Schrödinger operators with potential V(n)=N^(-y) cos (2πn^p)

Let H be the Schrödinger operator with potential V(n) = n^(−γ) cos(2πn^ρ), where ρ ∈ (1,2) and γ ∈ (0, 1/2 − ρ−1/2). I show that for almost every boundary condition H has pure-point spectrum

Cantor polynomials and some related classes of OPRL

We explore the spectral theory of the orthogonal polynomials associated to the classical Cantor measure and similar singular continuous measures. We prove regularity in the sense of Stahl–Totik with polynomial bounds on the transfer matrix. We present numerical evidence that the Jacobi parameters for this problem are asymptotically almost periodic and discuss the possible meaning of the isospectral torus and the Szegő class in this context

A family of Schr\"odinger operators whose spectrum is an interval

By approximation, I show that the spectrum of the Schr\"odinger operator with potential $V(n) = f(n\rho \pmod 1)$ for f continuous and $\rho > 0$, $\rho \notin \N$ is an interval.Comment: Comm. Math. Phys. (to appear

Exact dynamical decay rate for the almost Mathieu operator

We prove that the exponential decay rate in expectation is well defined and is equal to the Lyapunov exponent, for supercritical almost Mathieu operators with Diophantine frequencies

Relative oscillation theory for Sturm-Liouville operators

Ein Einstieg in die Relative Oszillationstheorie.An Introduction to relative oscillation theory, which allows you to compute the number of eigenvalues in an interval in terms of zeros of Wronskians

Orthogonal Polynomials on the Unit Circle with Verblunsky Coefficients defined by the Skew-Shift

I give an example of a family of orthogonal polynomials on the unit circle with Verblunsky coefficients given by the skew-shift for which the associated measures are supported on the entire unit circle and almost every Aleksandrov measure is pure point. Furthermore, I show in the case of the two-dimensional skew-shift that the zeros of para-orthogonal polynomials obey the same statistics as an appropriate irrational rotation. The proof is based on an analysis of the associated CMV matrices

Discrete Schrödinger Operators with Random Alloy-type Potential

We review recent results on localization for discrete alloy-type models based on the multiscale analysis and the fractional moment method, respectively. The discrete alloy-type model is a family of Schrödinger operator