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Timetable (PEPW03)

Almost-periodic and other ergodic problems

Tuesday 7th April 2015 to Friday 10th April 2015

Tuesday 7th April 2015
09:00 to 09:50 Registration
09:50 to 10:00 Welcome from John Toland (INI Director) INI 1
10:00 to 11:00 J You (Nanjing University)
Dry Ten Martini Problem in Non-Critical Case
We prove that the Dry Ten Martini Problem, i.e., all possible spectral gaps are open, holds for almost Mathieu operator with $(\lambda, \beta)\ne (\pm 1,0)$.
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Semi-classical analysis of non-self-adjoint transfer operators
Co-author: Margherita Disertori (Bonn)

We shall discuss the asymptotics of the top eigenvalues of non-self-adjoint integral operators in the semi-classical regime. The motivation comes from complex-valued one-dimensional statistical mechanics models, particularly, those arising in the study of random operators.

12:30 to 13:30 Lunch at Wolfson Court
13:30 to 14:30 Phase transitions for the almost Mathieu operator
Co-authors: Artur Avila (IMPA & Paris 7), Jiangong You (Nanjing University)

For the almost Mathieu operator with any fixed frequency, we locate the point where phase transition from singular continuous spectrum to pure point spectrum takes place, which settles Aubry-Andr\'e conjecture for all irrational frequencies, and also solves Avila and Jitomirskaya's conjectures. Together with former paper of Avila, we give a complete description of phase transitions for the almost Mathieu operator.

14:30 to 15:00 Afternoon Tea
15:00 to 15:25 Lyapunov exponents of quasi-periodic cocycles
Co-author: Pedro Duarte (University of Lisbon)

The purpose of this talk is to review some recent results concerning Lyapunov exponents of higher dimensional, analytic cocycles over a multi-frequency torus translation. Such cocycles appear naturally in the study of band lattice quasi-periodic Schrodinger operators.

The main new feature of this work is allowing a cocycle depending on several variables to have singularities, which requires a careful analysis involving pluri-subharmonic and analytic functions of several variables.

15:30 to 15:55 W Liu (Fudan University)
Arithmetic Spectral Transitions for the Maryland Model
In this talk, I will give a precise description of spectra of the Maryland model $ (h_{\lambda,\alpha,\theta}u) _n=u_{n+1}+u_{n-1}+ \lambda \tan \pi(\theta+n\alpha)u_n$ for all values of parameters. For Almost Mathieu Operator (H_{\lambda,\alpha,\theta}u) _n=u_{n+1}+u_{n-1}+ \lambda \cos 2\pi(\theta+n\alpha)u_n, the Lyapunov exponent can almost determine its spectral types(A.Avila, S.Jitomirskaya, J.You, Q.Zhou). When turn to Maryland model, I introduce an arithmetically defined index $\delta (\alpha, \theta)$ and show that for $\alpha\notin\mathbb{Q},$ $\sigma_{sc}(h_{\lambda,\alpha,\theta})=\overline{\{e:\gamma_{\lambda}(e)
16:00 to 16:25 Measure of the spectrum of the extended Harper's model
Co-author: Svetlana Jitomirskaya (University of California, Irvine)

In this talk, we will discuss the measure of the spectrum of the extended Harper's model(EHM). The measure of the spectrum of the Harper's model, which in mathematics is better known as the almost Mathieu operator(AMO), is know to be |4-2a| where a is the coupling constant. The way of calculating the measure mainly relies on the analysis of the AMO with rational frequency and the continuity argument. Here we focus on how to calculate the measure of the spectrum of the EHM with rational frequency, therefore implying the result of irrtional frequency.

16:30 to 17:30 Welcome Wine Reception
Wednesday 8th April 2015
10:00 to 11:00 Dynamics of two quasi-periodically perturbed systems
We will present results concerning the dynamics of two different quasi-periodically perturbed systems. One of the systems we consider is the quasi-periodic Schrödinger cocycle.
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Recurrent random walks in random and quasi-periodic environments on a strip
This is joint work with D. Dolgopyat

We prove that a recurrent random walk (RW) in random environment (RE) on a strip which does not obey the Sinai law exhibits the Central Limit asymptotic behaviour.

We also show that there exists a collection of proper sub-varieties in the space of transition probabilities such that

1. If RE is stationary and ergodic and the transition probabilities are concentrated on one of sub-varieties from our collection then the CLT holds; 2. If the environment is i.i.d then the above condition is also necessary for the CLT.

As an application of our techniques we prove the CLT for quasi-periodic environments with Diophantine frequencies. One-dimensional RWRE with bounded jumps are a particular case of the strip model.

12:30 to 13:30 Lunch at Wolfson Court
13:30 to 14:30 Spectral Properties of Schroedinger Operator with a Quasi-periodic Potential in Dimension Two
Co-author: Roman Shterenberg (UAB)

We consider $H=-\Delta+V(x)$ in dimension two, $V(x)$ being a quasi-periodic potential. We prove that the spectrum of $H$ contains a semiaxis (Bethe-Sommerfeld conjecture) and that there is a family of generalized eigenfunctions at every point of this semiaxis with the following properties. First, the eigenfunctions are close to plane waves $e^{i\langle \vec k,\vec x\rangle }$ at the high energy region. Second, the isoenergetic curves in the space of momenta $\vec k$ corresponding to these eigenfunctions have a form of slightly distorted circles with holes (Cantor type structure). It is shown that the spectrum corresponding to these eigenfunctions is absolutely continuous. A method of multiscale analysis in the momentum space is developed to prove the results.

14:30 to 15:00 Afternoon Tea
15:00 to 16:00 Continuity of Lyapunov Exponents and Cantor spectrum for a class of $C^2$ Quasiperiodic Schr\"odinger Cocycles
Co-author: Zhenghe Zhang (Rice University)

We show that for a class of $C^2$ quasiperiodic potentials and for any fixed \emph{Diophantine} frequency, the Lyapunov exponents of the corresponding Schr\"odinger cocycles are uniformly positive and weakly H\"older continuous as function of energies. Moreover, we show that the spectrum is Cantor. Our approach is of purely dynamical systems, which depends on a detailed analysis of asymptotic stable and unstable directions. We also apply it to general quasiperiodic $\mathrm{SL}(2,\R)$ cocycles.

16:00 to 16:25 S Morozov (Ludwig-Maximilians-Universität München)
High energy asymptotics of the integrated density of states of almost periodic pseudo-differential operators
The existence of complete asymptotic expansion for the integrated density of states in the high energy regime was long conjectured for periodic Schrödinger operators. I will discuss the history of the subject and present an eventual solution in the multidimensional situation. It turns out that the result applies to a big class of almost periodic pseudo-differential operators with smooth symbols. The proof is based on an application of the gauge transform discussed in the minicourse of A. Sobolev during the introductory workshop. The talk is based on a joint work with L. Parnovski and R. Shterenberg.
Thursday 9th April 2015
10:00 to 11:00 S Nakamura (University of Tokyo)
Microlocal properties of scattering matrices
We consider scattering theory for a pair of operators $H_0$ and $H=H_0+V$ on $L^2(M,m)$, where $M$ is a Riemannian manifold, $H_0$ is a multiplication operator on $M$ and $V$ is a pseudodifferential operator of order $-\mu$, $\mu>1$. We show that a time-dependent scattering theory can be constructed, and the scattering matrix is a pseudodifferential operator on each energy surface. Moreover, the principal symbol of the scattering matrix is given by a Born approximation type function. The main motivation of the study comes from applications to discrete Schr\"odigner operators, but it also applies to various differential operators with constant coefficients and short-range perturbations on Euclidean spaces.
11:00 to 11:30 Morning Coffee
11:30 to 12:30 M Shamis (Weizmann Institute of Science)
Wegner estimates for deformed Gaussian ensembles
Co-authors: Michael Aizenman (Princeton University), Ron Peled (Tel Aviv University), Jeff Schenker (Michigan State University ), Sasha Sodin (Tel Aviv University)

The deformed Gaussian ensembles are obtained by adding a deterministic Hermitian matrix to a random matrix drawn from the Gaussian Orthogonal, Unitary (or Symplectic) ensembles. We shall discuss several Wegner-type estimates for these models. As an application, we establish localization at strong disorder for the Wegner orbital model, with sharp dependence of the localization threshold on the number of orbitals.

12:30 to 13:30 Lunch at Wolfson Court
13:30 to 14:30 M Goldstein (University of Toronto)
On gaps and bands of quasi-periodic operators
This talk will review results on quasi-periodic Sturm-Liouville operators developed in recent joint works with D.Damanik and M.Lukic and also on joint work with D.Damanik, W.Schlag, M.Voda. The central feature to be discussed is the following relations between gaps and bands.
14:30 to 15:00 Afternoon Tea
15:00 to 15:25 Random nonmonotonic multichannel Schr\"{o}dinger operators
Co-author: John Imbrie (University of Virginia)

Anderson localization is by now well understood for the standard random Schrodinger operator. On the other hand the motivation for the problem, which lies in many body systems still lacks a developed theory. For our part we consider several aspects arising in more-than-one body systems which prevent an immediate application of the methods of one body systems. In systems such as random Ising models, energy levels of the system may depend analytically on (finite truncations of) random parameters. Of course in the standard Anderson model the dependence of the energy levels on the random parameters is linear which leads to the celebrated Wegner estimate which allows the usual multiscale analysis. In our talk, we consider a single body model with potentials depending analytically on the random parameters. In multichannel Schrodinger models, the potentials at each site of the lattice are matrices which may depend analytically on the random parameters, eg, these models can be realized as tight binding models in $Z^D$ with dilute randomness. In the multichannel model, we utilize the transversality of the system's energies with respect to the random environment, this allows some control of the probabilities of resonances. Finally, we discuss new methods of localization proofs, for the multichannel model we obtain stretched exponential localization of eigenfunction correlations.

15:30 to 15:55 C Rojas-Molina (Ludwig-Maximilians-Universität München)
Ergodic properties and localization for Delone-Anderson models
Co-authors: F. Germinet (U. de Cergy-Pontoise) and P. Müller (Ludwig-Maximilians-Universität München)

Delone-Anderson models arise in the study of wave localization in random media, where the underlying configuration of impurities in space is aperiodic, as for example, in disordered quasicrystals. The lack of translation invariance in the model yields a break of ergodicity, and the loss of properties linked to it. In this talk we will present results on the existence of the integrated density of states, the ergodic properties of these models and results on dynamical localization.

16:00 to 16:25 C Sadel (Institute of Science and Technology (IST Austria))
Anderson transition at 2D growth-rate for the Anderson model on antitrees with normalized edge weights
An antitree is a discrete graph that is split into countably many shells $S_n$ consisting of finitely many vertices so that all vertices in $S_n$ are connected with all vertices in the adjacent shells $S_{n+1}$ and $S_{n-1}$. We normalize the edges between $S_n$ and $S_{n+1}$ with weights to have a bounded adjacency operator and add an iid random potential. We are interested in the case where the number of vertices $\# S_n$ in the $n$-th shell grows like $n^a$. In a particular set of energies we obtain a transition of the spectral type from pure point to partly s.c. to a.c. spectrum at $a=1$ which corresponds to the growth-rate in 2 dimensions.
19:30 to 22:00 Conference Dinner at Gonville and Caius College
Friday 10th April 2015
10:00 to 11:00 Reflectionless property and related problems on 1D Schrödinger operators
Reflectionless property for 1D Schrödinger operators is defined by using their Weyl functions or Green functions. The property is especially important when potentials of Schrödinger operators are ergodic, and it is proved that the reflectionless property holds on their absolutely continuous spectra. On the other hand Remling showed the deterministic version. They are related to the shift operation of potentials. In this talk we discus the capability of its extension to KdV equation and propose several open problems.
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Level statistics for 1-dimensional Schr\"odinger operator and beta-ensemble
A part of this talk is based on joint work with Prof. Kotani. We consider the following two classes of 1-dimensional random Schr\"odinger operators : (1) operators with decaying random potential, and (2) operators whose coupling constants decay as the system size becomes large. Our problem is to identify the limit $¥xi_{¥infty}$ of the point process consisting of rescaled eigenvalues. The result is : (1) for slow decay, $¥xi_{¥infty}$ is the clock process ; for critical decay $¥xi_{¥infty}$ is the $Sine_{¥beta}$ process, (2) for slow decay, $¥xi_{¥infty}$ is the deterministic clock process ; for critical decay $¥xi_{¥infty}$ is the $Sch_{¥tau}$ process. As a byproduct of (1), we have a proof of coincidence of the scaling limits of circular and Gaussian beta ensembles.
12:30 to 13:30 Lunch at Wolfson Court
13:30 to 14:30 A Gorodetski (University of California, Irvine)
The Fibonacci Hamiltonian
Co-authors: David Damanik (Rice University), William Yessen (Rice University)

In the talk we will consider the discrete Schrodinger operator with potential given by the Fibonacci substitution sequence (the Fibonacci Hamiltonian) and provide a detailed description of its spectrum and spectral characteristics (namely, the optimal Holder exponent of the integrated density of states, the dimension of the density of states measure, the dimension of the spectrum, and the upper transport exponent) for all values of the coupling constant. In particular, we will establish strict inequalities between the four spectral characteristics in question, and discuss the exact small and large coupling asymptotics of these spectral characteristics. A crucial ingredient is the relation between spectral properties of the Fibonacci Hamiltonian and dynamical properties of the Fibonacci trace map (such as dimensional characteristics of the non-wandering hyperbolic set and its measure of maximal entropy as well as other equilibrium measures, topological entropy, multipliers of periodic orbits). We will establish exact identities relating the spectral and dynamical quantities, and show the connection between the spectral quantities and the thermodynamic pressure function.

14:30 to 15:00 Afternoon Tea
15:00 to 15:25 The isospectral torus of quasi-periodic Schrodinger operators via periodic approximations
Co-authors: David Damanik (Rice University), Michael Goldstein (University of Toronto)

This talk describes joint work with D.\ Damanik and M.\ Goldstein. We study quasi-periodic Schr\"odinger operators $H = -\frac{d^2}{dx^2} +V$ in the regime of analytic sampling function and small coupling. More precisely, the potential is \[ V(x)=\sum_{m\in \mathbb{Z}^\nu} c(m) \exp(2\pi i m \omega x) \] with $|c(m)|\le \epsilon \exp(-\kappa |m|)$. Our main result is that any reflectionless potential $Q$ isospectral with $V$ is also quasi-periodic and in the same regime, with the same Diophantine frequency $\omega$, i.e. \[ Q(x)=\sum_{m\in \mathbb{Z}^\nu} d(m) \exp(2\pi i m \omega x) \] with $|d(m)|\le \sqrt{2\epsilon} \exp(-\frac{\kappa}2 |m|)$. The proof relies on approximation by periodic potentials $\tilde V$, which are obtained by replacing the frequency $\omega$ by rational approximants $\tilde \omega$. We adapt the multiscale analysis, developed by Damanik--Goldstein for $V$, so that it applies to the periodic approximants $\tilde V$. This allows us to establish estimates for gap lengths and Fourier coefficients of $\tilde V$ which are independent of period, unlike the standard estimates known in the theory of periodic Schr\"odinger operators. Starting from these estimates, we obtain the main result by comparing the isospectral tori and translation flows of $\tilde V$ and $V$.

15:30 to 15:55 Spectral packing dimension for 1-dimensional quasiperiodic Schrodinger operators
Co-author: Svetlana Jitomirskaya (UC Irvine)

In this talk, we are going to discuss the packing dimension of the spectral measure of 1-dimensional quasiperiodic Schrodinger operators. We prove that if the base frequency is Liouville, the packing dimension of the spectral measure will be one. As a direct consequence, we show that for the critical and supercritical Almost Mathieu Operator, the spectral measure has different Hausdorff and packing dimension.

16:00 to 16:25 Homogeneous Spectrum for Limit-Periodic Operators
Co-author: Milivoje Lukic (University of Toronto)

We will discuss the spectra of limit-periodic Schr\"odinger operators. Specifically, the spectrum of a limit-periodic operator which obeys the Pastur-Tkachenko condition is homogeneous in the sense of Carleson. When combined with work of Gesztesy-Yuditskii, our theorem implies that the spectrum of a continuum Schr\"odinger operator with Pastur--Tkachenko potential has infinite gap length whenever the potential fails to be uniformly almost periodic.

16:30 to 16:55 On the Homogeneity of the Spectrum for Quasi-Periodic Schroedinger Operators
Co-authors: David Damanik (Rice University), Michael Goldstein (University of Toronto), Wilhelm Schlag (University of Chicago)

I will discuss a recent result showing that the spectrum of discrete one-dimensional quasi-periodic Schroedinger operators is homogeneous in the regime of positive Lyapunov exponent. The homogeneity is in the sense of Carleson, as used in the study of the inverse spectral problem for reflectionless potentials. The talk is based on joint work with David Damanik, Michael Goldstein, and Wilhelm Schlag.

University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons