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

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 …

Prepotential approach to solvable rational potentials and exceptional orthogonal polynomials

We show how all the quantal systems related to the exceptional Laguerre and Jacobi polynomials can be constructed in a direct and systematic way, without the need of shape invariance and Darboux-Crum transformation. Furthermore, the prepotential need not be assumed a priori. The prepotential, the deforming function, the potential, the eigenfunctions and eigenvalues are all derived within the same framework. The exceptional polynomials are expressible as a bilinear combination of a deformation function and its derivative.

高斯求积的普遍性性质，导数规则以及一种新的Stieltjes反演方法

Universality Properties of Gaussian Quadrature, The Derivative Rule, and a Novel Approach to Stieltjes Inversion

本文研究了高斯求积的普遍性性质，导数规则以及一种新的Stieltjes反演方法，通过分析经典的正交多项式和非经典的Pollaczek多项式，展示了该方法在数值Stieltjes反演中的高效性，并探讨了其在离散谱分解中的应用。

特征向量的先验 tanθ 定理

The a priori tanθtheorem for eigenvectors

本文证明当自伴算子A的谱间隙d满足扰动V的范数小于√2 d时，特征向量f与投影到未扰动子空间上的夹角θ满足tanθ ≤ ‖V‖/d，且该界是紧的。