Titchmarsh-Weyl formula for the spectral density of a class of Jacobi matrices in the critical case

Abstract

We consider a class of Jacobi matrices with unbounded entries in the so called critical (double root, Jordan box) case. We prove a formula for the spectral density of the matrix which relates its spectral density to the asymptotics of orthogonal polynomials associated with the matrix.

Asymptotic behavior of orthogonal polynomials. Singular critical case

“…Our goal is to study the critical case for rapidly growing coefficients a n , b n when the Carleman condition is not satisfied. The asymptotic formulas we obtain are quite different from those of the papers [10,13].…”

“…For sufficiently general coefficients a n , b n , the critical case was studied in the papers [10,13] (see also the references therein) where the Carleman condition (1.3) was required. Our goal is to study the critical case for rapidly growing coefficients a n , b n when the Carleman condition is not satisfied.…”

“…Now formulas (2.8) and (2.9) are true (and hence we are in the singular case) if σ > 3/2. The regular situation was studied in the papers [10,13].…”

“…13. Let a n = (n + 1)(n + x)(n + y)(n + x + y) and b n = 2n 2 + (2x + 2y − 1)n + xy (6.21) where parameters x, y ∈ R + .…”

About essential spectra of unbounded Jacobi matrices

“…It is also rich enough to allow building an intuition about the general case. In particular, in this class there are examples of Jacobi matrices with purely absolutely continuous spectrum filling the whole real line (see [15,17,35,38,41]), having a bounded gap in absolutely continuous spectrum (see [5-7, 9, 10, 12, 14, 19, 27]), having absolutely continuous spectrum on the half-line (see [4,8,16,18,[23][24][25][26]33]), having purely singular continuous spectral measure with explicit Hausdorff dimension (see [2]), having a dense point spectrum on the real line (see [2]), and having an empty essential spectrum (see [11,[29][30][31]42]).…”