Orthogonal polynomials on the unit circle: New results

We announce numerous new results in the theory of orthogonal polynomials on the unit circle

Fine Structure of the Zeros of Orthogonal Polynomials: A Review

We review recent work on zeros of orthogonal polynomials

Tosio Kato's Work on Non-Relativistic Quantum Mechanics: An Outline

Based at a talk given at the Kato Centennial Symposium in Sept. 2017, this article discusses the scientific life and some of the scientific work of T. Kato.Comment: 15 pages. Based on a much longer review article (of 200 plus pages) still in prpearatio

Fine structure of the zeros of orthogonal polynomials, I. A tale of two pictures

Mhaskar-Saff found a kind of universal behavior for the bulk structure of the zeros of orthogonal polynomials for large n. Motivated by two plots, we look at the finer structure for the case of the Verblunsky coefficients and for what we call the BLS condition: αn = Cb^n + O ((bΔ)^n). In the former case, we describe the results of Stoiciu. In the latter case, we prove asymptotically equal spacing for the bulk of zeros

Analogs of the M-Function in the Theory of Orthogonal Polynomials on the Unit Circle

We show that the multitude of applications of the Weyl-Titchmarsh m-function leads to a multitude of different functions in the theory of orthogonal polynomials on the unit circle that serve as analogs of the m-function

Zeros of OPUC and Long Time Asymptotics of Schur and Related Flows

We provide a complete analysis of the asymptotics for the semi-infinite Schur flow: $\alpha_j(t)=(1- |\alpha_j(t)|^2) (\alpha_{j+1}(t)-\alpha_{j-1}(t))$ for $\alpha_{-1}(t)= 1$ boundary conditions and $n=0,1,2,...$, with initial condition $\alpha_j(0)\in (-1,1)$. We also provide examples with $\alpha_j(0)\in\mathbb{D}$ for which $\alpha_0(t)$ does not have a limit. The proofs depend on the solution via a direct/inverse spectral transform