One-dimensional inverse scattering and spectral problems.
Date
2011-06-03
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
Abstract
Inverse scattering and spectral one-dimensional problems are discussed systematically in a selfcontained way. Many novel results due to the author are presented. The classical results are often presented in a new way. Several highlights of the new results include:
- Analysis of the invertibility of the steps in the Gel’fand-Levitan and Marchenko inversion procedures,
- Theory of the inverse problem with I-function as the data and its applications;
- Proof of the property C for ordinary differential operators, numerous applications of property C;
- Inverse problems with “incomplete” data;
- Spherically symmetric inverse scattering problem with fixed-energy data: analysis of the Newton- Sabatier (NS) scheme for inversion of fixed-energy phase shifts is given. This analysis shows that the NS scheme is fundamentally wrong, and is not a valid inversion method.
- Complete presentation of the Krein inverse scattering theory is given. Consistency of this theory is proved.
- Quarkonium systems;
- A study of the properties of I-function;
- Some new inverse problems for the heat and wave equations are studied.
- A study of inverse scattering problem for an inhomogeneous Schr¨odinger equation;
Description
Keywords
Property C for ODE, Inverse spectral and scattering problems, Inverse problems for PDE and ODE, Spectral and scattering theory