Here you can find some information on the research project M1309 The Camassa-Holm equation and indefinite spectral problems funded by the Austrian Science Fund (FWF).

The aim of this project is to investigate the blow up phenomena for the Camassa-Holm equation by employing the inverse scattering transform approach. It is well known that the blow-up may occur if the corresponding isospectral problem is indefinite. This isospectral problem is given by the weighted Sturm-Liouville equation. We want to develop direct and inverse scattering theory for the isospectral problem associated with the Camassa-Holm equation.

The project started in September 2011 and finished on August 2013. For a continuation see my Stand-Alone project P26060.

• Sergio Albeverio (HCM, Germany)
• Rainer Brunnhuber (University of Klagenfurt, Austria),
• Branko Ćurgus (Western Washington University, USA),
• Jonathan Eckhardt (Cardiff University, UK),
• Andreas Fleige,
• Mark Malamud (IAMM of NASU, Ukraine),
• Hagen Neidhardt (WIASS, Germany),
• Gerald Teschl (University of Vienna, Austria)

1. On a necessary aspect for the Riesz basis property for indefinite Sturm-Liouville problems, Math. Nachr. 287, no. 14-15, 1710-1732 (2014) (arXiv:1202.2444)
2. Schrödinger operators with δ'-interactions on a Cantor type set, (with J. Eckhardt, M. Malamud and G. Teschl), J. Differential Equations 257, 415-449 (2014) (arXiv:1401.7581)
3. The similarity problem for indefinite Sturm-Liouville operators and the HELP inequality, Adv. Math. 246, 368-413 (2013) (arXiv:1207.2586)
4. Singular Weyl-Titchmarsh-Kodaira theory for one-dimensional Dirac operators, (with R. Brunnhuber, J. Eckhardt, and G. Teschl), Monatsh. Math. 174, 515-547 (2014). (arXiv:1305.3099)
5. The Riesz basis property of an indefinite Sturm-Liouville problem with non-separated boundary conditions, (with B. Ćurgus and A. Fleige), Integr. Equat. Oper. Theory 77, 533-557 (2013) (arXiv:1306.1329)
6. An isospectral problem for global conservative multi-peakon solutions of the Camassa-Holm equation, (with J. Eckhardt), Comm. Math. Phys. 239, 893-918 (2014) (arXiv:1406.3702)
7. Spherical Schrödinger operators with δ-interactions, (with S. Albeverio, M. Malamud, and H. Neidhardt), J. Math. Phys. 54, Art ID: 052103 (2013), 24 pages. (arXiv:1211.4048)
8. Spectral analysis of semibounded Schrödinger operators with δ'-interactions, (with M. Malamud), Ann. Henri Poincare 15, 501-541 (2014) (arXiv:1212.1691)
9. Spectral analysis of indefinite Sturm–Liouville operators, Funct. Anal. Appl. 48, no.3, 88-92 (2014).
10. Spectral asymptotics for perturbed spherical Schrödinger operators and applications to quantum scattering, (with G. Teschl) Comm. Math. Phys. 322, 255-275 (2013) (arXiv:1205.5049)
11. 1-D Schrödinger operators with local point interactions: a review, (with M. Malamud), in "Spectral Analysis, Integrable Systems, and Ordinary Differential Equations", H. Holden et al. (eds), Proceedings of Symposia in Pure Mathematics 87, Amer. Math. Soc., (2013), 235-262 (arXiv:1303.4055)
12. Inverse uniqueness results for one-dimensional weighted Dirac operators, (with J. Eckhardt and G. Teschl, ), in "Spectral Theory and Differential Equations: V.A. Marchenko 90th Anniversary Collection", E. Khruslov, L. Pastur, and D. Shepelsky (eds), 117-133, Advances in the Mathematical Sciences 233, Amer. Math. Soc., Providence, 2014. arXiv:1305.3100

• Habilitation: Spectral Theory of Singular Sturm-Liouville Operators, October 2012