no. 3
Transfer matrices and transport for Schrödinger operators
[Matrices de transfert et transport pour des opérateurs de Schrödinger]
Germinet, François1 ; Kiselev, Alexander ; Tcheremchantsev, Serguei
1 Université de Cergy-Pontoise, laboratoire AGM, CNRS-UMR 8088, Dépt. de Mathématiques, 95302 Cergy-Pontoise Cédex, (France), University of Wisconsin, Department of Mathematics, Madison, WI 53706, (USA), Université d'Orléans, Laboratoire MAPMO, CNRS-UMR 6628, B.P. 6759, 45067 Orléans Cédex, (France)
Annales de l'Institut Fourier, Tome 54 (2004) no. 3, pp. 787-830.

Nous fournissons une borne inférieure générale pour la dynamique des opérateurs de Schrödinger unidimensionnels en fonction des matrices de transfert. En particulier, cela donne une borne inférieure non triviale pour les exposants de transport dès que la norme des matrices de transfert ne croît pas plus vite que polynômialement sur un ensemble d'énergie de mesure de Lebesgue pleine, et ce indépendamment de la nature du spectre. Des applications avec des hamiltoniens avec des potentiels a) épars, b) quasi-périodique, c) aléatoires décroissant sont données. De plus, nous développons dans un contexte général une analyse des paquets d'ondes qui permet de caractériser les exposants de transport à petit et grand moments.

We provide a general lower bound on the dynamics of one dimensional Schrödinger operators in terms of transfer matrices. In particular it yields a non trivial lower bound on the transport exponents as soon as the norm of transfer matrices does not grow faster than polynomially on a set of energies of full Lebesgue measure, and regardless of the nature of the spectrum. Applications to Hamiltonians with a) sparse, b) quasi-periodic, c) random decaying potential are provided. We also develop some general analysis of wave- packets that enables one to characterize transports exponents at low and large moments.

DOI : 10.5802/aif.2034
Classification : 81Q10, 47N50
Keywords: Schrödinger operators, transfer matrices, transport exponents
Mot clés : opérateur de Schrödinger, matrices de transfert, exposants de transport

Germinet, François 1 ; Kiselev, Alexander  ; Tcheremchantsev, Serguei 

1 Université de Cergy-Pontoise, laboratoire AGM, CNRS-UMR 8088, Dépt. de Mathématiques, 95302 Cergy-Pontoise Cédex, (France), University of Wisconsin, Department of Mathematics, Madison, WI 53706, (USA), Université d'Orléans, Laboratoire MAPMO, CNRS-UMR 6628, B.P. 6759, 45067 Orléans Cédex, (France) 
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Germinet, François; Kiselev, Alexander; Tcheremchantsev, Serguei. Transfer matrices and transport for Schrödinger operators. Annales de l'Institut Fourier, Tome 54 (2004) no. 3, pp. 787-830. doi : 10.5802/aif.2034. https://aif.centre-mersenne.org/articles/10.5802/aif.2034/

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