On the spacing of zeros of paraorthogonal polynomials for singular measures

Eigenvalue spacing for 1D singular Schrödinger operators

The aim of this paper is to provide uniform estimates for the eigenvalue spacings of one-dimensional semiclassical Schrödinger operators with singular potentials on the half-line. We introduce a new development of semiclassical measures related to families of Schrödinger operators that provides a means of establishing uniform non-concentration estimates within that class of operators. This dramatically simplifies analysis that would typically require detailed WKB expansions near the turning point, near the singular point and several gluing type results to connect various regions in the domain.

衰减势能的特征值间距上界

Upper bounds on eigenvalue spacing for decaying potentials

本文研究了衰减势能下Dirichlet特征值间距的上界，揭示了势能衰减率与特征值间距的关系。

Spacing properties of the zeros of orthogonal polynomials on Cantor sets via a sequence of polynomial mappings

Let $μ$ be a probability measure with an infinite compact support on $\mathbb{R}$. Let us further assume that $(F_n)_{n=1}^\infty$ is a sequence of orthogonal polynomials for $μ$ where $(f_n)_{n=1}^\infty$ is a sequence of nonlinear polynomials and $F_n:=f_n\circ\dots\circ f_1$ for all $n\in\mathbb{N}$. We prove that if there is an $s_0\in\mathbb{N}$ such that $0$ is a root of $f_n^\prime$ for each $n>s_0$ then the distance between any two zeros of an orthogonal polynomial for $μ$ of a given degree greater than $1$ has a lower bound in terms of the distance between the set of critical points a…