One-dimensional inverse scattering and spectral problems

Abstract

Inverse scattering and spectral one-dimensional problems are discussed systematically in a self-contained 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) Analisys 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 Schrödinger equation.