Krieger J., Martel Y., Raphaël P.: Two-soliton solutions to the three-dimensional gravitational Hartree equation. Commun. Pure Appl. Math. 62(11), 1501-1550 (2009).
33. Lions P.-L., Paul T.: Sur les mesures de Wigner. Rev. Mat. Iberoamericana 9(3), 553-618 (1993).
34. Littlejohn R. G.: The semiclassical evolution of wave packets. Phys. Rep. 138(4-5), 193-291 (1986).
35. Ozawa T.: Long range scattering for nonlinear Schrödinger equations in one space dimension. Commun. Math. Phys. 139, 479-493 (1991).
36. Paul, T.: Semi-classical methods with emphasis on coherent states. In: Quasiclassical methods (Minneapolis, MN, 1995), IMA Vol. Math. Appl., Vol. 95, New York: Springer, 1997, pp. 51-88.
37. Robert, D.: On the Herman-Kluk semiclassical approximation. http://arXiv. org/abs/0908. 0847v1 [math-ph], 2009.
38. Rousse, V.: Semiclassical simple initial value representations. Ark. för Mat. (2009, to appear).
39. Serkin V. N., Hasegawa A., Belyaeva T. L.: Nonautonomous solitons in external potentials. Phys. Rev. Lett. 98(7), 074102 (2007).
40. Swart T., Rousse V.: A mathematical justification for the Herman-Kluk propagator. Commun. Math. Phys. 286(2), 725-750 (2009).
41. Trèves, F.: Introduction to pseudodifferential and Fourier integral operators. Vol. 1, New York: Plenum Press, 1980.
42. Yajima K.: Existence of solutions for Schrödinger evolution equations. Commun. Math. Phys. 110, 415-426 (1987).