@article{SEDP_2006-2007____A2_0, author = {Castella, Fran\c{c}ois}, title = {Time averaging for the strongly confined nonlinear {Schr\"odinger} equation}, journal = {S\'eminaire \'Equations aux d\'eriv\'ees partielles (Polytechnique) dit aussi "S\'eminaire Goulaouic-Schwartz"}, note = {talk:2}, pages = {1--22}, publisher = {Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique}, year = {2006-2007}, mrnumber = {2385189}, language = {en}, url = {http://www.numdam.org/item/SEDP_2006-2007____A2_0/} }
TY - JOUR AU - Castella, François TI - Time averaging for the strongly confined nonlinear Schrödinger equation JO - Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" N1 - talk:2 PY - 2006-2007 SP - 1 EP - 22 PB - Centre de mathématiques Laurent Schwartz, École polytechnique UR - http://www.numdam.org/item/SEDP_2006-2007____A2_0/ LA - en ID - SEDP_2006-2007____A2_0 ER -
%0 Journal Article %A Castella, François %T Time averaging for the strongly confined nonlinear Schrödinger equation %J Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" %Z talk:2 %D 2006-2007 %P 1-22 %I Centre de mathématiques Laurent Schwartz, École polytechnique %U http://www.numdam.org/item/SEDP_2006-2007____A2_0/ %G en %F SEDP_2006-2007____A2_0
Castella, François. Time averaging for the strongly confined nonlinear Schrödinger equation. Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2006-2007), Exposé no. 2, 22 p. http://www.numdam.org/item/SEDP_2006-2007____A2_0/
[Ba] G. Bastard, Wave mechanics applied to semiconductor heterostructures, Les éditions de physique, 1996.
[BaMSW] W. Bao, P. Markowich, C. Schmeiser, R. Weishäupl, On the Gross-Pitaevski equation with strongly anisotropic confinement: formal asymptotics and numerical experiments, preprint (2004). | MR | Zbl
[BM] N. Ben Abdallah, F. Méhats, Semiclassical analysis of the Schrödinger equation with a partially confining potential, J. Math. Pures Appl. 84, pp. 580-614 (2005). | MR | Zbl
[BMSW] N. Ben Abdallah, F. Méhats, C. Schmeiser, R. M. Weishäupl, The nonlinear Schrödinger equation with strong anisotropic harmonic potential, to appear in SIAM J. Math. Anal. | MR | Zbl
[BMP] N. Ben Abdallah, F. Méhats, O. Pinaud, The adiabatic approximation of the Schrödinger Poisson system with a partial confinement, SIAM J. Math. Anal. 36, N.3, pp. 986-1013 (2005). | MR | Zbl
[BCD] B. Bidégaray, F. Castella, P. Degond, From Bloch model to the rate equations, Disc. Cont. Dyn. Sys., Vol. 11, N. 1, pp. 1-26 (2004). | MR | Zbl
[BCDG] B. Bidégaray-Fesquet, F. Castella, E. Dumas, M. Gisclon, From Bloch model to the rate equations II: the case of almost degenerate energy levels, Math. Mod. Meth. Appl. Sci., Vol. 14, pp. 1785-1817 (2004). | MR | Zbl
[BC] J.-M. Bony, J.-Y. Chemin, Espaces fonctionnels associés au calcul de Weyl-Hörmander, Bull. Soc. Math. France, Vol. 122, N. 1, pp. 77-118 (1994). | Numdam | MR | Zbl
[CDG1] F. Castella, P. Degond, T. Goudon, Diffusion dynamics of classical systems driven by an oscillatory force, J. Stat. Phys., Vol. 124, N. 2-4, pp. 913-950 (2006). | MR | Zbl
[CDG2] F. Castella, P. Degond, T. Goudon, Large time dynamics of a classical system subject to a fast varying force, to appear in Comm. Math. Phys. (2007). | MR | Zbl
[FG] D. K. Ferry, S. M. Goodnick, Transport in nanostructures, Cambridge Univ. Press, 1997.
[Gr] E. Grenier, Oscillatory perturbations of the Navier-Stokes equations, J. Math. Pures Appl., Vol. 76, pp. 477-498 (1997). | MR | Zbl
[HJ] G. A. Hagedorn, A. Joye, A time-dependent Born-Oppenheimer approximation with exponentially small error estimates, Comm. Math. Phys. 223, N. 3, pp. 583-626 (2001). | MR | Zbl
[He] B. Helffer, Théorie spectrale pour des opérateurs globalement elliptiques, Astérisque, Vol. 112, Soc. Math. de France (1984). | Numdam | MR | Zbl
[HN] B. Helffer, F. Nier, Hypoelliptic estimates and spectral theory for Fokker-Planck operators and Witten Lalacians, Springer, to appear (2005). | MR | Zbl
[HR] B. Helffer, D. Robert, Caclcul fonctionnel par la transformation de Mellin et opérateurs admissibles, J. Funct. Anal., Vol. 53, N. 3, pp. 246-268 (1983). | MR | Zbl
[Ho] L. Hörmander, The analysis of linear partial differential operators, Springer (1985).
[K] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, Heidelberg, 1966. | MR | Zbl
[LZ] B.M. Levitan, V.V. Zhikov, Almost periodic functions and differential equations, Cambridge University Press (1982). | MR | Zbl
[LM] P. Lochak, C. Meunier, Multiphase averaging for classical systems. With applications to adiabatic theorems, Applied Mathematical Sciences, Vol. 72. Springer-Verlag. (1988). | MR | Zbl
[Ma] A. Martinez, An introduction to Semicassical and Microlocal Analysis, Universitext, Springer-Verlag, New York (2002). | MR | Zbl
[MS] G. Métivier, S. Schochet, Averaging Theorems for conservative systems and the weakly compressible Euler equations, J. Diff. Eq., Vol. 187, N. 1, pp. 106-183 (2003). | MR | Zbl
[Mi] K. A. Mitchell, Geometric phase, curvature, and Extrapotentials in Constrained Quantum Systems, preprint arXiv:quant-ph/0001059, 18 Jan 2000.
[P] O. Pinaud, Adiabatic approximation of the Schrödinger-Poisson system with a partial confinement: the stationary case, J. Math. Phys. 45, N. 5, pp. 2029-2050 (2004). | MR | Zbl
[PB] E. Polizzi, N. Ben Abdallah, Self-consistent three dimensional model for quantum ballistic transport in open systems, Phys. Rev B. 66, pp. 245301-245309 (2002).
[RS] M. Reed, B. Simon, Methods of Modern Mathematical Physics, Academic Press, New York, San Francisco and London, 1975. | MR
[SV] J.A. Sanders, F. Verhulst, Averaging methods in nonlinear dynamical systems, Applied Mathematical Sciences, Vol. 59 (Springer-Verlag, 1985). | MR | Zbl
[Sc] S. Schochet, Fast singular limits of hyperbolic PDEs, J. Diff. Eq., Vol. 114, N. 2, pp. 476-512 (1994). | MR | Zbl
[ST] H. Spohn, S. Teufel, Adiabatic decoupling and time-dependent Born-Oppenheimer theory, Comm. Math. Phys. 224, N. 1, pp. 113-132. (2001). | MR | Zbl
[T] S. Teufel, Adiabatic Perturbation Theory in Quantum Dynamics, Habilitationsschrift, Zentrum Mathematik Technische Universität München, 2002. | MR
[W1] W.-M. Wang, Pure point spectrum of the Floquet Hamiltonian for the quantum harmonic oscillator under time quasi-periodic perturbations, Preprint. | MR
[W2] W.-M. Wang, Quasi-periodic solutions of nonlinearly perturbed quantum harmonic oscillator, Preprint.