Scattering Theory for Jacobi Operators

The aim of this project is to develop direct and inverse scattering theory for Jacobi operators which are short-range perturbations of finite-gap quasiperiodic operators. We want to derive the corresponding Gelfand-Levitan- Marchenko equations and find necessary and sufficient conditions for the scattering data to uniquely determine the coefficients. Then the results will be applied to study the corresponding initial value problems of the Toda and Kac-van Moerbeke hierarchies and, in particular, their long time asymptotics via a Riemann-Hilbert approach.

Direct and inverse scattering theory for one-dimensional Schrödinger and Jacobi (discrete Schrödinger) operators is a classical topic in quantum mechanics. The principal goal is to recover the perturbation by comparing the unperturbed motion with the perturbed one. The first aim of the project was to develop direct and inverse scattering theory for Jacobi operators which are short- range perturbations of periodic operators, and to apply these results to study the corresponding initial value problems of the Toda and Kac-van Moerbeke hierarchies. Among our main results are minimal scattering data which determine the perturbed operator uniquely in the above mentioned situations and a complete analysis of the inverse scattering transform for the entire Toda hierarchy in these cases. The second aim was to perform a stability analysis of solitons of the Toda lattice on a periodic background solution. So far, it was generally believed that a perturbed periodic integrable system splits asymptotically into a number of solitons plus a decaying radiation part, a situation similar to that observed for perturbations of the constant solution. We showed that this is not the case; instead the radiation part does not decay, but manifests itself asymptotically as a modulation of the periodic solution which undergoes a continuous phase transition in the isospectral class of the periodic background solution. We could provide an explicit formula for this modulated solution in terms of Abelian integrals on the underlying hyperelliptic Riemann surface and provide numerical evidence for its validity. We used the Toda lattice as a model but the same methods and ideas are applicable to all soliton equations in one space dimension (e.g. the Korteweg- de Vries equation).