On nonperturbative localization with quasi-periodic potential

By Jean Bourgain, Michael Goldstein. 2000

265 citations recorded by the source corpus.

The two main results of the article are concerned with Anderson Localization for one-dimensional lattice Schroedinger operators with quasi-periodic potentials with d frequencies. First, in the case d = 1 or 2, it is proved that the spectrum is pure-point with exponentially decaying eigenfunctions for all potentials (defined in terms of a trigonometric polynomial on the d-dimensional torus) for which the Lyapounov exponents are strictly positive for all frequencies and all energies. Second, for every non-constant real-analytic potential and with a Diophantine set of d frequencies, a lower bound is given for the Lyapounov exponents for the same potential rescaled by a sufficiently large constant.

Subfields: Atomic and Molecular Physics, and Optics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/0f12b15309adcdeb8f454eed68feb427d59f1336

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In Atomic and Molecular Physics, and Optics

1. David Damanik
2. Artur Avila
3. Kristian Bjerklöv
4. Yiqian Wang
5. Zhenghe Zhang

In Mathematical Physics

1. David Damanik
2. Artur Avila
3. Yiqian Wang
4. Zhenghe Zhang
5. Kristian Bjerklöv
6. A. Zorich
7. Marcelo Viana
8. Jiagang Yang

In Statistical and Nonlinear Physics

1. David Damanik
2. Artur Avila
3. Yiqian Wang
4. Zhenghe Zhang
5. Kristian Bjerklöv
6. Marcelo Viana
7. A. Zorich
8. Jiangong You

Citations in the corpus

• Positive Lyapunov exponent and minimality for a class of one-dimensional quasi-periodic Schrödinger equations (2005, background): “In [4], Bourgain and Goldstein show Anderson localization for analytic potential functions defined on T2 in the large coupling regime.”
• Schrödinger operators with dynamically defined potentials (2014, background): “The concept of non-perturbative localization was further explored in the more general setting of analytic sampling functions by Bourgain and Goldstein [38]; compare Theorem 5.”
• Strictly Ergodic Subshifts and Associated Operators (2005, background): “This is known for trigonometric potentials [76], analytic potentials [21, 70, 129], and Gevrey potentials [97].”
• Continuity of the Lyapunov Exponent for Quasiperiodic Operators with Analytic Potential (2001, unknown): “ˆ(0) = LN(E) As shown in [ 6 ] (see also [5], Ch. IV)”
• Uniform Positivity and Continuity of Lyapunov Exponents for a Class of $C^2$ Quasiperiodic Schrödinger Cocycles (2013, background): “valanche Principle, and proved Ho¨lder or weak Ho¨lder continuity of L(E) in the regime of positive LE. Notice that LDT for real analytic potentials with Diophantine frequency was first established in [BoGo] in order to get Anderson Localization. This also illustrates the power and importance of LDT.”
• Almost reducibility and absolute continuity I (2010, background): “is left with the problem of describing the stable regimes. As it turns out, the understanding of supercriticality is quite developed, through the works of Bourgain, Goldstein, Jitomirskaya and Schlag [BG], [GS1], [BJ1], [GS2], [GS3]. Subcriticality on the other hand, is a relatively new concept, whi”