Sum Rules and the Szegö Condition for Jacobi Matrices

Author: Zlatoš, Andrej

Year: 2003

Degree: Dissertation (Ph.D.)

Advisor: Simon, Barry M.

Committee Members: Simon, Barry M.; Makarov, Nikolai G.; Killip, Rowan; Schlag, Wilhelm

Option: Mathematics

DOI: 10.7907/DBVE-VF23

Abstract

We consider Jacobi matrices J with real b_n on the diagonal, positive a_n on the next two diagonals, and with u'(x) the density of the absolutely continuous part of the spectral measure. In particular, we are interested in compact perturbations of the free matrix J_0, that is, such that the a_n go to 1 and b_n go to 0. We study the Case sum rules for such matrices. These are trace formulae relating sums involving the a_n's and b_n's on one side and certain quantities in terms of the spectral measure on the other. We establish situations where the sum rules are valid, extending results of Case and Killip-Simon.

The matrix J is said to satisfy the Szego condition whenever the integral

int_{0}^{pi} log [u'(2 cos t)] dt,

which appears in the sum rules, is finite. Applications of our results include an extension of Shohat's classification of certain Jacobi matrices satisfying the Szego condition to cases with an infinite point spectrum, and a proof that if n(a_n - 1) go to a, nb_n go to b, and 2a < |b|, then the Szego condition fails. Related to this, we resolve a conjecture by Askey on the Szego condition for Jacobi matrices which are Coulomb perturbations of J_0. More generally, we prove that if

a_n = 1 + a/n^c + O(n^{-1-eps}) and b_n = b/n^c + O(n^{-1-eps})

with 0 < γ ≤ 1 and eps > 0, then the Szego condition is satisfied if and only if 2a ≥|b|

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