Deift Problem and Spectral Theory

Percy Deift is one of the best-known mathematicians due to his work on spectral theory, integrable systems, random matrix theory. At his 60th birthday conference in 2005, he was asked to present a list of unsolved problems. He updated this list ten years later. As the number one unsolved problem in both lists we still have the following conjecture: a solution of the KdV equation is almost periodic in time, as soon as the initial data function is almost periodic with respect to the space variable. Deift himself mentioned several recent spectacular and seminal works by Binder, Damanik, Goldstein, Lukic, e.g. [Duke Math. J., 2018], that provided partial affirmative solutions of this conjecture. We would add to this our most recent result [Trans. Amer. Math. Soc., 2019]. Nevertheless, as the main hypothesis of this project we state the following proposition: there exists arbitrary smooth almost periodic initial data such that the solution of the KdV equation is not almost periodic in time. Clearly the aim is extremely ambitious since it deals with a challenging problem explicitly posed by one of the leading experts in the field, and has been unsolved for decades. As a suitable output of the project we would consider partial results in developing the following popular topics: spectral theory of almost periodic operators, including canonical systems; asymptotics of Chebyshev polynomials, including extremal problems on Cantor type sets. Any comprehensive theory in these directions would be highly recognized in the international mathematical community. We have a rather developed plan to solve the first problem in the Deift list. It is based on a new theory of reflectionless operators, which was developed in our joint paper with Volberg [Invent. Math., 2014], in connection with the Kotani problem. We found a certain analytic condition (DCT) on the resolvent domain, which can be used as a criterion of almost periodicity for generic Widom domains: if DCT holds in the domain, then every reflectionless potential with the given resolvent domain is almost periodic, and if it fails, then every reflectionless potential is not almost periodic. In the project we present solid arguments in support of the following hypothesis: among degenerated domains there are such that DCT fails, but among associated reflectionless potentials there is an almost periodic one. Adding a certain attribute of stability for such domains, by the Volberg-Yuditskii argument, we would get a solution of the KdV equation, which is not almost periodic in time with almost periodic initial data.

This is a report on two years research project. The project was devoted to a highly ambitious goal solving the number one Deift problem in the theory of integrable systems. Definitely pandemic situation has an impact on its performance: all mutual visits were canceled as well as a special conference that supposed to bring in Linz many key specialists in spectral and approximation theories and in related fields of functional and harmonic analysis. For this reason, in the first year we concentrate on a closely related but less ambition problem, which realization was more realistic to perform with communications restricted to Zoom and Skype. In late 90-th Sodin and Yuditskii found a closed to optimal condition for a description of almost periodic Jacobi matrices with absolutely continuous spectrum. This result was widely appreciated. Simultaneously they described almost periodic 1-D Schrödinger operators with similar properties. However, the mentioned above continual systems required additional spectral restrictions related to accumulations of the spectrum at infinity, so called finite gap length conditions. Note that Christiansen, Simon, Yuditskii and Zinchenko adopted this theory to describe asymptotics of Chebyshev polynomials [Duke Math. J. 2019], while the problem on a parallel general theory for continuous systems remained open. According to Potapov and de Branges this theory has to describe canonical systems, however the concept of almost periodicity was not introduced until appearance of the recent ideas of Ch. Remling. Following this idea, in collaboration with R. Bessonov and M. Lukic, we proved that canonical systems in Arov gauge possess all required properties (in particular we introduced the concept of almost periodicity for such systems). Already as consequences of this general result, we found additional spectral conditions that are required to have almost periodic Schrödinger and Dirac operators and canonical systems in the Potapov- de Branges gauge, which was common in the previous investigations. In the second year it became clear that the pandemic situation will not improve, so we concentrated on the Deift problem, coordinating mutual efforts in Zoom contacts. In this period Prof. A. Volberg joint our research group (Damanik, Milivoje, Yuditskii). An extended report with proved theorems and still open conjectures is presented in a mutual manuscript "The Deift conjecture: a program to construct a counterexample", arXiv:2111.09345. Due to a retirement of the principal investigator the work on the problem was stopped. In particular, the method of infinitesimal variations of comb domains found in the work on the project was successfully applied in solving the well-known Andrievsky problem in approximation theory.