Bulk Universality and Clock Spacing of Zeros for Ergodic Jacobi Matrices with A.C. Spectrum

双边雅可比矩阵的时钟间距

Clock spacing for two-sided Jacobi matrices

研究具有两个可移动端点的有限截断双边雅可比矩阵的局部特征值间距，证明点态无反射条件下时钟间距的类似性质，并得到含可移动起点的Christoffel-Darboux核的新标度极限。

Fine Structure of the Zeros of Orthogonal Polynomials, IV. A Priori Bounds and Clock Behavior

We prove locally uniform spacing for the zeros of orthogonal polynomials on the real line under weak conditions (Jacobi parameters approach the free ones and are of bounded variation). We prove that for ergodic discrete Schrodinger operators, Poisson behavior implies positive Lyapunov exponent. Both results depend on a priori bounds on eigenvalue spacings for which we provide several proofs.

On the spacing of zeros of paraorthogonal polynomials for singular measures

We prove a lower bound on the spacing of zeros of paraorthogonal polynomials on the unit circle, based on continuity of the underlying measure as measured by Hausdorff dimensions. We complement this with the analog of the result from arXiv:1011.3159 showing that clock spacing holds even for certain singular continuous measures.

Fine structure of the zeros of orthogonal polynomials, III. Periodic recursion coefficients

We discuss asymptotics of the zeros of orthogonal polynomials on the real line and on the unit circle when the recursion coefficients are periodic. The zeros on or near the absolutely continuous spectrum have a clock structure with spacings inverse to the density of zeros. Zeros away from the a.c. spectrum have limit points mod p and only finitely many of them.

Universality Limits of a Reproducing Kernel for a Half-Line Schrödinger Operator and Clock Behavior of Eigenvalues

We extend some recent results of Lubinsky, Levin, Simon, and Totik from measures with compact support to spectral measures of Schrödinger operators on the half-line. In particular, we define a reproducing kernel $S_L$ for Schrödinger operators and we use it to study the fine spacing of eigenvalues in a box of the half-line Schrödinger operator with perturbed periodic potential. We show that if solutions $u(ξ, x)$ are bounded in $x$ by $e^{εx}$ uniformly for $ξ$ near the spectrum in an average sense and the spectral measure is positive and absolutely continuous in a bounded interval $I$ in the …

Asymptotics for Christoffel functions associated to continuum Schrödinger operators

We prove asymptotics of the Christoffel function, $λ_L(ξ)$, of a continuum Schrödinger operator for points in the interior of the essential spectrum under some mild conditions on the spectral measure. It is shown that $Lλ_L(ξ)$ has a limit and that this limit is given by the Radon--Nikodym derivative of the spectral measure with respect to the Martin measure. Combining this with a recently developed local criterion for universality limits at scale $λ_L(ξ)$, we compute universality limits for continuum Schrödinger operators at scale $L$ and obtain clock spacing of the eigenvalues of the finite …