Orthogonal Polynomials with Recursion Coefficients of Generalized Bounded Variation

TL;DR

This paper studies probability measures with specific recursion coefficient conditions, proving the absence of singular continuous parts and continuity of the density on certain intervals, extending results to Schrödinger operators with complex potentials.

Contribution

It introduces a generalized bounded variation condition for recursion coefficients and proves spectral properties for measures satisfying this condition, including the absence of singular continuous spectrum.

Findings

Measures have no singular continuous spectrum on (-2,2).

The density function is continuous and non-vanishing outside a finite set.

Results extend to Schrödinger operators with Wigner--von Neumann type potentials.

Abstract

We consider probability measures on the real line or unit circle with Jacobi or Verblunsky coefficients satisfying an $\ell^p$ condition and a generalized bounded variation condition. This latter condition requires that a sequence can be expressed as a sum of sequences $\beta^{(l)}$, each of which has rotated bounded variation, i.e., $\sum_{n=0}^\infty | e^{i\phi_l} \beta_{n+1}^{(l)} - \beta_n^{(l)} |$ is finite for some $\phi_l$. This includes discrete Schr\"odinger operators on a half-line or line with finite linear combinations of Wigner--von Neumann type potentials. For the real line, we prove that in the Lebesgue decomposition $d\mu=f dm + d\mu_s$ of such measures, the intersection of (-2,2) with the support of $d\mu_s$ is contained in an explicit finite set S (thus, $d\mu$ has no singular continuous part), and f is continuous and non-vanishing on $(-2,2) \setminus S$. The results for the unit circle are analogous, with (-2,2) replaced by the unit circle.