Hardy spaces in Kotani-Last and other spectral problems

Recent developments in the spectral theory of Schrödinger operators, Jacobi and CMV matrices reported in top- ranked mathematical journals and our own new level of understanding of the function theory in infinitely connected domains provide a solid basis for tackling some old and new problems, including: 1) Kotani-Last problem, 2) Killip-Simon problem for general finite-gap sets, 3) Mixed inverse spectral problems for reflectionless Jacobi matrices, 4) Parametric description of spectral surfaces of periodic multi-diagonal operators, 5) Widom condition for resolvent domains of Schrödinger operators. The first two were posed by the leading experts in the field, and we concentrate on their description. Kotani-Last problem requires proof that the presence of an absolutely continuous component in the spectrum of an ergodic operator implies that it is almost periodic. According to Avila, "this problem has been for a while, and became a central topic of the theory, after recent popularization (by Simon, Jitomirskaya, and Damanik)". He answered negatively to this conjecture. Naturally, it is important and highly challenging not only to prove or disprove the conjecture, but also to explain this interesting phenomenon. There are at least two programs related to the subject: by Kotani, on Grassmann manifold and spectral theory of 1-D Schrödinger operators, and by Remling, on reflectionless Jacobi matrices. Our approach is dual to that of Avila and deals with methods of the inverse spectral theory. Consider a real compact in a generic position in which (1) all reflectionless Jacobi matrices with corresponding spectrum have no singular component, and (2) its complement is a Widom domain. We claim that such operators are ergodic, and there is a kind of tumbler with two positions: Direct Cauchy Theorem holds in the domain, or it fails. In the first case, all reflectionless matrices are almost periodic, and we expect that in the second case all of them are not. We can provide examples of such compact sets thus showing classes of ergodic matrices with purely a.c. spectrum without almost periodicity. The Killip-Simon theorem is probably one of the main achievements in the spectral theory of Jacobi matrices and orthogonal polynomials in the last decade. In collaboration with Damanik, they generalized this theorem to the periodic case. Although several important partial results were obtained, the problem of an extension of Damanik- Killip-Simon theorem to the general finite gap non-periodic case remains open. The proof of the original theorem is based on Sum Rules, and its generalization to the periodic case on the so-called "Magic Formulas". We suggest a way to find new Sum Rules and a counterpart of the Magic Formula in the non-periodic case.

The project suggested a program to solve two quite famous problems in the spectral theory of ergodic and close to them operators, namely the Kotani-Last and the Killip-Simon problems. For the first problem one asks if the existence of an absolutely continuous component in the spectrum of an ergodic family of 1-D Schrödinger operators implies their almost periodicity. In fact, jointly with A. Volberg we developed a comprehensive theory to answer this question. Under the assumption of three natural axioms, we found an analytic condition on the joint resolvent domain of the given ergodic family completely responsible for almost periodicity. As soon as (Direct) Cauchy Theorem (DCT) holds in the given domain, all reflectionless operators with the given spectrum are almost periodic and if it fails they form an ergodic family but none of them is almost periodic. We investigated deeply the DCT condition and were able to show that even if all reflectionless measures on the boundary of the domain are absolutely continuous, the DCT may be violated. Note that all previously known examples of its violation were related to a presence of singular components in a reflectionless measure. Thus the Kotani-Last problem received a complete (negative) solution. The main result was published in Inventiones Mathematicae. The original theorem of Killip and Simon describes how Hilbert-Schmidt class perturbations of the free discrete Schrödinger operator affect its spectrum and vice versa. Jointly with Damanik they were able to generalize this result to perturbations of periodic Jacobi matrices. Both results were published in Annals of Mathematics. Their approach was based on the characterization of all periodic Jacobi matrices via certain polynomial operator identity, associated to a common spectral set. They called this identity the magic formula. We suggested a new magic formula, which is valid for an arbitrary finite system of intervals but required to use rational functions and a new class of periodic operators, which we called GMP matrices. We got a generalization of the Killip-Simon theorem for GMP matrices. To extend it to the required case of Jacobi matrices we found a new integrable system, which we called Jacobi flow on GMP matrices. Both new mathematical objects (GMP matrices and the Jacobi flow on them) lead to a complete (positive) solution of the Killip-Simon problem. The corresponding paper is under consideration in Annals of Mathematics, but its results were e.g. presented at the Isaac Newton Institute, Cambridge, and on a one-month mini-seminar joint with Barry Simon at the Hebrew University of Jerusalem. Given investigations generated a new approved by FWF project (P29363-N32). Ph.D. student B. Eichinger wrote his dissertation, which deals with the spectral theory of periodic GMP matrices and asymptotics for Chebyshev polynomials.