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.
The project started in December 2004 and finished in December 2006. For a continuation see my START project.
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).
- Joint meeting of AMS, DMV, ÖMG in Mainz 2005: Special session on Spectral Analysis of Differential and Difference Operators, organized by Evgeny Korotyaev (Potsdam and HU Berlin), Boris Mityagin (Ohio State), and Gerald Teschl (Vienna).
- I. Egorova, J. Michor, and G. Teschl, Scattering theory for Jacobi operators with quasi-periodic background, Comm. Math. Phys. 264-3, 811-842 (2006).
- J. Michor and G. Teschl, Trace formulas for Jacobi operators in connection with scattering theory for quasi-periodic background, in Operator Theory, Analysis and Mathematical Physics, J. Janas (ed.) et al., 69-76, Oper. Theory Adv. Appl., 174, Birkhäuser, Basel, 2007.
- D. Damanik and G. Teschl, Bound states of discrete Schrödinger operators with super-critical inverse square potentials, Proc. Amer. Math. Soc. 135, 1123-1127 (2007).
- A. Sakhnovich, Skew-self-adjoint discrete and continuous Dirac type systems: inverse problems and Borg-Marchenko theorems, Inverse Problems 22, 2083-2101 (2006).
- I. Egorova, J. Michor, and G. Teschl, Inverse scattering transform for the Toda hierarchy with quasi-periodic background, Proc. Amer. Math. Soc. 135, 1817-1827 (2007).
- F. Gesztesy, H. Holden, and G. Teschl, The algebro-geometric Toda hierarchy initial value problem for complex-valued initial data, Rev. Mat. Iberoamericana 24-1, 117-182 (2008).
- A. Sakhnovich, Harmonic maps, Bäcklund-Darboux transformations and `line solution' analogues, J.Phys. A: Math.Gen. 39, 15379-15390 (2006)
- S. Kamvissis and G. Teschl, Stability of periodic soliton equations under short range perturbations, Phys. Lett. A 364-6, 480-483 (2007).
- I. Egorova, J. Michor, and G. Teschl, Soliton solutions of the Toda hierarchy on quasi-periodic backgrounds revisited, Math. Nachr. 282:4, 526--539 (2009).
- I. Egorova, J. Michor, and G. Teschl, Scattering theory for Jacobi operators with steplike quasi-periodic background, Inverse Problems 23, 905-918 (2007).
- G. Teschl, Algebro-geometric constraints on solitons with respect to quasi-periodic backgrounds, Bull. London Math. Soc. 39-4, 677-684 (2007).
- S. Kamvissis and G. Teschl, Long-time asymptotics of the periodic Toda lattice under short-range perturbations, J. Math. Phys. 53, 073706 (2012).
- Johanna Michor, Scattering Theory for Jacobi Operators and Applications to Completely Integrable Systems, May 2005 (awarded the Studienpreis of the Austrian Mathematical Society 2006)