# Timetable (PEPW02)

## Periodic and other ergodic problems

Monday 23rd March 2015 to Friday 27th March 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 N Saveliev (University of Miami)Index theory on end-periodic manifolds Co-authors: Tomasz Mrowka (MIT), Daniel Ruberman (Brandeis University) End-periodic manifolds are non-compact Riemannian manifolds whose ends are modeled on an infinite cyclic cover of a closed manifold; an important special case are manifolds with cylindrical ends. We extend some of the classical index theorems to this setting, including the Atiyah-Patodi-Singer theorem computing the index of Dirac-type operators. Our theorem expresses this index in terms of a new periodic eta-invariant which equals the classical eta-invariant in the cylindrical end setting. INI 1 11:00 to 11:30 Morning Coffee 11:30 to 12:30 Uncertainty relations and Wegner estimates for random breather potentials Co-authors: Ivica Nakic (Zagreb University), Matthias Täufer (TU Chemnitz), Martin Tautenhahn (TU Chemnitz) We present a new scale-free, quantitative unique continuation estimate for Schroedinger operators in multidimensional space. Depending on the context such estimates are sometimes called uncertainty relations, observations inequalities or spectral inequalities. To illustrate its power we prove a Wegner estimate for Schroedinger operators with random breather potentials. Here we encounter a non-linear dependence on the random coupling constants, preventing the use of standard perturbation theory. The proofs rely on an analysis of the level sets of the random potential, and can be extended to a rather general framework. INI 1 12:30 to 13:30 Lunch at Wolfson Court 13:30 to 14:30 A Wegner estimate and localisation for alloy-type models with sign-changing exponentially decaying single-site potentials Co-authors: Martin Tautenhahn (TU Chemnitz, Germany), Ivan Veselic (Tu Chemnitz, Germany), Karsten Leohardt (MPIPKS, Dresden, Germany) In this talk, we will consider discrete Schroedinger operators on the d-dimensional Euclidean lattice with random potential of alloy-type. The single site potential is exponentially decaying and allowed to be sign changing. The main aim is to prove a Wegner estimate, which is polynomial in the size of the box and linear in the size of the energy interval. Our result generalises earlier ones obtained by Veselic. Our Wegner estimate is of a type which can be used for the multiscale analysis proof of localisation in all energy regions, where the initial scale estimate holds. Concerning localisation, it should be mentioned that Krueger has obtained localisation results for a class of discrete alloy-type models which include ours. This is joint work with Karsten Leonhardt (MPIPKS, Dresden), Martin Tautenhahn (TU Chemnitz), and Ivan Veselic (TU Chemnitz). Bibliography: H. Krueger: Localization for random operators with non-monotone potentials with exponentially decaying correlations, Ann. Henri Poincare 13 (3), 543-598, 2012. I. Veselic: Wegner estimates for discrete alloy-type models, Ann. Henri Poincaree 11 (5), 991-1005, 2010. INI 1 14:30 to 15:00 Afternoon Tea 15:00 to 16:00 T Kappeler (Universität Zürich)Spectral asymptotics of Zahkarov Shabat operators and their application to the nonlinear Schrödinger equation on the circle In this talk I will present asymptotics of various spectral quantities of Zahkarov Shabat operators and show how to apply them for proving that the nonlinear Fourier transform of the defocusing nonlinear Schrödinger equation on the circle (Birkhoff map)is the linear Fourier transform up to a nonlinear part which is 1-smoothing.Various implications will be discussed. This is joint work with Beat Schaad and Peter Topalov. INI 1 16:00 to 17:00 Entanglement in the disordered XY spin chain and open problems for random block operators Random block operators appear as effective one-particle Hamiltonians in the study of the anisotropic XY spin chain. We will discuss that dynamical localization of the effective Hamiltonian implies a uniform area law for the entanglement of all eigenstates of the XY chain in random field. An open problem in this context is the regularity of the Lyapunov exponents for the associated random block operators. A difficulty arises due to the break down of irreducibility of transfer matrices at zero energy. INI 1 17:00 to 18:00 Welcome Wine Reception
 10:00 to 11:00 S Jitomirskaya (University of California, Irvine)Diophantine properties and the spectral theory of explicit quasiperiodic models The development of the spectral theory of quasiperiodic operators has been largely centered around and driven by several explicit models, all coming from physics. In this talk we will review the highlights of the current state-of-the-art of the spectral theory for the following three models: almost Mathieu operator, extended Harper's model and Maryland model, focusing on arithmetically driven spectral transitions, measure of the spectrum, and the Cantor nature of the spectrum. Those models all demonstrate interesting dependence on the arithmetics of parameters (even in some cases when the final conclusion does not have such dependence) and have traditionally been approached through KAM-type schemes. Even when the KAM arguments have been replaced by the non-perturbative ones allowing to treat more couplings, frequencies that are neither far from nor close enough to rationals presented a challenge as for them there was nothing left to perturb about. A remarkable relatively recent development concerning the explicit models is that very precise results have become possible: not only many facts have been established for a.e. frequencies and phases, but in many cases it has become possible to go deeper in the arithmetics and either establish precise arithmetic transitions or even obtain results for all values of parameters. More detailed talks on some of the covered topics will be given in the April workshop by J. You, Q. Zhou, R. Han, and W. Liu. INI 1 11:00 to 11:30 Morning Coffee 11:30 to 12:30 Anderson localization for one-dimensional ergodic Schrödinger operators with piecewise monotonic sampling functions Co-author: Svetlana Jitomirskaya (University of California, Irvine) We consider the one-dimensional ergodic operator families $$\label{h_def} (H_{\alpha,\lambda}(x) \Psi)_m=\Psi_{m+1}+\Psi_{m-1}+\lambda v(x+\alpha m) \Psi_m,\quad m\in \mathbb Z,$$ in $l^2(\mathbb Z)$. Such operators are well studied for analytic $v$, where they undergo a metal-insulator transition from absolutely continuous spectra (for small $\lambda$) to purely point spectra with exponentially decaying eigenfunctions (for large $\lambda$); the latter is usually called Anderson localization. Very little is known for general continuous of smooth $v$. However, there are several well developed models with discontinuous $v$, such as Maryland model and the Fibonacci Hamiltonian. We study the family $H_{\alpha,\lambda}(x)$ with $v$ satisfying a bi-Lipshitz type condition (for example, $v(x)=\{x\}$). It turns out that for every $\lambda$, for almost every $\alpha$ and all $x$ the spectrum of the operator $H_{\alpha,\lambda}(x)$ is pure point. This is the first example of pure point spectrum at small coupling for bounded quasiperiodic-type operators, or more generally for ergodic operators with underlying systems of low disorder. We also show that the Lyapunov exponent of this system is continuous in energy for all $\lambda$ and is uniformly positive for $\lambda$ sufficiently (but nonperturbatively) large. In the regime of uniformly positive Lyapunov exponent, our result gives uniform localization, thus providing the first natural example of an operator with this property. This is a joint result with Svetlana Jitomirskaya, University of California, Irvine. INI 1 12:30 to 13:30 Lunch at Wolfson Court 13:30 to 14:30 Perturbative methods for Schrödinger operator: from periodic to quasiperiodic potentials. Co-authors: Young-Ran Lee (Sogang University), Roman Shterenberg (UAB) We consider Schrödinger operator in dimension two and discuss perturbative methods and spectral results for periodic, limit-periodic and quasi-periodic potentials. We start with methods for periodic potentials, and then discuss their development for limit-periodic potentials, and, eventually, multiscale analysis in the momentum space for quasi-periodic potentials. INI 1 14:30 to 15:00 Afternoon Tea 15:00 to 16:00 C Joyner (Queen Mary, University of London)Spectral statistics of Bernoulli matrix ensembles - a random walk approach Co-author: Uzy Smilansky (Weizmann Institute of Science) We investigate the eigenvalue statistics of random Bernoulli matrices, where the matrix elements are chosen independently from a binary set with equal probability. This is achieved by initiating a discrete random walk process over the space of matrices and analysing the induced random motion of the eigenvalues - an approach which is similar to Dyson's Brownian motion model but with important modifications. In particular, we show our process is described by a Fokker-Planck equation, up to an error margin which vanishes in the limit of large matrix dimension. The stationary solution of which corresponds to the joint probability density function of certain well-known fixed trace Gaussian ensembles. INI 1 16:00 to 17:00 On Level Spacings for Jacobi Operators The talk will review some results concerning the spacings of eigenvalues for restrictions of self-adjoint Jacobi operators to large "boxes" and their connections with other spectral properties of these operators. A central focus will be the phenomenon of "clock behavior" often associated with absolutely continuous spectrum. INI 1
 09:00 to 17:00 Day dedicated to the memory of B.M. Levitan 10:00 to 11:00 D Yafaev (Université de Rennes 1)Surface waves and scattering by unbounded obstacles Consider the Laplace operator $H=-\Delta$ in the exterior $\Omega$ of a parabolic region in ${\bf R}^d$, and let $H_{0}=-\Delta$ be the operator in the space $L^2 ({\bf R}^d)$. The wave operators for the pair $H_{0}$, $H$ exist for an arbitrary self-adjoint boundary condition on $\partial\Omega$. For the case of the Dirichlet boundary condition, the wave operators are unitary which excludes the existence of surface waves on $\partial\Omega$. For the Neumann boundary condition, the existence of surface waves is an open problem, and we are going to discuss it. INI 1 11:00 to 11:30 Morning Coffee 11:30 to 12:30 Some connections between Weyl-Titchmarsh theory, oscillation theory, and density of states We intend to discuss connections between Weyl-Titchmarsh theory, oscillation theory, and density of states for certain classes of one-dimensional Schroedinger operators. INI 1 12:30 to 13:30 Lunch at Wolfson Court 13:30 to 14:30 Local Density of States and the Spectral Function for Quasi-Periodic Operators INI 1 14:30 to 15:00 Afternoon Tea 15:00 to 16:00 Periodic spectral problem for the massless Dirac operator Co-author: Michael Levitin (University of Reading) Periodic spectral problems are normally formulated in terms of the Schrodinger operator. The aim of the talk is to examine issues that arise if one formulates a periodic spectral problem in terms of the Dirac operator. The motivation for the particular model considered in the talk does not come from solid state physics. Instead, we imagine a single massless neutrino living in a compact 3-dimensional universe without boundary. There is no electromagnetic field in our model because a neutrino does not carry an electric charge and cannot interact (directly) with an electromagnetic field. The role of the electromagnetic covector potential is therefore taken over by the metric. In other words, we are interested in understanding how the curvature of space affects the energy levels of the neutrino. More specifically, we consider the massless Dirac operator on a 3-torus equipped with Euclidean metric and standard spin structure. It is known that the eigenvalues can be calculated explicitly: the spectrum is symmetric about zero and zero itself is a double eigenvalue. Our aim is to develop a perturbation theory for the eigenvalue with smallest modulus with respect to perturbations of the metric. Here the application of perturbation techniques is hindered by the fact that eigenvalues of the massless Dirac operator have even multiplicity, which is a consequence of this operator commuting with the antilinear operator of charge conjugation (a peculiar feature of dimension 3). We derive an asymptotic formula for the eigenvalue with smallest modulus for arbitrary perturbations of the metric and present two particular families of Riemannian metrics for which the eigenvalue with smallest modulus can be evaluated explicitly. We also establish a relation between our asymptotic formu la and the eta invariant. [1] R.J.Downes, M.Levitin and D.Vassiliev, Spectral asymmetry of the massless Dirac operator on a 3-torus, Journal of Mathematical Physics, 2013, vol. 54, article 111503. INI 1 16:00 to 17:00 Chambers formulas and semiclassical analysis for generalized Harper's butterflies If the first mathematical results were obtained more than 30 years ago with the interpretation of the celebrated Hofstadter butterfly, more recent experiments in Bose-Einstein theory suggest new questions. I will start with a partial survey on old results of Helffer-Sjöstrand and Kerdelhue´ and then discuss more recent questions related to generalized butterflies (Dalibard and coauthors, Hou, Kerdelhue´--Royo-Letelier). These new questions are strongly related to Harper on triangular or hexagonal lattices (in connection with the now very popular graphene). Our historics is focused on the mathematical results. INI 1
 10:00 to 11:00 Kinetic transport in crystals and quasicrystals The Lorentz gas is one of the simplest, most widely used models to study the transport properties of rarified gases in matter. It describes the dynamics of a cloud of non-interacting point particles in an infinite array of fixed spherical scatterers. More than one hundred years after its conception, it is still a major challenge to understand the nature of the kinetic transport equation that governs the macroscopic particle dynamics in the limit of low scatterer density (the Boltzmann-Grad limit). Lorentz suggested that this equation should be the linear Boltzmann equation. This was confirmed in three celebrated papers by Gallavotti, Spohn, and Boldrighini, Bunimovich and Sinai, under the assumption that the distribution of scatterers is sufficiently disordered. In the case of strongly correlated scatterer configurations (such as crystals or quasicrystals), we now understand why the linear Boltzmann equation fails and what to substitute it with. A particularly striking featur e of the periodic Lorentz gas is a heavy tail for the distribution of free path lengths, with a diverging second moment, and superdiffusive transport in the limit of large times. Joint work with A. Strombergsson and B. Toth. INI 1 11:00 to 11:30 Morning Coffee 11:30 to 12:30 Two-scale 'micro-resonant' homogenisation of periodic (and some ergodic) problems Co-author: Ilia Kamotski (University College London) There has been lot of recent interest in composite materials whose macroscopic physical properties can be radically different from those of conventional materials, often due to effects of the so-called "micro-resonances". Mathematically this leads to studying high-contrast homogenization of (periodic or not) problems with a `critically’ scaled high contrast, where the resulting two-scale asymptotic behaviour appears to display a number of interesting effects. Mathematical analysis of these problems requires development of "two-scale" versions of operator and spectral convergences, of compactness, etc. We will review some background, as well as some more recent generalizations and applications. One is two-scale analysis of general "partially-degenerating" periodic problems, where strong two-scale resolvent convergence appears to hold under a rather generic decomposition assumptions, implying in particular (two-scale) convergence of se migroups with applications to a wide class of micro-resonant dynamic problems. Another is two-scale homogenization with random micro-resonances, which appears to yield macroscopic dynamics effects akin to Anderson localization. Some of the work is joined with Ilia Kamotski. INI 1 12:30 to 13:30 Lunch at Wolfson Court 19:30 to 22:00 Conference Dinner at Christ's College
 10:00 to 11:00 T Suslina (Saint Petersburg State University)Operator error estimates for homogenization of elliptic systems with periodic coefficients We study a wide class of matrix elliptic second order differential operators $A_\varepsilon$ in a bounded domain with the Dirichlet or Neumann boundary conditions. The coefficients are assumed to be periodic and depend on $x/\varepsilon$. We are interested in the behavior of the resolvent of $A_\varepsilon$ for small $\varepsilon$. Approximations of this resolvent in the $L_2\to L_2$ and $L_2 \to H^1$ operator norms are obtained. In particular, a sharp order estimate $$\| (A_\varepsilon - \zeta I)^{-1} - (A^0 - \zeta I)^{-1} \|_{L_2 \to L_2} \le C\varepsilon$$ is proved. Here $A^0$ is the effective operator with constant coefficients. INI 1 11:00 to 11:30 Morning Coffee 11:30 to 12:30 Wannier functions for periodic Schrödinger operators and harmonic maps into the unitary group Co-author: Adriano Pisante ("La Sapienza" University of Rome) The localization of electrons in crystalline solids is often expressed in terms of the Wannier functions, which provide an orthonormal basis of L2(Rd) canonically associated to a given periodic Schrödinger operator. A very popular tool in theoretical and computational solid-state physics are the maximally localized Wannier functions, which are defined as the minimizers (in a suitable space of Wannier functions) of a localization functional introduced by Marzari and Vanderbilt in 1997. While early confirmed by numerical evidence, the exponential localization of such minimizers has remained an open question until recently. In the talk, the concept of Wannier basis will be reviewed in detail, with emphasis on its geometric counterpart (Bloch frame). Then a recent result proving the existence and the exponential localization of the minimizers, under suitable assumptions, will be presented (joint work with A. Pisante). The proof exploits methods and techniques from the regularity theory of harmonic maps into the unitary group and the so-called "decomposition into unitons" of such maps. INI 1 12:30 to 13:30 Lunch at Wolfson Court 13:30 to 14:30 Everywhere discontinuous anisotropy of thin periodic composite plates Co-author: Kirill Cherednichenko (University of Bath) We consider an elastic periodic composite plate in full bending regime, i.e. when the displacement of the plate is of finite order. Both the thickness of the plate $h$ and the period of the composite structure $\varepsilon$ are small parameters. We start from the non-linear elasticity setting. Passing to the limit as $h, \varepsilon \to 0$ we carry out simultaneous dimension reduction and homogenisation to obtain an effective limit elastic functional which describes the asymptotic properties of the composite plate. We show, in particular, that in the regime $h INI 1 14:30 to 15:00 Afternoon Tea 15:00 to 16:00 Maryland equation, renormalization formulas and mimimal meromorphic solutions to difference equations Co-author: Fedor Sandomirskyi (Saint Petersburg State University) Consider the difference Schrödinger equation$\psi_{k+1}+\psi_{k-1}+\lambda\ {cotan} (\pi\omega k+\theta)\psi_k=E\psi_k,\quad k\in{\mathbb Z}$,where$\lambda$,$\omega$,$\theta$and$E$are parameters. If$\omega$is irrational, this equation is quasi-periodic. It was introduced by specialists in solid state physics from Maryland and is now called the Maryland equation. Computer calculations show that, for large$k$, its eigenfunctions have a multiscale, "mutltifractal" structure. We obtained renormalization formulas that express the solutions to the input Marryland equation for large$k$in terms of solutions to the Marryland equation with new parameters for bounded$k$. The proof is based on the theory of meromorphic solutions of difference equations on the complex plane, and on ideas of the monodromization met hod -- the renormalization approach first suggested by V.S.Buslaev and A.A. Fedotov. Our formulas are close to the renormalization formulas from the theory of the Gaussian exponential sums$S(N)=\sum_{n=0}^N\,e^{2\pi i (\omega n^2+\theta n)}$, where$\omega$and$\theta$are parametrs. For large$N\$, these sums also have a multiscale behavior. The renormalization formulas lead to a natural explanation of the famous mutiscale structure that appears to reflect certain quasi-classical asymptotic effects (Fedotov-Klopp, 2012). INI 1 16:00 to 17:00 On the gaps in the spectrum of the periodic Maxwell operator Co-authors: S. Cooper (University of Bath), V. Smyshlyaev (UCL) We demonstrate the existence of the gaps in the spectrum of the periodic Maxwell operator with medium contrast coefficients. We discuss the location of the gaps and their dependence on the geometry of the media. INI 1
 12:30 to 13:30 S Klein ([Department of Mathematical Sciences, NTNU])Continuity of Lyapunov Exponents via Large Deviations Large deviation type (LDT) estimates for transfer matrices are important tools in the study of discrete, one dimensional, quasi-periodic Schrodinger operators. They have been used to establish positivity of the Lyapunov exponent, continuity properties of the Lyapunov exponent and of the integrated density of states, estimates on the Green's function, Anderson localization. We prove - in a general, abstract setting - that the availability of appropriate LDT estimates implies continuity of the Lyapunov exponents, with a modulus of continuity depending explicitly on the strength of the LDT. The devil is of course in the details, hidden here behind the words "availability" and "appropriate". We show that the study of the Lyapunov exponents associated with a band lattice quasi-periodic Schrodinger operator fits this abstract setting, provided the potential is a real analytic function of (one or of) several variables and that the frequency vector is Diophantine. Co-authored with: P Duarte INI 2