Spectral theory, abelian coverings and Iterations

The proposal Spectral Theory, Abelian Coverings and Iterations deals with spectral theory of ergodic difference/differential operators of the second order. The importance of the topic was recently recognized at the highest level by the mathematical community: Avila devoted his whole Fields Medal talk to quasi-periodic Schrödinger operators. Roughly speaking the theory uses direct and inverse methods in its development. Both approaches have their positive and negative aspects, weak and strong sides. In a broad sense the goal of the project is to construct a quite comprehensive theory for classes of operators that are typical in the direct spectral theory based on the inverse spectral theory approach. As a sample of a comprehensive theory we can mention our joint with Volberg result on operators with absolutely continuous spectrum (Kotani-Last problem). In this case we are looking for a theory of such operators with singular spectrum, and more specifically, we would like to clarify up to which extend an essential dependence of frequencies (quasi- and limit-periodic classes of operators) is important to observe this spectral phenomena. Even in this clarification the goal is very ambitious. Indeed, we know that every reflectionless Jacobi matrix whose spectrum is a Cantor set of positive Lebesgue measure is almost periodic, but what one can say if the spectrum is the classical Cantor set? (Carleson). Here is a quotation from a recent paper by Krüger and Simon: in this case, we mainly have conjectures, discussion, and some numerical experiments At the moment we restrict ourselves by two basic ideas: (1) to use Abelian Coverings and related functional spaces of analytic functions in application to ergodic operators with singular continuous spectrum; (2) to use GMP matrices and iteration theory for rational functions to study properties of Jacobi matrices associated to the corresponding Julia sets. The first idea deals with a well-known result of Lyons and McKean, who demonstrated a dramatic extension of allowable classes of analytic functions on Riemann surfaces in passing to their Abelian Coverings. GMP matrices were recently found and used successfully to solve Killip-Simon problem. Note that goal (2) can be considered as a certain intermediate case between a comparably well-studied (in this setting) problem on Julia sets associated to iterations of polynomials (Bellissard, Bessis, Moussa)and the mentioned Carleson question on the standard Cantor set. With respect to an analytic property that might play a crucial role in the general context, Sodin conjectured that the resolvent domain of an almost periodic family should be uniformly perfect. Pommerenke described properties of such sets in terms of averaging operators acting on a universal covering. In this way one gets rich families of character automorphic functions. Note that corresponding classes of analytic in the unit disc functions were studied by Korenblum and his followers.

This project deals with spectral theory of ergodic and almost periodic difference/differential operators of the second order (probably the most famous object in the spectral theory due to their relations to quantum mechanics and integrable dynamical systems). Typically their resolvent sets are Denjoy domains with Cantor type boundaries. As one of the main research goals it was planed to study and use Abelian Coverings of Denjoy domains and related functional models on Riemann surfaces. The most essential development of this theory was presented in the publication "KdV hierarchy via Abelian coverings and operator identities" [Transection of the American Mathematical Society, 2019]. We were interested in an application to the KdV hierarchy first because of its broad appeal. Recent works of Binder, Damanik, Goldstein, and Lukic has, under stronger assumptions, proven something similar only for the KdV equation itself, and Kotani has announced a related result under an integer moment condition. In short, we demonstrated that the KdV hierarchy equation for a broad class of almost periodic initial data is a consequence of a trivial commutant relation for two multiplication operators acting in the Hardy spaces on this Abelian cover. Seems, the successful completion of the project is clearly evidenced by another publication of its results in the Duke Mathematical Journal - one of the world's leading mathematical journals (joint work with J.S. Christiansen, B. Simon and M. Zinchenko). We proved Szegö-Widom asymptotics for the Chebyshev polynomials of a compact subset of R, which obeys the Parreau-Widom and DCT (Direct Cauchy Theorem) conditions. These asymptotics were complemented by very important inverse statements. In particular, in a generic position the domain should be of Parreau-Widom type with DCT as soon as the least deviation for the Chebyshev polynomials on the given compact behaves almost periodic. The project findings created a promising and solid base for solving the so-called Number One Problem in the theory of integrable systems, according to the list of Percy Deift. He conjectured that a solution of the KdV equations is almost periodic in time as soon as the initial data is almost periodic in space. Our program was already supported as FWF Stand-alone project "Deift Problem and Spectral Theory". The project allowed increasing the reputation of PI, his department and institute, improving international contacts with leading experts in the area. One of them led to a solution of the widely studied, but long time open, Remez problem for trigonometric polynomials (jointly with S. Tikhonov). Working in the project B. Eichinger prepared his own research project and applied successfully for Erwin Schrödinger fellowship (awarded by FWF).