Here you can find some information on the research project P26060 The Camassa-Holm equation and indefinite spectral problems funded by the Austrian Science Fund (FWF).

The aim of this project is to investigate the Camassa-Holm equation by employing the inverse scattering/spectral transform approach. Motivated by our recent study of conservative multi-peakon solutions, we suggest a new generalized spectral problem, which is quadratic in a spectral parameter, as an isospectral problem for the conservative CH equation. The aims of the project are to develop direct and inverse scattering theory for this generalized indefinite spectral problem and to study the blow up phenomena for the CH equation with the help of the inverse scattering transform.

The project started in September 2013.

• Jonathan Eckhardt (University of Vienna, Austria),
• Fritz Gesztesy (Baylor University, USA),
• Helge Holden (Norwegian University of Science and Technology, Norway),
• Markus Holzleitner (University of Vienna, Austria),
• Tom H. Koornwinder (KdV Institute, Amsterdam, Netherlands),
• Mark Malamud (IAMM of NASU, Ukraine),
• Gerald Teschl (University of Vienna, Austria)

1. Dispersion estimates for spherical Schrödinger equations with critical angular momentum, (with M. Holzleitner and G. Teschl), in "Partial Differential Equations, Mathematical Physics, and Stochastic Analysis", F. Gesztesy et al. (eds), EMS Congress Reportspp. 319-348, 2018. (arXiv:1611.05210)
2. Jacobi polynomials, Bernstein-type inequalities and dispersion estimates for the discrete Laguerre operator, (with T. Koornwinder and G. Teschl), Adv. Math. 333, 796-821 (2018) (arXiv:1602.08626)
3. Real-valued algebro-geometric solutions of the two-component Camassa-Holm hierarchy, (with J. Eckhardt, F. Gesztesy, H. Holden, and G. Teschl), Ann. Inst. Fourier (Grenoble) 67, no.3, 1185-1230 (2017) (arXiv:1512.03956)
4. Quadratic operator pencils associated with the conservative Camassa-Holm flow, (with J. Eckhardt), Bull. Soc. Math. France 145, no.1, 47-95 (2017) (arXiv:1406.3703)
5. The Camassa-Holm equation and the string density problem, (with J. Eckhardt and G. Teschl), Intern. Math. Nachr. 233, 1-24 (2016) (arXiv:1701.03598)
6. Dispersion estimates for spherical Schrödinger equations: The effect of boundary conditions, (with M. Holzleitner and G. Teschl), Opuscula Math. 36, 769-786 (2016) (arXiv:1601.01638)
7. Dispersion estimates for the discrete Laguerre operator, (with G. Teschl), Lett. Math. Phys. 106, 545-555 (2016) (arXiv:1510.07019)
8. Schrödinger operators with δ-interactions in a space of vector-valued functions, (with M. Malamud and D. Natiagailo), Math. Notes 100, no.1, 59-77 (2016) (arXiv:1603.00594)
9. Dispersion estimates for spherical Schrödinger equations, (with G. Teschl and J. H. Toloza), Ann. Henri Poincaré 17, no.11, 3147-3176 (2016) (arXiv:1504.03015)
10. Spectral asymptotics for canonical systems, (with J. Eckhardt and G. Teschl), J. reine angew. Math., to appear (arXiv:1412.0277)
11. On spectral deformations and singular Weyl functions for one-dimensional Dirac operators, (with A. Beigl, J. Eckhardt and G. Teschl), J. Math. Phys. 56, Art ID 012102 (2015) (arXiv:1410.1152)
12. The inverse spectral problem for indefinite strings, (with J. Eckhardt), Invent. Math. 204, no.3, 939-977 (2016) (arXiv:1409.0139)
13. A note on J-positive block operator matrices, Integr. Equat. Oper. Theory 81, 113-125 (2015) (arXiv:1403.2406)