Transfer matrices and transport for Schrödinger operators
Annales de l'Institut Fourier, Volume 54 (2004) no. 3, p. 787-830

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.

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.

Classification:  81Q10,  47N50
Keywords: Schrödinger operators, transfer matrices, transport exponents
     author = {Germinet, Fran\c cois and Kiselev, Alexander and Tcheremchantsev, Serguei},
     title = {Transfer matrices and transport for Schr\"odinger operators},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {54},
     number = {3},
     year = {2004},
     pages = {787-830},
     doi = {10.5802/aif.2034},
     mrnumber = {2097423},
     zbl = {1074.81019},
     language = {en},
Germinet, François; Kiselev, Alexander; Tcheremchantsev, Serguei. Transfer matrices and transport for Schrödinger operators. Annales de l'Institut Fourier, Volume 54 (2004) no. 3, pp. 787-830. doi : 10.5802/aif.2034.

I. Guarneri On an estimate concerning quantum diffusion in the presence of a fractal spectrum, Europhys. Lett., Tome 21 (1993), pp. 729-733 | MR 1470878 | Zbl 0893.47048

G. Mantica Wave propagation in almost-periodic structures, Physica D, Tome 109 (1997), pp. 113-127 | MR 2054797 | Zbl 1053.81028

[BCM] J. M. Barbaroux; R. Montcho Remarks on the relation between quantum dynamics and fractal spectra, J. Math. Anal. Appl, Tome 213 (1997) no. 2, pp. 698-722 | MR 1910829 | Zbl 1014.82021

[BGK] J.-M. Bouclet; F. Germinet; A. Klein Sub-exponential decay of operator kernels for functions of generalized Schrödinger operators (to appear in Proc. Amer. Math. Soc) | MR 1861091 | Zbl 1012.81018

[BGSB] J. Bellissard; I. Guarneri; H. Schulz-Baldes Phase-averaged transport for quasi-periodic Hamiltonians, Comm. Math. Phys, Tome 227 (2002) no. 3, pp. 515-539

[BGT2] J.-M. Barbaroux; F. Germinet; S. Tcheremchantsev Quantum diffusion and generalized fractal dimensions: the $\hbox{L^2(\RR^d)}$ case, Actes des journées EDP de Nantes (2000) | MR 1876760 | Zbl 1050.28006

[BGT1] J.-M. Barbaroux; F. Germinet; S. Tcheremchantsev Fractal dimensions and the phenomenon of intermittency in quantum dynamics, Duke Math. J, Tome 110 (2001), pp. 161-193 | MR 1762667 | Zbl 0962.82030

[BGT3] J.-M. Barbaroux; F. Germinet; S. Tcheremchantsev Generalized fractal dimensions: equivalence and basic properties, J. Math. Pure et Appl, Tome 80 (2001), pp. 977-1012 | Zbl 0797.35136

[BSB] J. Bellissard; H. Schulz-Baldes Subdiffusive quantum transport for 3-D Hamiltonians with absolutely continuous spectra, J. Stat. Phys., Tome 99 (2000), pp. 587-594 | Zbl 0619.47005

[C] J.-M. Combes; Eds. W.F. Ames, E.M. Harrel, J.V. Herod Connection between quantum dynamics and spectral properties of time evolution operators, Differential Equations and Applications in Mathematical Physics (1993), pp. 59-69 | MR 1102675 | Zbl 0717.60074

[CFKS] H. Cycon; R. Froese; W. Kirsch; B. Simon Schrödinger Operators, Springer-Verlag, 1987 | Zbl 0979.81035

[CL] R. Carmona; J. Lacroix Spectral theory of random Schrödinger operators, Birkhaüser, Boston, 1990 | MR 1349825 | Zbl 0893.47004

[CM] J.M. Combes; G. Mantica Fractal Dimensions and Quantum Evolution Associated with Sparse Potential Jacobi Matrices, Long time behaviour of classical and quantum systems, (Bologna, 1999) (Ser. Concr. Appl. Math.) Tome 1 (2001), pp. 107-123

[Da] E.B. Davies Spectral Theory and Differential Operators, Cambridge University Press, 1995 | MR 1428099 | Zbl 0908.47002

[DR1] R. Del Rio; S. Jitomirskaya; Y. Last; B. Simon What is localization?, Phys. Rev. Lett., Tome 75 (1995), pp. 117-119 | MR 1298942 | Zbl 1055.47500

[DR2] R. Del Rio; S. Jitomirskaya; Y. Last; and B. Simon Operators with singular continuous spectrum. IV. Hausdorff dimensions, rank one perturbations and localization, J. Anal. Math., Tome 69 (1996), pp. 153-200 | MR 2021200 | Zbl 1033.81032

[DRMS] R. Del Rio; N. Makarov; B. Simon Operators with singular continuous spectrum. II. Rank one operators, Comm. Math. Phys, Tome 165 (1994), pp. 59-67

[DT] D. Damanik; S. Tcheremchantsev Power-law bounds on transfer matrices and quantum dynamics in one dimension, Comm. Math. Phys, Tome 236 (2003), pp. 513-534

[G] I. Guarneri Spectral properties of quantum diffusion on discrete lattices, Europhys. Lett, Tome 10 (1989), pp. 95-100 | MR 1937430 | Zbl 1013.81009

[GK2] F. Germinet; A. Klein A characterization of the Anderson metal-insulator transport transition (to appear in Duke Math. J) | MR 2042531 | Zbl 1062.82020

[GK1] F. Germinet; A. Klein Decay of operator-valued kernels of functions of Schrödinger and other operators, Proc. Amer. Math. Soc, Tome 131 (2003), pp. 911-920 | MR 2042531 | Zbl 02065046

[GK3] F. Germinet; A. Klein The Anderson metal-insulator transport transition, Contemp. Math, Tome 339 (2003), pp. 43-57 | MR 915965 | Zbl 0666.34023

[GP] D.J. Gilbert; D.B. Pearson On subordinacy and analysis of the spectrum of one-dimensional Schrödinger operators, J. Math. Anal. Appl, Tome 128 (1987), pp. 30-56 | MR 1663518 | Zbl 0910.47059

[GSB1] I. Guarneri; H. Schulz-Baldes Lower bounds on wave packet propagation by packing dimensions of spectral measures, Math. Phys. Elec. J, Tome 5 (1999) no. paper 1 | MR 1749574 | Zbl 1001.81019

[GSB2] I. Guarneri; H. Schulz-Baldes Intermittent lower bound on quantum diffusion, Lett. Math. Phys, Tome 49 (1999), pp. 317-324 | Zbl 05025523

[GT] F. Germinet; S. Tcheremchantsev Generalized fractal dimensions on the negative axis for compactly supported measures (preprint) | Zbl 0699.35189

[HS] B. Helffer; J. Sjöstrand; H. Holden and A. Jensen, eds. Equation de Schrödinger avec champ magnétique et équation de Harper in Schrödinger Operators (Lectures Notes in Physics) Tome 345 (1989), pp. 118-197 | MR 1738043 | Zbl 0991.81021

[JL] S. Jitomirskaya; Y. Last Power-law subordinacy and singular spectra. I. Half-line operators, Acta Math, Tome 183 (1999), pp. 171-189 | MR 1957731 | Zbl 1013.82027

[JSBS] S. Jitomirskaya; H. Schulz-Baldes; G. Stolz Delocalization in polymer models, Comm. Math. Phys, Tome 233 (2003), pp. 27-48 | MR 1895534 | Zbl 1043.35097

[KKS] A. Koines; A. Klein; M. Seifert Generalized Eigenfunctions for Waves in Inhomogeneous Media, J. Funct. Anal, Tome 190 (2002), pp. 255-291 | MR 1741780 | Zbl 0951.35033

[KL] A. Kiselev; Y. Last Solutions, spectrum, and dynamics for Schrödinger operators on infinite domains, Duke Math. J., Tome 102 (2000), pp. 125-150 | MR 1628290 | Zbl 0912.34074

[KLS] A. Kiselev; Y. Last; B. Simon Modified Prüfer and EFGP Transforms and the Spectral Analysis of One-Dimensional Schrödinger Operators, Commun. Math. Phys, Tome 194 (1997), pp. 1-45 | MR 1866165 | Zbl 01731919

[KrR] D. Krutikov; C. Remling Schrödinger operators with sparse potentials: asymptotics of the Fourier transform of the spectral measure, Comm. Math. Phys (2001), pp. 509-532 | MR 1423040 | Zbl 0905.47059

[La] Y. Last Quantum dynamics and decomposition of singular continuous spectrum, J. Funct. Anal, Tome 142 (1996), pp. 406-445 | MR 1666767 | Zbl 0931.34066

[LS] Y. Last; B. Simon Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators, Invent. Math., Tome 135 (1999), pp. 329-367

[Ma] G. Mantica Quantum intermittency in almost periodic systems derived from their spectral properties, Physica D, Tome 103 (1997), pp. 576-589 | Zbl 0925.58041

[P1] D. Pearson Singular continuous measures in scattering theory, Comm. Math. Phys, Tome 60 (1978) no. 1, pp. 13-36 | MR 1489237 | Zbl 0895.58033

[P] Pesin Dimension Theory in Dynamical Systems: Contemporary Views and Applications, Univ. Chicago Press, 1996 | MR 484145 | Zbl 0451.47013

[PF] L. Pastur; A. Figotin Spectra of Random and Almost-Periodic Operators, Springer-Verlag, Heidelberg, 1992 | MR 1223779 | Zbl 0752.47002

[SBB] H. Schulz-Baldes; J. Bellissard Anomalous transport: a mathematical framework, Rev. Math. Phys, Tome 10 (1998), pp. 1-46 | MR 1606847 | Zbl 0908.47066

[Si2] B. Simon; J. Feldman, R. Froese, L. Rosen, eds. Spectral Analysis and rank one perturbations and applications (CRM Lecture Notes) Tome 8 (1995), pp. 109-149 | MR 1350963 | Zbl 0944.34064

[Si1] B. Simon Bounded eigenfunctions and absolutely continuous spectra for one-dimensional Schrödinger operators, Proc. AMS, Tome 124 (1996), pp. 3361-3369 | Zbl 0824.47019

[SiSp] B. Simon; T. Spencer Trace class perturbations and the absence of absolutely continuous spectra, Comm. Math. Phys, Tome 125 (1989) no. 1, pp. 113-125 | MR 1017742 | Zbl 0684.47010

[SiSt] B. Simon; G. Stolz Operators with singular continuous spectrum. V. Sparse potentials, Proc. Amer. Math. Soc, Tome 124 (1996) no. 7, pp. 2073-2080 | MR 1342046 | Zbl 0979.34063

[T] E.C. Titchmarsh Eigenfunction Expansions, Oxford University Press, Oxford, 1962 | MR 176151 | Zbl 0099.05201

[Tc2] S. Tcheremchantsev Dynamical analysis of Schrödinger operators with growing sparse potentials (to appear in Commun. Math. Phys) | MR 1957683 | Zbl 1060.47070

[Tc1] S. Tcheremchantsev Mixed lower bounds in quantum dynamics, J. Funct. Anal, Tome 197 (2003), pp. 247-282 | MR 2105642 | Zbl 1100.47027

[We] J. Weidmann Spectral Theory of Ordinary Differential Operators, Lecture Notes in Mathematics, Tome 1258, Springer-Verlag, 1987 | MR 923320 | Zbl 0647.47052

[Z] A. Zlatos Sparse potentials with fractional Hausdorff dimension (to appear in J. Funct. Anal) | MR 2027640 | Zbl 1038.47026