Monotone Jacobi parameters and non-Szegő weights

Abstract

We relate asymptotics of Jacobi parameters to asymptotics of the spectral weights near the edges. Typical of our results is that for a n ≡ 1, b n = −Cn −β (0 < β < 2 3 ), one has dµ(x) = w(x) dx on (−2, 2), and near x = 2, w(x) = e −2Q(x) where(1 + O((2 − x))).

Spectral edge behavior for eventually monotone Jacobi and Verblunsky coefficients

“…Much of the effort in the proof of Theorem 3 goes towards controlling the higher-order terms in (1.12), (1.13). Lest we neglect the most important special cases, and recalling that [6] described the case a n ≡ 1, b n = −Cn −β , we consider the case a n = 1 − Cn −τ , b n = 0 (1.18) and compute explicitly the expansion to order O(log(2 − x)).…”

“…(note that for n = N 0 , the undefined quantity γ N 0 −1 (x) appears in some of the equations, but in a term multiplied by Φ N 0 −1 (x) = 0 so this doesn't present any ambiguity; to avoid undefined quantities let us set γ N 0 −1 (x) = 0). In the following proposition we use the argument of [6], starting from n = N 0 instead of n = 1; the estimates are proved in [6, Section 3] for the case b n ≡ 0 but, with the additional input of Lemma 2.3, the arguments work in our level of generality.…”

“…By continuity, there exists Using (2.11) for n = N (x) and n = N (x) + 1 gives Now we need to control the oscillatory part of the solutions. Here we can follow the method of [6], with modifications to allow both nontrivial sequences of diagonal and off-diagonal Jacobi coefficients; indeed, the proofs of Lemma 2.6, Prop. 2.8, and Lemma 2.9 are straightforward generalizations of arguments from [6].…”

“…[6] considered separately two cases, a n ≡ 1 and b n ≡ 0, while our theorem allows both sequences to be non-constant, unifying and generalizing their arguments. [6] also assumed monotonicity from N 0 = 1, while we only assume monotonicity from some arbitrary N 0 ; while compactly supported perturbations are often easy to handle in spectral theory, in this problem having eventual monotonicity rather than monotonicity requires us to control the two-dimensional space of eigensolutions, instead of just the Dirichlet eigensolution, and introduces the complications related to subordinacy. Remark 1.1.…”

“…Pollaczek [11,12,13] considered Jacobi parameters a n , b n given by rational functions with numerators and denominators of the same degree (which can be rewritten in the above form, with all exponents negative integers). Kreimer-Last-Simon [6] considered the case a n = 0, b n = −Cn −β and developed some of the techniques we will use. We view (1.12), (1.13) as a natural class of polynomially decaying perturbations; it includes, for instance, linear combinations and products of sequences such as (n + n 0 ) −γ , γ > 0.…”