Seminars (PEPW02)

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.

We consider the one-dimensional ergodic operator families \begin{equation} \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, \end{equation} 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.

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.

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.

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.

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).