Extremal polynomials on subsets of the unit circle

We plan to investigate two different classes of extremal polynomials. That is, we plan to study Chebyshev- and orthogonal polynomials. The Chebyshev polynomials of degree n associated to a compact subset of the complex plane are those polynomials that are bounded by one in modulus on the given set and have maximal leading coefficient. In a seminal paper (1969) Widom has given a complete description of the asymptotics of Chebyshev polynomials if the given set is a disjoint union of Jordan regions. He was not able to complete his studies if the set also contains Jordan arcs. Almost 50 years later a corresponding theory is still lacking. Instead of Chebyshev polynomials, we propose to study a more general problem. That is, for a given point in the complementary domain to find the degree n polynomial that is bounded by one on the set and has maximal value at the given point. A conjecture of Yuditskii states that the limit of the extremal value as a function on the complementary domain should be the diagonal of a reproducing kernel. This would lead to a complete new understanding of the asymptotics of Chebyshev polynomials. We suggest to study this question for union of arcs on the unit circle and for a spiral curve. The Killip-Simon theorem describes how Hilbert-Schmidt perturbations of the free discrete Schr\ödinger operator affect its spectrum and vice versa, how changes in the spectrum affect the coefficients of the corresponding operator. The interval [-2,2] is the spectrum of the free discrete Schrödinger operator. The Killip-Simon theorem has been generalized to arbitrary union of intervals. The Killip-Simon theorem is the real-line version of a famous theorem of Szegö for the unit circle and the corresponding unitary operators, which are called CMV matrices. A generalization of Szegö`s theorem for arbitrary union of arcs on the unit circle is still unknown. We plan to study the methods that lead to the generalization of the Killip-Simon theorem also for the unit circle and are confident that this serves as a solid basis to prove a generalization of Szegö`s theorem. The Korteweg-de-Vries equation is a partial differential equation in time and space. It is known that periodic initial conditions lead to almost periodic solutions in time direction. Deift asked the question whether this is also the case for almost periodic initial conditions. On the basis of a work of Binder, Damanik, Goldstein and Lukic we plan to give an affirmative answer to this question for a class of initial conditions that seem natural in this setting.

We describe two major problems that were solved during this project: The Korteweg-de-Vries equation is a partial differential equation in time and space. It is known that periodic initial conditions lead to almost periodic solutions in time direction. Deift asked the question whether this is also the case for almost periodic initial conditions. Under a very natural moment condition on the spectrum, we constructed for almost periodic initial data a solution which is also almost periodic in time direction and thus provide an affirmative answer to this question. Given a bounded set in the plane one may ask how a charge will distribute in order to find an equilibrium, assuming it can move freely on the set. If the charge distribution is modeled by a probability measure supported on the set, this leads to minimizing the logarithmic energy among all admissible weights. In general there is a unique charge distribution corresponding to an equilibrium state. It is natural to expect (and well known) that the corresponding logarithmic potential is constant on the set. This constant value is called Robin constant and is relevant in many areas of mathematical research. For instance, it enters as a rescaling parameter in results describing asymptotic behaviors of extremal polynomials. In particular it is relevant in the spectral theory of Jacobi matrices. Clearly the minimization problem described above is not well posed if the set is unbounded. However, in contrast to difference operators such as Jacobi matrices, differential operators such as continuum Schrödinger operators usually have unbounded spectra. Thus, it is desirable to have a notion of Robin constant also for unbounded sets. In the case of spectra of continuum Schrödinger operators and Dirac operators we were able to find the right analog in the course of this project. In fact, this was only the starting point of a whole theory of Stahl--Totik regularity for continuum Schrödinger and Dirac operators, respectively. This theory could be summarized as a characterization when solutions of these differential operators with fixed initial values (so called Dirichlet solutions) have a regular growth behavior in spacial direction.