Spectral theory for generalized bounded variation perturbations of orthogonal polynomials and Schrodinger operators

Abstract

The purpose of this text is to present some new results in the spectral theory of orthogonal polynomials and Schrodinger operators. These results concern perturbations of the free Schrodinger operator and of the free case for orthogonal polynomials on the unit circle (which corresponds to Verblunsky coefficients equal to 0) and the real line (which corresponds to off-diagonal Jacobi coefficients equal to 1 and diagonal Jacobi coefficients equal to 0). The condition central to our results is that of generalized bounded variation. This class consists of finite linear combinations of sequences of rotated bounded variation with an L^1 perturbation. This generalizes both usual bounded variation and expressions of the form lambda(x) cos(phi x + alpha) with lambda(x) of bounded variation (and, in particular, with lambda(x) = x^gamma, Wigner-von Neumann potentials) as well as their finite linear combinations. Assuming generalized bounded variation and an L^p condition (with any finite p) on the perturbation, our results show preservation of absolutely continuous spectrum, absence of singular continuous spectrum, and that embedded pure points in the continuous spectrum can only occur in an explicit finite set.