4:00 pm Wednesday, February 22, 2012
Geometry-Analysis Seminar: **Spectral Theory and Parreau-Widom Sets**
by **
Jacob Stordal Christiansen** (University of Copenhagen) in HB 227
In the talk, I'll discuss almost periodic Jacobi operators on $\ell^2(\Z)$. Such operators are selfadjoint and tend to have Cantor spectrum. Throughout, we shall focus on the class of operators whose spectrum (or essential spectrum) is an infinite gap set, $E$, of Parreau-Widom type. This notion is suitably defined via conformal mappings of $\C_+$ onto comb-like domains and it includes Cantor sets of positive measure. An all-important role will be played by the set $T_E$ of reflectionless operators on $E$. By a result of Remling, they form the natural limiting object for operators with a.c. spectrum on $E$. As will be explained, $T_E$ is topologically an infinite dimensional torus for most Parreau-Widom sets. We shall introduce the Szego class for E and show that all its elements are asymptotically almost periodic operators. It also follows that the associated orthogonal polynomials admit a power asymptotic behavior, aka Szego asymptotics. Host Department: Rice University-Mathematics Submitted by damanik@rice.edu |