Department of Mathematics
Slowly decaying perturbations of Jacobi and Schrodinger operators
The spectrum of an operator is a generalization of the notion of the set of eigenvalues of a finite matrix. We are concerned with spectral properties of certain classes of operators, such as Schrodinger operators (central to quantum mechanics) and Jacobi matrices (tied to orthogonal polynomials). Work of Deift—-Killip and Killip--Simon shows that L^2 perturbations of the free Schrodinger operator or free Jacobi matrix preserve a.c. (absolutely continuous) spectrum. This result is optimal on the L^p scale, so spectral properties of slower decaying perturbations can only be established under additional assumptions. In this talk, we will discuss several recent results on slowly decaying perturbations. Some of our results solve an open problem about a class of oscillatory decaying perturbations which includes almost periodic times decaying sequences. In another approach, we describe the spectral consequences of L^2 bounded variation conditions. Finally, we discuss our recent contributions to higher-order Szego theorems; this includes the disproving of a conjecture of Simon and the first equivalence result in the regime of arbitrarily slow decay.
Sponsored by the Department of Mathematical Sciences
Host: Vladimir Dragovic
Refreshments will be served in FO 2.610F 30 minutes before the talk begins.