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

DOI: 10.5802/aif.2034
Classification: 81Q10,  47N50
Keywords: Schrödinger operators, transfer matrices, transport exponents
@article{AIF_2004__54_3_787_0,
     author = {Germinet, Fran\c{c}ois and Kiselev, Alexander and Tcheremchantsev, Serguei},
     title = {Transfer matrices and transport for {Schr\"odinger} operators},
     journal = {Annales de l'Institut Fourier},
     pages = {787--830},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {54},
     number = {3},
     year = {2004},
     doi = {10.5802/aif.2034},
     mrnumber = {2097423},
     zbl = {1074.81019},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2034/}
}
TY  - JOUR
TI  - Transfer matrices and transport for Schrödinger operators
JO  - Annales de l'Institut Fourier
PY  - 2004
DA  - 2004///
SP  - 787
EP  - 830
VL  - 54
IS  - 3
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2034/
UR  - https://www.ams.org/mathscinet-getitem?mr=2097423
UR  - https://zbmath.org/?q=an%3A1074.81019
UR  - https://doi.org/10.5802/aif.2034
DO  - 10.5802/aif.2034
LA  - en
ID  - AIF_2004__54_3_787_0
ER  - 
%0 Journal Article
%T Transfer matrices and transport for Schrödinger operators
%J Annales de l'Institut Fourier
%D 2004
%P 787-830
%V 54
%N 3
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.2034
%R 10.5802/aif.2034
%G en
%F AIF_2004__54_3_787_0
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. https://aif.centre-mersenne.org/articles/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 | Zbl

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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

[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

[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 | Zbl

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

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

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

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

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

[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 | Zbl

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

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

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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

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

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

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

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

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

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

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

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

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

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

Cited by Sources: