Finite Gap Jacobi Matrices, I. The Isospectral Torus

Abstract

Abstract. Let e ⊂ R be a finite union of disjoint closed intervals. In the study of OPRL with measures whose essential support is e, a fundamental role is played by the isospectral torus. In this paper, we use a covering map formalism to define and study this isospectral torus. Our goal is to make a coherent presentation of properties and bounds for this special class as a tool for ourselves and others to study perturbations. One important result is the expression of Jost functions for the torus in terms of theta functions.

Two extensions of Lubinsky’s universality theorem

“…The point of this remark is that the construction in Section 2 relies on Jost solutions. For E's with rational harmonic measures, the Jacobi parameters are periodic; and Jost solutions can be constructed with Floquet theory rather than the more elaborate methods of [48,36,26,5].…”

“…We also note there that it suffices to prove the results in Section 2 when each interval has rational harmonic measure, so that one can use Floquet theory in place of the more subtle analysis of [48,36,26,5].…”

“…Our model is the measure associated to a point in the isospectral torus associated to E, where the analysis depends on results of Widom [48], Sodin-Yuditskii [36], Peherstorfer-Yuditskii [26], and Christiansen-SimonZinchenko [5].…”

Lieb–Thirring Inequalities for Finite and Infinite Gap Jacobi Matrices

“…In the setting of periodic and almost periodic parameters, the role of J 0 as a natural limiting point is taken over by the so-called isospectral torus, denoted T E . See, e.g., [3,4,22] for a deeper discussion of this object. The finite gap version of (1.4) with p = 1/2 says that if E is a finite gap set (i.e., a finite union of disjoint, compact intervals) and J is a trace class perturbation of an element…”

Finite Gap Jacobi Matrices and the Schottky–Klein Prime Function

“…This paper is motivated by recent work of Christiansen et al [9] who describe a covering map formalism to define and study the isospectral torus associated with finite gap Jacobi matrices. In turn, that paper drew on earlier work of Sodin and Yuditskii [29] and Peherstorfer and Yuditskii [27] which concentrated on infinite gap sets.…”