One-dimensional inverse scattering and spectral problems.

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: 1) Analysis of the invertibility of the steps in the Gel’fand-Levitan and Marchenko inversion procedures, 2) Theory of the inverse problem with I-function as the data and its applications; 3) Proof of the property C for ordinary differential operators, numerous applications of property C; 4) Inverse problems with “incomplete” data; 5) 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. 6) Complete presentation of the Krein inverse scattering theory is given. Consistency of this theory is proved. 7) Quarkonium systems; 8) A study of the properties of I-function; 9) Some new inverse problems for the heat and wave equations are studied. 10) A study of inverse scattering problem for an inhomogeneous Schr¨odinger equation;