# Finite Gap Jacobi Matrices: A Review

@article{Christiansen2013FiniteGJ,
title={Finite Gap Jacobi Matrices: A Review},
author={Jacob S. Christiansen and Barry Simon and Maxim Zinchenko},
journal={arXiv: Spectral Theory},
year={2013}
}
• Published 22 January 2013
• Mathematics
• arXiv: Spectral Theory
Perhaps the most common theme in Fritz Gesztesy's broad opus is the study of problems with periodic or almost periodic finite gap differential and difference equations, especially those connected to integrable systems. The present paper reviews recent progress in the understanding of finite gap Jacobi matrices and their perturbations. We'd like to acknowledge our debt to Fritz as a collaborator and friend. We hope Fritz enjoys this birthday bouquet!
15 Citations
Mini-Workshop: Reflectionless Operators: The Deift and Simon Conjectures
Reflectionless operators in one dimension are particularly amenable to inverse scattering and are intimately related to integrable systems like KdV and Toda. Recent work has indicated a strong (but
Numerical computation of the isospectral torus of finite gap sets and of IFS Cantor sets
We describe a numerical procedure to compute the so-called isospectral torus of finite gap sets, that is, the set of Jacobi matrices whose essential spectrum is composed of finitely many intervals.
Killip-Simon-classes of Jacobi matrices with essential spectrum on two symmetric and of SMP matrices on two arbitrary intervals
• Mathematics, Computer Science
• 2013
The parametric description of SMP matrices of the Killip-Simon-class with their essential spectrum on two arbitrary intervals is the main result of this paper.
Jacobi Flow on SMP Matrices and Killip–Simon Problem on Two Disjoint Intervals
• Mathematics
• 2016
We give a free parametric representation for the coefficient sequences of Jacobi matrices whose spectral measures satisfy the Killip–Simon condition with respect to two (arbitrary) disjoint
Orthogonal polynomials on Cantor sets of zero Lebesgue measure
In this survey article, we review some results and conjectures related to orthogonal polynomials on Cantor sets. The main purpose of this paper is to emphasize the role of equilibrium measures in
Asymptotics of extremal polynomials for some special cases
ASYMPTOTICS OF EXTREMAL POLYNOMIALS FOR SOME SPECIAL CASES Gökalp Alpan Ph.D. in Mathematics Advisor: Alexandre Goncharov May 2017 We study the asymptotics of orthogonal and Chebyshev polynomials on
Orthogonal polynomials for the weakly equilibrium Cantor sets
• Mathematics
• 2015
Let $K(\gamma)$ be the weakly equilibrium Cantor type set introduced in [10]. It is proven that the monic orthogonal polynomials $Q_{2^s}$ with respect to the equilibrium measure of $K(\gamma)$

## References

SHOWING 1-10 OF 85 REFERENCES
Finite Gap Jacobi Matrices, I. The Isospectral Torus
• Mathematics
• 2008
AbstractLet $\frak{e}\subset\mathbb{R}$ be a finite union of disjoint closed intervals. In the study of orthogonal polynomials on the real line with measures whose essential support is  $\frak{e}$
Sum rules for Jacobi matrices and their applications to spectral theory
• Mathematics
• 2001
We discuss the proof of and systematic application of Case's sum rules for Jacobi matrices. Of special interest is a linear combination of two of his sum rules which has strictly positive terms.
Critical Lieb-Thirring Bounds in Gaps and the Generalized Nevai Conjecture for Finite Gap Jacobi Matrices
• Mathematics
• 2010
We prove bounds of the form $\sum_{e\in I\cap\sigma_\di (H)} \dist (e,\sigma_\e (H))^{1/2} \leq L^1$-norm of a perturbation, where $I$ is a gap. Included are gaps in continuum one-dimensional
On the Direct Cauchy Theorem in Widom Domains: Positive and Negative Examples Peter Yuditskii
We discuss several questions which remained open in our joint work with M. Sodin “Almost periodic Jacobi matrices with homogeneous spectrum, infinite-dimensional Jacobi inversion, and Hardy spaces of
Perturbations of orthogonal polynomials with periodic recursion coefficients
• Mathematics
• 2007
The results of Denisov-Rakhmanov, Szegő-Shohat-Nevai, and Killip-Simon are extended from asymptotically constant orthogonal polynomials on the real line (OPRL) and unit circle (OPUC) to
Algebro-geometric quasi-periodic finite-gap solutions of the Toda and Kac-van Moerbeke hierarchies
• Mathematics
• 1998
Introduction The Toda hierarchy, recursion relations, and hyperelliptic curves The stationary Baker-Akhiezer function Spectral theory for finite-gap Jacobi operators Quasi-periodic finite-gap
Probabilistic Averages of Jacobi Operators
I study the Lyapunov exponent and the integrated density of states for general Jacobi operators. The main result is that questions about these can be reduced to questions about ergodic Jacobi
Subordinacy and spectral theory for infinite matrices
• Mathematics
• 1992
The notion of subordinacy, previously used as a tool in the spectral theory of ordinary differential operators, is extended and applied to solutions of three-term recurrence relations. The resulting