La théorie de Mourre est un outil puissant pour étudier le spectre continu d’opérateurs auto-adjoints et pour développer une théorie de la diffusion. Dans cet exposé nous proposons d’un nouveau regard sur la théorie de Mourre en donnant une nouvelle approche du résultat principal de la théorie : le principe d’aborption limite (PAL) obtenu à partir de l’estimation de Mourre. Nous donnons alors une nouvelle interprétation de ce résultat. Cet exposé a aussi pour but d’être une introduction à la théorie.
@article{JEDP_2006____A5_0, author = {Gol\'enia, Sylvain}, title = {Un nouveau regard sur l'estimation de Mourre}, journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles}, year = {2006}, pages = {1-12}, doi = {10.5802/jedp.32}, language = {fr}, url = {http://dml.mathdoc.fr/item/JEDP_2006____A5_0} }
Golénia, Sylvain. Un nouveau regard sur l’estimation de Mourre. Journées équations aux dérivées partielles, (2006), pp. 1-12. doi : 10.5802/jedp.32. http://gdmltest.u-ga.fr/item/JEDP_2006____A5_0/
[ABG] W.O. Amrein, A. Boutet de Monvel and V. Georgescu : -groups, commutator methods and spectral theory of -body hamiltonians., Birkhäuser 1996. | MR 1388037 | Zbl 0962.47500
[BCHM] J-F. Bony, R. Carles, D. Haeffner, L. Michel : Scattering theory for the Schrödinger equation with repulsive potential. J. Math. Pures Appl. 84, no. 5, 509-579, 2005. | MR 2134848 | Zbl 1082.35130
[B] N. Burq : Semiclassical estimates for the resolvent in non trapping geometries. Int. Math. Res. Notices 2002, no 5, 221–241. | MR 1876933 | Zbl 01719337
[CJ] F. Castella, Th. Jecko : Besov estimates in the high-frequency Helmholtz equation, for a non-trapping and potential. To appear in J. Diff. Eq. | MR 2289541 | Zbl 1105.35091
[CGH] L. Cattaneo, G. M. Graf and W. Hunziker : A general resonance theory based on Mourre’s inequality, math-ph/0507063.
[DG] J. Dereziński and C. Gérard : Scattering theory of classical and quantum N-particle systems. Springer-Verlag 1997. | MR 1459161 | Zbl 0899.47007
[DJ] J. Dereziński and V. Jakšić : Spectral theory of Pauli-Fierz operators. J. Funct. Anal. 180, no 2, pp. 243-327, 2001. | MR 1814991 | Zbl 1034.81016
[GGé] V. Georgescu and C. Gérard : On the Virial Theorem in Quantum Mechanics, Commun. Math. Phys. 208, 275–281, (1999). | MR 1729087 | Zbl 0961.81009
[GGM1] V. Georgescu, C. Gérard, and J.S. Møller : Commutators, -semigroups and resolvent estimates, J. Funct. Anal. 216, no 2, pp. 303-361, 2004. | MR 2095686 | Zbl 1073.47518
[GGM2] V. Georgescu, C. Gérard, and J.S. Møller : Spectral theory of massless Pauli-Fierz models, Comm. Math. Phys. 249, no 1, pp. 29-78, 2004. | MR 2077252 | Zbl 1091.81059
[GGo] V. Georgescu, S. Golénia : Isometries, Fock spaces and spectral analysis of Schrödinger operators on trees., Journal of Functional Analysis 227 (2005), 389-429. | MR 2168080 | Zbl 1106.47006
[GJ] S. Golénia, T. Jecko : A new look at Mourre’s commutator theorem., preprint mp_arc 06-138, submitted to publication.
[HeS] B. Helffer and J. Sjöstrand : Opérateurs de Schrödinger avec champs magnétiques faibles et constants. Exposé No. XII, Séminaire EDP, février 1989, Ecole Polytechnique. | Numdam | MR 1032288 | Zbl 0702.35185
[HuS] W. Hunziker and I.M. Sigal : The quantum -body problem, J. Math. Phys. 41 (6), 3448–3510, 2000. | MR 1768629 | Zbl 0981.81026
[J1] Th. Jecko : From classical to semiclassical non-trapping behaviour, C. R. Acad. Sci. Paris, Ser. I, 338, p. 545–548, 2004. | MR 2057027 | Zbl 1046.81037
[J2] Th. Jecko : Non-trapping condition for semiclassical Schrödinger operators with matrix-valued potentials. Math. Phys. Electronic Journal, No. 2, vol. 11, 2005. | MR 2122361 | Zbl 1067.81037
[M] E. Mourre : Absence of singular continuous spectrum for certain self-adjoint operators. Commun. in Math. Phys. 78, 391–408, 1981. | MR 603501 | Zbl 0489.47010
[RS4] M. Reed, B. Simon : Methods of Modern Mathematical Physics, Tome IV : Analysis of operators. Academic Press. | Zbl 0401.47001
[S] J. Sahbani :The conjugate operator method for locally regular Hamiltonians. J. Oper. Theory 38, No. 2, 297–322 (1997). | MR 1606944 | Zbl 0905.47003