Fine structure of the zeros of orthogonal polynomials IV: A priori bounds and clock behavior

Abstract

Abstract. 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 Schrödinger operators, Poisson behavior implies positive Lyapunov exponent. Both results depend on a priori bounds on eigenvalue spacings for which we provide several proofs.

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

“…We adapt the definition from [LS08] (Background Section 2.5.3): Definition 1.3.2. Fix ξ * in an interval I, and number the zeros ξ N of u (−, L) with increasing positive integers to the right of ξ * and decreasing negative integers to the left so that ... < ξ −1 < ξ * ≤ ξ 0 < .... We say there is strong clock behavior of zeros of u at ξ * on an interval I if the density of states ρ(ξ)dξ is continuous and nonvanishing on I and for fixed n…”

The Christoffel-Darboux kernel

“…The following theorem of Last-Simon [57], based on ideas of Golinskii [41], can be used on successive zeros (see [57] for the proof).…”

“…Without knowing of Freud's work, Simon, in a series of papers (one joint with Last) [82,83,84,57], focused on this behavior, called it clock spacing, and proved it in a variety of situations (not using universality or the CD kernel). After Lubinsky's work on universality, Levin [59] has changed slightly from Section 6.…”

Two extensions of Lubinsky’s universality theorem

“…Following Levin-Lubinsky, it also implies uniform clock behavior of the zeros in I in the sense of Last-Simon [17] (if a < b).…”

“…Jost asymptotics are the key to proving clock behavior for zeros in [30,31,17]. In a sense, using the Levin-Lubinsky strategy, we can regard (1.16) as a kind of infinitesimal Jost asymptotics.…”