Localization for the one-dimensional Anderson model via positivity and large deviations for the Lyapunov exponent
HTML articles powered by AMS MathViewer
- by Valmir Bucaj, David Damanik, Jake Fillman, Vitaly Gerbuz, Tom VandenBoom, Fengpeng Wang and Zhenghe Zhang PDF
- Trans. Amer. Math. Soc. 372 (2019), 3619-3667
Abstract:
We provide a complete and self-contained proof of spectral and dynamical localization for the one-dimensional Anderson model, starting from the positivity of the Lyapunov exponent provided by Fürstenberg’s theorem. That is, a Schrödinger operator in $\ell ^2(\mathbb {Z})$ whose potential is given by independent, identically distributed (i.i.d.) random variables almost surely has pure point spectrum with exponentially decaying eigenfunctions, and its unitary group exhibits exponential off-diagonal decay, uniformly in time. We also explain how to obtain analogous statements for extended CMV matrices whose Verblunsky coefficients are i.i.d., as well as for half-line analogues of these models.References
- Andre Ahlbrecht, Volkher B. Scholz, and Albert H. Werner, Disordered quantum walks in one lattice dimension, J. Math. Phys. 52 (2011), no. 10, 102201, 48. MR 2894584, DOI 10.1063/1.3643768
- Michael Aizenman, Localization at weak disorder: some elementary bounds, Rev. Math. Phys. 6 (1994), no. 5A, 1163–1182. Special issue dedicated to Elliott H. Lieb. MR 1301371, DOI 10.1142/S0129055X94000419
- Michael Aizenman and Stanislav Molchanov, Localization at large disorder and at extreme energies: an elementary derivation, Comm. Math. Phys. 157 (1993), no. 2, 245–278. MR 1244867
- Michael Aizenman, Jeffrey H. Schenker, Roland M. Friedrich, and Dirk Hundertmark, Finite-volume fractional-moment criteria for Anderson localization, Comm. Math. Phys. 224 (2001), no. 1, 219–253. Dedicated to Joel L. Lebowitz. MR 1868998, DOI 10.1007/s002200100441
- Joseph Avron and Barry Simon, Almost periodic Schrödinger operators. II. The integrated density of states, Duke Math. J. 50 (1983), no. 1, 369–391. MR 700145, DOI 10.1215/S0012-7094-83-05016-0
- Ilia Binder, Michael Goldstein, and Mircea Voda, On fluctuations and localization length for the Anderson model on a strip, J. Spectr. Theory 5 (2015), no. 1, 193–225. MR 3340179, DOI 10.4171/JST/94
- Ilia Binder, Michael Goldstein, and Mircea Voda, On the sum of the non-negative Lyapunov exponents for some cocycles related to the Anderson model, Ergodic Theory Dynam. Systems 37 (2017), no. 2, 369–388. MR 3614029, DOI 10.1017/etds.2015.59
- Philippe Bougerol and Jean Lacroix, Products of random matrices with applications to Schrödinger operators, Progress in Probability and Statistics, vol. 8, Birkhäuser Boston, Inc., Boston, MA, 1985. MR 886674, DOI 10.1007/978-1-4684-9172-2
- J. Bourgain and M. Goldstein, On nonperturbative localization with quasi-periodic potential, Ann. of Math. (2) 152 (2000), no. 3, 835–879. MR 1815703, DOI 10.2307/2661356
- Jean Bourgain and Wilhelm Schlag, Anderson localization for Schrödinger operators on $\bf Z$ with strongly mixing potentials, Comm. Math. Phys. 215 (2000), no. 1, 143–175. MR 1800921, DOI 10.1007/PL00005538
- María-José Cantero, F. Alberto Grünbaum, Leandro Moral, and Luis Velázquez, Matrix-valued Szegő polynomials and quantum random walks, Comm. Pure Appl. Math. 63 (2010), no. 4, 464–507. MR 2604869
- René Carmona, Abel Klein, and Fabio Martinelli, Anderson localization for Bernoulli and other singular potentials, Comm. Math. Phys. 108 (1987), no. 1, 41–66. MR 872140
- W. Craig and B. Simon, Subharmonicity of the Lyaponov index, Duke Math. J. 50 (1983), no. 2, 551–560. MR 705040, DOI 10.1215/S0012-7094-83-05025-1
- D. Damanik, A short course on one-dimensional random Schrödinger operators, arXiv:1107.1094 (2011).
- D. Damanik and J. Fillman, Spectral theory of discrete one-dimensional ergodic Schrödinger operators (in preparation).
- David Damanik, Jake Fillman, Milivoje Lukic, and William Yessen, Characterizations of uniform hyperbolicity and spectra of CMV matrices, Discrete Contin. Dyn. Syst. Ser. S 9 (2016), no. 4, 1009–1023. MR 3543643, DOI 10.3934/dcdss.2016039
- David Damanik and Anton Gorodetski, An extension of the Kunz-Souillard approach to localization in one dimension and applications to almost-periodic Schrödinger operators, Adv. Math. 297 (2016), 149–173. MR 3498796, DOI 10.1016/j.aim.2016.04.006
- D. Damanik and P. Stollmann, Multi-scale analysis implies strong dynamical localization, Geom. Funct. Anal. 11 (2001), no. 1, 11–29. MR 1829640, DOI 10.1007/PL00001666
- E. B. Davies and B. Simon, Eigenvalue estimates for non-normal matrices and the zeros of random orthogonal polynomials on the unit circle, J. Approx. Theory 141 (2006), no. 2, 189–213. MR 2252099, DOI 10.1016/j.jat.2006.03.006
- 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. 69 (1996), 153–200. MR 1428099, DOI 10.1007/BF02787106
- François Delyon, Hervé Kunz, and Bernard Souillard, One-dimensional wave equations in disordered media, J. Phys. A 16 (1983), no. 1, 25–42. MR 700179
- J. Ding and C. K. Smart, Localization near the edge for the Anderson Bernoulli model on the two dimensional lattice, arXiv:1809.09041 (2018).
- Henrique von Dreifus and Abel Klein, A new proof of localization in the Anderson tight binding model, Comm. Math. Phys. 124 (1989), no. 2, 285–299. MR 1012868
- Alexander Elgart and Abel Klein, An eigensystem approach to Anderson localization, J. Funct. Anal. 271 (2016), no. 12, 3465–3512. MR 3558248, DOI 10.1016/j.jfa.2016.09.008
- J. Fröhlich, F. Martinelli, E. Scoppola, and T. Spencer, Constructive proof of localization in the Anderson tight binding model, Comm. Math. Phys. 101 (1985), no. 1, 21–46. MR 814541
- Jürg Fröhlich and Thomas Spencer, Absence of diffusion in the Anderson tight binding model for large disorder or low energy, Comm. Math. Phys. 88 (1983), no. 2, 151–184. MR 696803
- Harry Furstenberg, Noncommuting random products, Trans. Amer. Math. Soc. 108 (1963), 377–428. MR 163345, DOI 10.1090/S0002-9947-1963-0163345-0
- H. Furstenberg and Y. Kifer, Random matrix products and measures on projective spaces, Israel J. Math. 46 (1983), no. 1-2, 12–32. MR 727020, DOI 10.1007/BF02760620
- F. Germinet and S. De Bièvre, Dynamical localization for discrete and continuous random Schrödinger operators, Comm. Math. Phys. 194 (1998), no. 2, 323–341. MR 1627657, DOI 10.1007/s002200050360
- François Germinet and Abel Klein, Bootstrap multiscale analysis and localization in random media, Comm. Math. Phys. 222 (2001), no. 2, 415–448. MR 1859605, DOI 10.1007/s002200100518
- François Germinet and Abel Klein, A characterization of the Anderson metal-insulator transport transition, Duke Math. J. 124 (2004), no. 2, 309–350. MR 2078370, DOI 10.1215/S0012-7094-04-12423-6
- Francois Germinet and Abel Klein, New characterizations of the region of complete localization for random Schrödinger operators, J. Stat. Phys. 122 (2006), no. 1, 73–94. MR 2203782, DOI 10.1007/s10955-005-8068-9
- Fritz Gesztesy and Maxim Zinchenko, Weyl-Titchmarsh theory for CMV operators associated with orthogonal polynomials on the unit circle, J. Approx. Theory 139 (2006), no. 1-2, 172–213. MR 2220038, DOI 10.1016/j.jat.2005.08.002
- Michael Goldstein and Wilhelm Schlag, Hölder continuity of the integrated density of states for quasi-periodic Schrödinger equations and averages of shifts of subharmonic functions, Ann. of Math. (2) 154 (2001), no. 1, 155–203. MR 1847592, DOI 10.2307/3062114
- A. Gorodetski and V. Kleptsyn, Parametric Fürstenberg theorem on random products of $\mathrm {SL}(2,\mathbb {R})$ matrices, arXiv:1809.00416 (2018).
- Eman Hamza, Alain Joye, and Günter Stolz, Localization for random unitary operators, Lett. Math. Phys. 75 (2006), no. 3, 255–272. MR 2211031, DOI 10.1007/s11005-005-0044-4
- S. Jitomirskaya and X. Zhu, Large deviations of the Lyapunov exponent and localization for the 1D Anderson model, arxiv:1803.10697 (2018); Comm. Math. Phys. (to appear).
- Alain Joye, Fractional moment estimates for random unitary operators, Lett. Math. Phys. 72 (2005), no. 1, 51–64. MR 2150261, DOI 10.1007/s11005-005-3256-8
- Werner Kirsch, An invitation to random Schrödinger operators, Random Schrödinger operators, Panor. Synthèses, vol. 25, Soc. Math. France, Paris, 2008, pp. 1–119 (English, with English and French summaries). With an appendix by Frédéric Klopp. MR 2509110
- Helge Krüger, Orthogonal polynomials on the unit circle with Verblunsky coefficients defined by the skew-shift, Int. Math. Res. Not. IMRN 18 (2013), 4135–4169. MR 3106885, DOI 10.1093/imrn/rns173
- Hervé Kunz and Bernard Souillard, Sur le spectre des opérateurs aux différences finies aléatoires, Comm. Math. Phys. 78 (1980/81), no. 2, 201–246 (French, with English summary). MR 597748
- Émile Le Page, Théorèmes limites pour les produits de matrices aléatoires, Probability measures on groups (Oberwolfach, 1981) Lecture Notes in Math., vol. 928, Springer, Berlin-New York, 1982, pp. 258–303 (French). MR 669072
- Émile Le Page, Régularité du plus grand exposant caractéristique des produits de matrices aléatoires indépendantes et applications, Ann. Inst. H. Poincaré Probab. Statist. 25 (1989), no. 2, 109–142 (French, with English summary). MR 1001021
- L. A. Pastur, Spectral properties of disordered systems in the one-body approximation, Comm. Math. Phys. 75 (1980), no. 2, 179–196. MR 582507
- David Ruelle, Ergodic theory of differentiable dynamical systems, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 27–58. MR 556581
- I. É. Šnol′, On the behavior of eigenfunctions, Doklady Akad. Nauk SSSR (N.S.) 94 (1954), 389–392 (Russian). MR 0062902
- C. Shubin, R. Vakilian, and T. Wolff, Some harmonic analysis questions suggested by Anderson-Bernoulli models, Geom. Funct. Anal. 8 (1998), no. 5, 932–964. MR 1650106, DOI 10.1007/s000390050078
- Barry Simon, Spectrum and continuum eigenfunctions of Schrödinger operators, J. Functional Analysis 42 (1981), no. 3, 347–355. MR 626449, DOI 10.1016/0022-1236(81)90094-X
- Barry Simon, Localization in general one-dimensional random systems. I. Jacobi matrices, Comm. Math. Phys. 102 (1985), no. 2, 327–336. MR 820578
- Barry Simon, Orthogonal polynomials on the unit circle. Part 1, American Mathematical Society Colloquium Publications, vol. 54, American Mathematical Society, Providence, RI, 2005. Classical theory. MR 2105088, DOI 10.1090/coll054.1
- Barry Simon, Orthogonal polynomials on the unit circle. Part 2, American Mathematical Society Colloquium Publications, vol. 54, American Mathematical Society, Providence, RI, 2005. Spectral theory. MR 2105089, DOI 10.1090/coll/054.2/01
- Barry Simon and Tom Wolff, Singular continuous spectrum under rank one perturbations and localization for random Hamiltonians, Comm. Pure Appl. Math. 39 (1986), no. 1, 75–90. MR 820340, DOI 10.1002/cpa.3160390105
- A. V. Teplyaev, The pure point spectrum of random orthogonal polynomials on the circle, Dokl. Akad. Nauk SSSR 320 (1991), no. 1, 49–53 (Russian); English transl., Soviet Math. Dokl. 44 (1992), no. 2, 407–411. MR 1151514
- D. Thouless, A relation between the density of states and range of localization for one-dimensional systems, J. Phys. C 5 (1972), 77–81.
- Jhishen Tsay, Some uniform estimates in products of random matrices, Taiwanese J. Math. 3 (1999), no. 3, 291–302. MR 1705982, DOI 10.11650/twjm/1500407129
- Fengpeng Wang and David Damanik, Anderson localization for quasi-periodic CMV matrices and quantum walks, J. Funct. Anal. 276 (2019), no. 6, 1978–2006. MR 3912798, DOI 10.1016/j.jfa.2018.10.016
- Z. Zhang, Uniform positivity of the Lyapunov exponent for monotone potentials generated by the doubling map, arXiv:1610.02137 (2016).
Additional Information
- Valmir Bucaj
- Affiliation: Department of Mathematics, United States Military Academy, West Point, New York 10996
- MR Author ID: 1039116
- Email: valmir.bucaj@westpoint.edu
- David Damanik
- Affiliation: Department of Mathematics, Rice University, Houston, Texas 77005
- MR Author ID: 621621
- Email: damanik@rice.edu
- Jake Fillman
- Affiliation: Department of Mathematics, Virginia Tech, 225 Stanger Street—0123, Blacksburg, Virginia 24061
- MR Author ID: 1065002
- Email: fillman@vt.edu
- Vitaly Gerbuz
- Affiliation: Department of Mathematics, Rice University, Houston, Texas 77005
- Email: vitaly.gerbuz@rice.edu
- Tom VandenBoom
- Affiliation: Department of Mathematics, Yale University, New Haven, Connecticut 06511
- MR Author ID: 1262719
- Email: thomas.vandenboom@yale.edu
- Fengpeng Wang
- Affiliation: School of Mathematics (Zhuhai), Sun Yat-sen University, Zhuhai, Guangdong 519082, People’s Republic of China
- MR Author ID: 1266120
- Email: wfpouc@163.com
- Zhenghe Zhang
- Affiliation: Department of Mathematics, University of California, Riverside, California 92521
- MR Author ID: 985480
- Email: zhenghe.zhang@ucr.edu
- Received by editor(s): November 30, 2018
- Received by editor(s) in revised form: February 19, 2019
- Published electronically: April 29, 2019
- Additional Notes: The first, second, fourth, and fifth authors were supported in part by NSF grant DMS-1361625.
The main idea of the new proof of the LDT in Section 3 was communicated to the second and seventh authors by Artur Avila while they were visiting IMPA, Rio de Janeiro. They would like to thank Artur Avila for sharing his idea, and IMPA for the hospitality.
The third author was supported in part by an AMS-Simons travel grant, 2016–2018
The sixth author was supported by CSC (No. 201606330003) and NSFC (No. 11571327).
The seventh author was supported in part by an AMS-Simons travel grant, 2014–2016 - © Copyright 2019 by the authors
- Journal: Trans. Amer. Math. Soc. 372 (2019), 3619-3667
- MSC (2010): Primary 35J10; Secondary 81Q10
- DOI: https://doi.org/10.1090/tran/7832
- MathSciNet review: 3988621