Hostname: page-component-7b7567c8c-7pwfn Total loading time: 0 Render date: 2026-11-15T21:22:34.525Z Has data issue: false hasContentIssue false
  • English
  • Français

Continuous spectrum for a class of smooth mixing Schrödinger operators

Published online by Cambridge University Press:  02 May 2017

BASSAM FAYAD and
YANHUI QUOpen the ORCID record for YANHUI QU[Opens in a new window]
BASSAM FAYAD
Affiliation:
CNRS, IMJ-PRG, Institut de Mathématiques de Jussieu, UMR7586 Bâtiment Sophie Germain, 75205 Paris Cedex 13, France email bassam.fayad@gmail.com
YANHUI QU
Affiliation:
Department of Mathematical Science, Tsinghua University, Beijing 100084, PR China email yhqu@math.tsinghua.edu.cn
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

We give the first example of a smooth volume preserving mixing dynamical system such that the discrete Schrödinger operators on the line defined with a potential generated by this system and a Hölder sampling function have almost surely a continuous spectrum.

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 2 , February 2019, pp. 357 - 369
Copyright
© Cambridge University Press, 2017 

References

Avila, A. and Forni, G.. Weak mixing for interval exchange transformations and translation flows. Ann. of Math. (2) 165(2) (2007), 637–664.Google Scholar
Boshernitzan, M. and Damanik, D.. Generic continuous spectrum for ergodic Schrödinger operators. Comm. Math. Phys. 283(3) (2008), 647–662.Google Scholar
Boshernitzan, M. and Damanik, D.. The repetition property for sequences on tori generated by polynomials or skew-shifts. Israel J. Math. 174 (2009), 189–202.Google Scholar
Bourgain, J.. On the spectrum of lattice Schrödinger operators with deterministic potential. J. Anal. Math. 87 (2002), 37–75.Google Scholar
Bourgain, J., Goldstein, M. and Schlag, W.. Anderson localization for Schrödinger operators on ℤ with potentials given by the skew-shift. Comm. Math. Phys. 220(3) (2001), 583–621.Google Scholar
Bourgain, J. and Schlag, W.. Anderson localization for Schrödinger operators on ℤ with strongly mixing potentials. Comm. Math. Phys. 215(1) (2000), 143–175.Google Scholar
Carmona, R. and Lacroix, J.. Spectral Theory of Random Schrödinger Operators (Probability and its Applications) . Birkhäuser, Boston, MA, 1990.Google Scholar
Cycon, H. L., Froese, R. G., Kirsch, W. and Simon, B.. Schrödinger Operators with Application to Quantum Mechanics and Global Geometry (Texts and Monographs in Physics) . Springer Study Edition. Springer, Berlin, 1987.Google Scholar
Fayad, B.. Analytic mixing reparametrizations of irrational flows. Ergod. Th. & Dynam. Sys. 22 (2002), 437–468.Google Scholar
Fayad, B.. Smooth mixing diffeomorphisms and flows with singular spectra. Duke Math. J. 132(2) (2006), 371–391.Google Scholar
Gordon, A.. On the point spectrum of the one-dimensional Schrödinger operator. Uspekhi Mat. Nauk 31 (1976), 257–258.Google Scholar
Katok, A. B.. Interval exchange transformations and some special flows are not mixing. Israel J. Math. 35 (1980), 301–310.Google Scholar
Katok, A. B.. Combinatorial Constructions in Ergodic Theory and Dynamics (University Lecture Series, 30) . American Mathematical Society, Providence, RI, 2003.Google Scholar
Kunz, H. and Souillard, B.. Sur le spectre des opérateurs aux différences finies aléatoires. Comm. Math. Phys. 78(2) (1980–1981), 201–246.Google Scholar
Parry, W.. Topics in Ergodic Theory (Cambridge Tracts in Mathematics, 75) . Cambridge University Press, Cambridge, 1981.Google Scholar
Pastur, L. and Figotin, A.. Spectra of Random and Almost-periodic Operators (Grundlehren der Mathematischen Wissenschaften, 297) . Springer, Berlin, 1992.Google Scholar
Yoccoz, J.-C.. Centralisateurs et conjugaison différentiable des difféomorphismes du cercle. Petits diviseurs en dimension 1. Astérisque 231 (1995), 89–242 (in French).Google Scholar