Autocorrelations of the characteristic polynomial of a random matrix under microscopic scaling

We calculate the autocorrelation function for the characteristic polynomial of a random matrix in the microscopic scaling regime. While results fitting this description have be proved before, we will cover all values of inverse temperature $\beta \in (0,\infty)$. The method also differs from prior work, relying on matrix models introduced by Killip and Nenciu

Sum Rules for Jacobi Matrices and Their Applications to Spectral Theory

We discuss the proof of and systematic application of Case's sum rules for Jacobi matrices. Of special interest is a linear combination of two of his sum rules which has strictly positive terms. Among our results are a complete classification of the spectral measures of all Jacobi matrices J for which J-J_0 is Hilbert--Schmidt, and a proof of Nevai's conjecture that the Szego condition holds if J-J_0 is trace class.Comment: 69 pages, published versio

Smooth solutions to the nonlinear wave equation can blow up on Cantor sets

We construct $C^\infty$ solutions to the one-dimensional nonlinear wave equation $$u_{tt} - u_{xx} - \tfrac{2(p+2)}{p^2} |u|^p u=0 \quad \text{with} \quad p>0$$ that blow up on any prescribed uniformly space-like $C^\infty$ hypersurface. As a corollary, we show that smooth solutions can blow up (at the first instant) on an arbitrary compact set. We also construct solutions that blow up on general space-like $C^k$ hypersurfaces, but only when $4/p$ is not an integer and $k > (3p+4)/p$

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

Matrix models and eigenvalue statistics for truncations of classical ensembles of random unitary matrices

We consider random non-normal matrices constructed by removing one row and column from samples from Dyson's circular ensembles or samples from the classical compact groups. We develop sparse matrix models whose spectral measures match these ensembles. This allows us to compute the joint law of the eigenvalues, which have a natural interpretation as resonances for open quantum systems or as electrostatic charges located in a dielectric medium. Our methods allow us to consider all values of $\beta>0$, not merely $\beta=1,2,4$.Comment: 36 pp, to appear in Comm. Math. Phy