Almost Everywhere Positivity of the Lyapunov Exponent for the Doubling Map

We show that discrete one-dimensional Schr\"odinger operators on the half-line with ergodic potentials generated by the doubling map on the circle, $V_\theta(n) = f(2^n \theta)$, may be realized as the half-line restrictions of a non-deterministic family of whole-line operators. As a consequence, the Lyapunov exponent is almost everywhere positive and the absolutely continuous spectrum is almost surely empty.Comment: 4 page

Half-line Schrodinger Operators With No Bound States

We consider Sch\"odinger operators on the half-line, both discrete and continuous, and show that the absence of bound states implies the absence of embedded singular spectrum. More precisely, in the discrete case we prove that if $\Delta + V$ has no spectrum outside of the interval $[-2,2]$, then it has purely absolutely continuous spectrum. In the continuum case we show that if both $-\Delta + V$ and $-\Delta - V$ have no spectrum outside $[0,\infty)$, then both operators are purely absolutely continuous. These results extend to operators with finitely many bound states.Comment: 34 page

Reflection symmetries of almost periodic functions

We study global reflection symmetries of almost periodic functions. In the non-limit periodic case, we establish an upper bound on the Haar measure of the set of those elements in the hull which are almost symmetric about the origin. As an application of this result we prove that in the non-limit periodic case, the criterion of Jitomirskaya and Simon ensuring absence of eigenvalues for almost periodic Schr\"odinger operators is only applicable on a set of zero Haar measure. We complement this by giving examples of limit periodic functions where the Jitomirskaya-Simon criterion can be applied to every element of the hull.Comment: 6 page

Who’s responsible for these Blues?: Reflecting on the murder of Armstrong Todd in Bebe Moore Campbell\u27s YOUR BLUES AIN\u27T LIKE MINE

This article is featured in the journal Tapestries: Interwoven voices of local and global identities, volume 4

Perturbations of Orthogonal Polynomials With Periodic Recursion Coefficients

We extend the results of Denisov-Rakhmanov, Szego-Shohat-Nevai, and Killip-Simon from asymptotically constant orthogonal polynomials on the real line (OPRL) and unit circle (OPUC) to asymptotically periodic OPRL and OPUC. The key tool is a characterization of the isospectral torus that is well adapted to the study of perturbations.Comment: 64 pages, to appear in Ann. of Mat

Cold galaxies

We use 350 mu angular diameter estimates from Planck to test the idea that some galaxies contain exceptionally cold (10-13 K) dust, since colder dust implies a lower surface brightness radiation field illuminating the dust, and hence a greater physical extent for a given luminosity. The galaxies identified from their spectral energy distributions as containing cold dust do indeed show the expected larger 350 mu diameters. For a few cold dust galaxies where Herschel data are available we are able to use submillimetre maps or surface brightness profiles to locate the cold dust, which as expected generally lies outside the optical galaxy.Comment: 9 pages, 15 figures. Accepted for publication MNRA

Necessary and Sufficient Conditions in the Spectral Theory of Jacobi Matrices and Schr\"odinger Operators

We announce three results in the theory of Jacobi matrices and Schr\"odinger operators. First, we give necessary and sufficient conditions for a measure to be the spectral measure of a Schr\"odinger operator -\f{d^2}{dx^2} +V(x) on $L^2 (0,\infty)$ with $V\in L^2 (0,\infty)$ and $u(0)=0$ boundary condition. Second, we give necessary and sufficient conditions on the Jacobi parameters for the associated orthogonal polynomials to have Szeg\H{o} asymptotics. Finally, we provide necessary and sufficient conditions on a measure to be the spectral measure of a Jacobi matrix with exponential decay at a given rate.Comment: 10 page

Variational Estimates for Discrete Schr\"odinger Operators with Potentials of Indefinite Sign

Let $H$ be a one-dimensional discrete Schr\"odinger operator. We prove that if \sigma_{\ess} (H)\subset [-2,2], then $H-H_0$ is compact and \sigma_{\ess}(H)=[-2,2]. We also prove that if $H_0 + \frac14 V^2$ has at least one bound state, then the same is true for $H_0 +V$. Further, if $H_0 + \frac14 V^2$ has infinitely many bound states, then so does $H_0 +V$. Consequences include the fact that for decaying potential $V$ with $\liminf_{|n|\to\infty} |nV(n)| > 1$, $H_0 +V$ has infinitely many bound states; the signs of $V$ are irrelevant. Higher-dimensional analogues are also discussed.Comment: 17 page