Special Session 89: DYNAMICS AND SPECTRA OF QUASIPERIODIC SCHRODINGER OPERATORS

Reducibility without KAM

Fernando Argentieri University of Zurich Switzerland Co-Author(s):    Fernando Argentieri and Bassam Fayad

Abstract: We prove reducibility results for close to constant 1-frequency quasi-periodic matrix valued cocycles in finite and smooth regularity without the use of a quadratic scheme. Reducibility results to rotations matrices are obtained regardless of the arithmetics of the irrational base frequency.

Semiclassical limits of quasiperiodic media

Simon Becker ETH Zurich Switzerland Co-Author(s):

Abstract: I will review the study of continuum models of periodic media in (almost) magnetic fields and outline how the semiclassical reduction leads to generalized discrete infinite range models that are perturbations of the standard representatives in the study of quasiperiodic operators. We will consider different scaling regimes leading to different spectral problems. I will then describe how effects like metal/insulator transitions manifest themselves in the continuum.

Regularizing effect of randomness on quasiperiodic dynamics

Ao Cai Soochow University Peoples Rep of China Co-Author(s):    Pedro Duarte and Silvius Klein

Abstract: We prove that there is an instant enhancement on the regularity, i.e. from discontinuity to H\older continuity, of the Lyapunov exponent of quasiperiodic Schr\odinger operators when randomness is involved in. This is based on a long-term project joint with Pedro Duarte (ULisboa) and Silvius Klein (PUC-Rio).

Exact local distribution of the absolutely continuous spectral measure

Xianzhe Li Nankai University Peoples Rep of China Co-Author(s):    Jiangong You and Qi Zhou

Abstract: It is well-established that the spectral measure for one-frequency Schr\odinger operators with Diophantine frequencies exhibits optimal $1/2$-H\older continuity within the absolutely continuous spectrum. This study extends these findings by precisely characterizing the local distribution of the spectral measure for dense small potentials, including a notable result for any subcritical almost Mathieu operators. Additionally, we investigate the stratified H\older continuity of the spectral measure at subcritical energies.

Reducibility of quasi-periodic symplectic cocycles

Yi Pan Institute of Science and Technology Austria Austria Co-Author(s):    Artur Avila, Raphael Krikorian

Abstract: Reducibility of quasi-periodic cocyles valued in symplectic groups is related to the spectrum of discrete Schrodinger operators on strips. We will talk about a global reducibility result: given one parameter family of such cocycles, for almost every parameter, either the maximal Lyapunov exponent is positive, or the cocycle is almost conjugate to some precise model. The techniques include Kotani theory, KAM theory and in particular study of hyperbolicity of renormalization operator. This is a joint work with Artur Avila and Raphael Krikorian.

Absolute continuity and Holder continuity of the integrated density of states (IDS) for the analytic quasiperiodic Schrodinger operators

Jing Wang Nanjing University of Science and Technology Peoples Rep of China Co-Author(s):    Xu Xu, Jiangong You, Qi Zhou

Abstract: In this talk, we will consider the continuity of the IDS for analytic quasiperiodic Schrodinger operators. In the subcritical region, we prove that the IDS and Lyapunov exponents are log-Holder continuous for operators with Liouvillean frequency; in the supercritical region, we prove that the IDS is absolutely continuous for operators with trigonometric potential and Diophantine frequency. The main tool is quantitative almost reducibility and Aubry duality. This talk is based on joint works with Xu Xu, Qi Zhou and Jiangong You.

Lyapunov spectrum and hyperbolicity of one frequency quasi-periodic Sp(4)-cocycle.

Disheng Xu Great Bay University Peoples Rep of China Co-Author(s):    Duxiao Wang, Qi Zhou

Abstract: We will talk about the Lyapunov exponents and hyperbolic behavior of one frequency quasi-periodic Sp(4) cocycle under certain assumptions. Joint work (in progress) with Duxiao Wang and Qi Zhou.

Delocalization of a general class of random block Schrodinger operators

Fan Yang Tsinghua University Peoples Rep of China Co-Author(s):    Changji Xu, Horng-Tzer Yau, Jun Yin

Abstract: Consider two generalizations of the famous Anderson model defined on a $d$-dimensional integer lattice of linear size $L$. The first generalization is the random band matrix model. In this model, the entries are independent centered complex Gaussian random variables, and the element $H_{xy}$ is nonzero only when the distance $|x-y|$ is less than the band width $W$. The second generalization is the block Anderson model. In this model, the i.i.d. diagonal potential in the Anderson model is replaced by an i.i.d. diagonal block potential with a coupling strength parameter $\lambda>0$, and the blocks have a linear size of $W$. Both models are non-mean-field models, where the parameter $W$ describes the length of local interactions. Furthermore, it is conjectured that these models exhibit Anderson transitions as $W$ or $\lambda$ varies. In this talk, I will present some of our recent results on the Anderson delocalization of these two models when $d\ge 7$ and $W\ge L^\delta$, where $\delta>0$ is a small constant. Additionally, I will discuss the quantum diffusion conjecture related to the delocalization of these models. The research is based on joint works with Changji Xu, Horng-Tzer Yau, and Jun Yin.

Anderson localization for potentials generated by hyperbolic transformations

Zhenghe Zhang UC Riverside USA Co-Author(s):    Artur Avila, David Damanik

Abstract: In this talk, I will prove Anderson localization for potentials generated by hyperbolic transformations via positivity and large deviations of the Lypapunov exponent. This is a joint work with Artur Avila and David Damanik.