Spectral Theory and the Sum Rules for Jacobi Matrices and Orthogonal Polynomials.

Abstract: The sum rules (aka trace formulae) for Jacobi matrices, a powerful tool of spectral theory, are relations between the coefficients of the matrix on one hand and its spectral measure on the other. I will discuss some recent developments in the area and their applications to questions about the spectral types of Jacobi matrices and Schrödinger operators. Related problems for orthogonal polynomials on the unit circle will also be discussed.