We discuss ultracold atomic gas with attractive interactions in a one-dimensional optical lattice. We find that its excitation spectrum displays a quantum soliton band, corresponding to N-particle bound states, and a continuum band of other, mostly extended, states. For a system of a finite size, the two branches are degenerate in energy for weak interactions, while a gap opens above a threshold value for the interaction strength. We find that the interplay between degenerate extended and bound states has important consequences for both static and dynamical properties of the system.
During the last 20 years there has been a considerable literature on a collection of related mathematical topics: higher degree versions of Poncelet’s Theorem, certain measures associated to some finite Blaschke products and the numerical range of finite dimensional completely non-unitary contractions with defect index 1. I will explain that without realizing it, the authors of these works were discussing OPUC. This will allow us to use OPUC methods to provide illuminating proofs of some of their results and in turn to allow the insights from this literature to tell us something about OPUC. In particular, I’ll discuss a Wendroff theorem for POPUC. This is joint work with Andrei Martínez-Finkelshtein and Brian Simanek.