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.
@article{1659,
title = {One-dimensional inverse scattering and spectral problems},
journal = {CUBO, A Mathematical Journal},
volume = {6},
year = {2004},
language = {en},
url = {http://dml.mathdoc.fr/item/1659}
}
Ramm, A.G. One-dimensional inverse scattering and spectral problems. CUBO, A Mathematical Journal, Tome 6 (2004) . http://gdmltest.u-ga.fr/item/1659/