Direct and Inverse Scattering on the Line

This book explores linear ordinary differential operators and the inverse problem, providing a comprehensive resource for mathematicians and physicists alike, particularly in scattering theory.

In Direct and Inverse Scattering on the Line, the authors delve into the theory of linear ordinary differential operators of arbitrary order. This work distinguishes itself from other treatments by emphasizing the construction of special eigenfunctions, known as generalized Jost solutions, rather than focusing solely on spectral theory. A significant aspect of the book is the exploration of the inverse problem, which involves reconstructing the operator from minimal data associated with these special eigenfunctions. The authors effectively bridge the gap between spectral theory and quantum mechanical scattering theory, particularly in the second-order case, where essential analysis revolves around bounded eigenfunctions.

The authors present a comprehensive and self-contained theory concerning the inverse problem for ordinary differential operators of any order. This theory not only linearizes associated nonlinear evolution equations, such as KdV and Boussinesq, but also discusses important concepts like Darboux-Bäcklund transformations and related first-order systems. The book outlines applications of these theories to spectral theory and quantum mechanical scattering, making it a valuable resource for those in the field.

Among its notable contributions, Direct and Inverse Scattering on the Line offers a new construction of normalized eigenfunctions and a thorough treatment of the self-adjoint inverse problem for orders greater than two. Additionally, the authors provide an analytic approach to the corresponding flows, detailing the phase space for Boussinesq and other equations. This book is designed for mathematicians, physicists, and engineers working in soliton equations, inverse scattering theory, or linear ordinary differential operators, making it suitable for graduate students and seminar reading.