Asymptotiques de Lifshitz

Klopp, Frédéric

Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2001-2002), Exposé no. 4, 12 p.

Cet exposé a pour but de présenter des résultats récents de l’auteur concernant les asymptotiques de Lifshitz pour des perturbations aléatoires d’opérateurs de Schrödinger périodiques. Certains de ces résultats ont été obtenus en collaboration avec T. Wolff.

[1] S. Agmon. On positive solutions of elliptic operators with periodic coefficients in $L^2\left({\mathbf{R}}^n\right)$, spectral results and extension to elliptic operators on Riemannian manifolds. Dans I. Knowles and R. Lewis, éditeurs, Differential equations, pages 7–17, Birmingham, 1984. North-Holland. | MR | Zbl

[2] J.-M. Barbaroux, J.-M. Combes, et P. Hislop. Landau hamiltonians with unbounded random potentials. Letters in Mathematical Physics, 40(4) :355–369, 1997. | MR | Zbl

[3] M. Sh Birman. Perturbations of operators with periodic coefficients. Dans Schrödinger Operators : standard and non-standard, Dubna, 1988.

[4] R. Carmona et J. Lacroix. Spectral Theory of Random Schrödinger Operators. Birkhäuser, Basel, 1990. | MR | Zbl

[5] A. Dembo et O. Zeitouni. Large deviation techniques and applications. Jones and Bartlett Publi-shers, Boston, 1992. | MR | Zbl

[6] J.-M. Deuschel et D. Stroock. Large deviations, volume 137 de Pure and applied Mathematics. Academic Press, 1989. | MR | Zbl

[7] M. Eastham. The spectral theory of periodic differential operators. Scottish Academic Press, Edinburgh, 1973. | Zbl

[8] F. Ghribi. Asymptotique de Lifshitz pour des opérateurs de Schrödinger à champ magnétique aléatoire. Thèse de doctorat, Université Paris 13, Villetaneuse. en préparation.

[9] B. Helffer et A. Mohamed. Asymptotic of the density of states for the Schrödinger operator with periodic electric potential. Duke Math. J., 92(1) :1–60, 1998. | MR | Zbl

[10] W. Kirsch. Random Schrödinger operators. Dans A. Jensen, H. Holden, éditeurs, Schrödinger Operators, number 345 in Lecture Notes in Physics, Berlin, 1989. Springer Verlag. Proceedings, Sonderborg, Denmark 1988. | MR | Zbl

[11] W. Kirsch et F. Martinelli. On the spectrum of Schrödinger operators with a random potential. Communications in Mathematical Physics, 85 :329–350, 1982. | MR | Zbl

[12] W. Kirsch et F. Martinelli. Large deviations and Lifshitz singularities of the integrated density of states of random hamiltonians. Communications in Mathematical Physics, 89 :27–40, 1983. | MR | Zbl

[13] W. Kirsch et B. Simon. Lifshitz tails for the Anderson model. Journal of Statistical Physics, 38 :65–76, 1985. | MR

[14] W. Kirsch, P. Stollmann, et G. Stolz. Localization for random perturbations of periodic Schrödinger operators. Random Oper. Stochastic Equations, 6(3) :241–268, 1998. | MR | Zbl

[15] F. Klopp. Une remarque à propos des asymptotiques de Lifshitz internes. Disponible sur http://zeus.math.univ-paris13.fr/~klopp/publi.html.

[16] F. Klopp. Localization for some continuous random Schrödinger operators. Communications in Mathematical Physics, 167 :553–570, 1995. | MR | Zbl

[17] F. Klopp. Band edge behaviour for the integrated density of states of random Jacobi matrices in dimension 1. Journal of Statistical Physics, 90(3-4) :927–947, 1998. | MR | Zbl

[18] F. Klopp et J. Ralston. Endpoints of the spectrum of periodic operators are generically simple. Methods and Applications of Analysis, 7(3) :459–464, 2000. | MR | Zbl

[19] F. Klopp et T. Wolff. Lifshitz tails for 2-dimensional random Schrödinger operators. Jour. d’Analyse Math. A paraître. | Zbl

[20] F. Klopp. Internal Lifshits tails for random perturbations of periodic Schrödinger operators. Duke Math. J., 98(2) :335–396, 1999. | MR | Zbl

[21] F. Klopp. Precise high energy asymptotics for the integrated density of states of an unbounded random Jacobi matrix. Rev. Math. Phys., 12(4) :575–620, 2000. | MR | Zbl

[22] F. Klopp. Internal lifshitz tails for schrödinger operators with random potentials. Jour. Math. Phys., 2002. A paraître. | MR | Zbl

[23] F. Klopp and L. Pastur. Lifshitz tails for random Schrödinger operators with negative singular Poisson potential. Comm. Math. Phys., 206(1) :57–103, 1999. | MR | Zbl

[24] P. Kuchment. Floquet theory for partial differential equations, volume 60 of Operator Theory : Advances and Applications. Birkhäuser, Basel, 1993. | MR | Zbl

[25] I. M. Lifshitz. Structure of the energy spectrum of impurity bands in disordered solid solutions. Soviet Physics JETP, 17 :1159–1170, 1963.

[26] I. M. Lifshitz. Energy spectrum structure and quantum states of disordered condensed systems. Soviet Physics Uspekhi, 7 :549–573, 1965. | MR

[27] I.M. Lifshitz, S.A. Gredeskul, et L.A. Pastur. Introduction to the theory of disordered systems. Wiley, New-York, 1988. | MR

[28] H. McKean et P. van Moerbeke. The spectrum of Hill’s equation. Inventiones Mathematicae, 30 :217–274, 1975. | Zbl

[29] H. P. McKean et E. Trubowitz. Hill’s surfaces and their theta functions. Bull. Amer. Math. Soc., 84(6) :1042–1085, 1978. | Zbl

[30] G. Mezincescu. Lifshitz singularities for periodic operators plus random potentials. Journal of Statistical Physics, 49 :1081–1090, 1987. | MR | Zbl

[31] G. Mezincescu. Internal Lifshits singularities for one dimensional Schrödinger operators. Communications in Mathematical Physics, 158 :315–325, 1993. | MR | Zbl

[32] H. Najar. Asymptotique de la densité d’états intégrée des modèles aléatoires continus. Thèse de doctorat, Université Paris 13, Villetaneuse, November 2000.

[33] S. Nakao. On the spectral distribution of the Schrödinger operator with random potential. Japan Journal of Mathematics, 3 :117–139, 1977. | MR | Zbl

[34] L. Pastur. Behaviour of some Wiener integrals as $t\to +\infty$ and the density of states of the Schrödinger equation with a random potential. Teor.-Mat.-Fiz, 32 :88–95, 1977. (in russian). | MR | Zbl

[35] L. Pastur et A. Figotin. Spectra of Random and Almost-Periodic Operators. Springer Verlag, Berlin, 1992. | MR | Zbl

[36] D. H. Phong et E. M. Stein. The Newton polyhedron and oscillatory integral operators. Acta Math., 179(1) :105–152, 1997. | MR | Zbl

[37] M. Reed et B. Simon. Methods of Modern Mathematical Physics, Vol IV : Analysis of Operators. Academic Press, New-York, 1978. | MR | Zbl

[38] O. Saad. Comportement en grandes énergies de la densité d’états du modèle d’Anderson non borné. Thèse de doctorat, Université de Paris 13, Villetaneuse. en préparation.

[39] J. Sjöstrand. Microlocal analysis for periodic magnetic Schrödinger equation and related questions. Dans Microlocal analysis and applications, volume 1495 des Lecture Notes in Mathematics, Berlin, 1991. Springer Verlag. | MR | Zbl

[40] M. M. Skriganov. Proof of the Bethe-Sommerfeld conjecture in dimension $2$. Dokl. Akad. Nauk SSSR, 248(1) :39–42, 1979. | MR | Zbl

[41] P. Stollmann. Lifshitz asymptotics via linear coupling of disorder. Math. Phys. Anal. Geom., 2(3) :279–289, 1999. | MR | Zbl

[42] A.S. Sznitman. Brownian motion, obstacles and random media. Springer-Verlag, Berlin, 1998. | MR | Zbl

[43] V. Varchenko. Newton polyhedra and estimations of oscillatory integrals. Functional Analysis and its Applications, 8 :175–196, 1976. | Zbl