Geodesic
On the analogues of Szegő's theorem for ergodic operators
Sbornik. Mathematics, Tome 206 (2015) no. 1, pp. 93-119 Cet article a éte moissonné depuis la source Math-Net.Ru

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Szegő's theorem on the asymptotic behaviour of the determinants of large Toeplitz matrices is generalized to the class of ergodic operators. The generalization is formulated in terms of a triple consisting of an ergodic operator and two functions, the symbol and the test function. It is shown that in the case of the one-dimensional discrete Schrödinger operator with random ergodic or quasiperiodic potential and various choices of the symbol and the test function this generalization leads to asymptotic formulae which have no analogues in the situation of Toeplitz operators. Bibliography: 22 titles.
Keywords: Szegő's theorem, random operators, limit theorems. 
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W. Kirsсh; L. Pastur. On the analogues of Szegő's theorem for ergodic operators. Sbornik. Mathematics, Tome 206 (2015) no. 1, pp. 93-119. http://geodesic.mathdoc.fr/item/SM_2015_206_1_a6/

[1] A. Böttcher, B. Silbermann, Analysis of Toeplitz Operators, Springer-Verlag, Berlin, 1990, 512 pp. | DOI | MR | Zbl

[2] P. Deift, A. Its, I. Krasovsky, “Toeplitz matrices and Toeplitz determinants under the impetus of the Ising model: some history and some recent results”, Comm. Pure Appl. Math., 66:9 (2013), 1360–1438 | DOI | MR | Zbl

[3] B. Simon, Szegő's theorem and its descendants. Spectral theory for $L^{2}$ perturbations of orthogonal polynomials, M. B. Porter Lectures, Princeton Univ. Press, Princeton, NJ, 2011, xii+650 pp. | MR | Zbl

[4] U. Grenander, G. Szegö, Töplitz forms and their applications, California Monographs in Mathematical Sciences, Univ. of California Press, Berkeley–Los Angeles, 1958, vii+245 pp. | MR | Zbl

[5] H. Widom, “Szegő expansions for operators with smooth or nonsmooth symbol”, Operator theory: operator algebras and applications, Part 1 (Durham, NH, 1988), Proc. Sympos. Pure Math., 51, Part 1, Amer. Math. Soc., Providence, RI, 1990, 599–615 | MR | Zbl

[6] A. V. Sobolev, Pseudodifferential operators with discontinuous symbols: Widom's conjecture, Mem. Amer. Math. Soc., 222, no. 1043, Amer. Math. Soc., Providence, RI, 2013, vi+104 pp. | DOI | MR | Zbl

[7] H. Widom, “On a class of integral operators with discontinuous symbol”, Toeplitz centennial (Tel Aviv, 1981), Operator Theory: Adv. Appl., 4, Birkhäuser, Basel–Boston, Mass., 1982, 477–500 | MR | Zbl

[8] A. V. Sobolev, “Quasi-classical asymptotics for pseudodifferential operators with discontinuous symbols: Widom's conjecture”, Funct. Anal. Appl., 44:4 (2010), 313–317 ; arXiv: 1004.2576 | DOI | DOI | MR | Zbl

[9] L. Pastur, A. Figotin, Spectra of random and almost-periodic operators, Grundlehren Math. Wiss., 297, Springer-Verlag, Berlin, 1992, viii+587 pp. | MR | Zbl

[10] A. Ya. Reznikova, “The central limit theorem for the spectrum of random Jacobi matrices”, Theory Probab. Appl., 25:3 (1981), 504–513 | DOI | MR | Zbl

[11] A. Ya. Reznikova, “Limit theorems for the spectra of 1D random Schrödinger operator”, Random Oper. Stochastic Equations, 12:3 (2004), 235–254 | DOI | MR | Zbl

[12] Y. Imry, Shang-Keng Ma, “Random-field instability of the ordered state of continuous symmetry”, Phys. Rev. Lett., 35 (1975), 1399–1401 | DOI

[13] J. Cardy, Scaling and renormalization in statistical physics, Cambridge Lecture Notes Phys., 5, Cambridge Univ. Press, Cambridge, 1996, xviii+238 pp. | MR | Zbl

[14] S. Jitomirskaya, “Ergodic Schrödinger operators (on one foot)”, Spectral theory and mathematical physics: a Festschrift in honor of Barry Simon's 60th birthday, Proc. Sympos. Pure Math., 76, Part 2, Amer. Math. Soc., Providence, RI, 2007, 613–647 | DOI | MR | Zbl

[15] F. Germinet, A. Taarabt, “Spectral properties of dynamical localization for Schrödinger operators”, Rev. Math. Phys., 25:9 (2013), 1350016, 28 pp. | DOI | MR | Zbl

[16] G. Stolz, “An introduction to the mathematics of Anderson localization”, Entropy and the quantum II, Contemp. Math., 552, Amer. Math. Soc., Providence, RI, 2011, 71–108 ; arXiv: 1104.2317 | DOI | MR | Zbl

[17] A. Ya. Khinchin, Continued fractions, Dover Publications, Inc., Mineola, NY, 1997, xii+95 pp. | MR | MR | Zbl | Zbl

[18] S. Jitomirskaya, H. Krüger, “Exponential dynamical localization for the almost Mathieu operator”, Comm. Math. Phys., 322:3 (2013), 877–882 | DOI | MR | Zbl

[19] S. Farkas, Z. Zimborás, “On the sharpness of the zero–entropy–density conjecture”, J. Math. Phys., 46:12 (2005), 123301, 8 pp. | DOI | MR | Zbl

[20] W. Kirsch, “An invitation to random Schrödnger operators”, Random Schrödinger operators, Panor. Synthèses, 25, Soc. Math. France, Paris, 2008, 1–119 | MR | Zbl

[21] L. Pastur, M. Shcherbina, Eigenvalue distribution of large random matrices, Math. Surveys Monogr., 171, Amer. Math. Soc., Providence, RI, 2011, xiv+632 pp. | DOI | MR | Zbl

[22] I. A. Ibragimov, Yu. V. Linnik, Independent and stationary sequences of random variables, Wolters–Noordhoff Publishing, Groningen, 1971, 443 pp. | MR | MR | Zbl | Zbl