Analytic Scattering Theory for Jacobi Operators and Bernstein–Szegö Asymptotics of Orthogonal Polynomials

We study semi-infinite Jacobi matrices [Formula: see text] corresponding to trace class perturbations [Formula: see text] of the “free” discrete Schrödinger operator [Formula: see text]. Our goal is to construct various spectral quantities of the operator [Formula: see text], such as the weight function, eigenfunctions of its continuous spectrum, the wave operators for the pair [Formula: see text], [Formula: see text], the scattering matrix, the spectral shift function, etc. This allows us to find the asymptotic behavior of the orthonormal polynomials [Formula: see text] associated to the Jacobi matrix [Formula: see text] as [Formula: see text]. In particular, we consider the case of [Formula: see text] inside the spectrum [Formula: see text] of [Formula: see text] when this asymptotic has an oscillating character of the Bernstein–Szegö type and the case of [Formula: see text] at the end points [Formula: see text].

Relative Szegő Asymptotics for Toeplitz Determinants

We study the asymptotic behaviour, as $n \to \infty$, of ratios of Toeplitz determinants $D_n({\rm e}^h {\rm d}\mu)/D_n({\rm d}\mu)$ defined by a measure $\mu$ on the unit circle and a sufficiently smooth function $h$. The approach we follow is based on the theory of orthogonal polynomials. We prove that the second order asymptotics depends on $h$ and only a few Verblunsky coefficients associated to $\mu$. As a result, we establish a relative version of the Strong Szegő Limit Theorem for a wide class of measures $\mu$ with essential support on a single arc. In particular, this allows the measure to have a singular component within or outside of the arc.

Szegö Asymptotics of Extremal Polynomials on the Segment [–1, +1]: The Case of a Measure with Finite Discrete Part

The strong asymptotics of monic extremal polynomials with respect to the norm 𝐿𝑝(σ) are studied. The measure σ is concentrated on the segment [–1, 1] plus a finite set of mass points in a region of the complex plane exterior to the segment [–1, 1].