This book deals with the theory of linear ordinary differential operators of
arbitrary order. Unlike treatments that focus on
spectral theory, this work centers on the construction of special
eigenfunctions (generalized Jost solutions) and on the inverse problem: the
problem of reconstructing the operator from minimal data associated to the
special eigenfunctions. In the second order case this program includes
spectral theory and is equivalent to quantum mechanical scattering theory; the
essential analysis involves only the bounded eigenfunctions. For higher order
operators, bounded eigenfunctions are again sufficient for spectral theory and
quantum scattering theory, but they are far from sufficient for a successful
inverse theory.
The authors give a complete and
self-contained theory of the inverse problem for an ordinary differential
operator of any order. The theory provides a linearization for the associated
nonlinear evolution equations, including KdV and Boussinesq. The authors also
discuss Darboux-Bäcklund transformations, related first-order systems and
their evolutions, and applications to spectral theory and quantum mechanical
scattering theory.
Among the book's most significant contributions are a new construction of
normalized eigenfunctions and the first complete treatment of the self-adjoint
inverse problem in order greater than two. In addition, the authors present
the first analytic treatment of the corresponding flows, including a detailed
description of the phase space for Boussinesq and other equations.
The book is intended for mathematicians, physicists, and engineers in the area
of soliton equations, as well as those interested in the analytical aspects of
inverse scattering or in the general theory of linear ordinary differential
operators. This book is likely to be a valuable resource to many.
Required background consists of a basic knowledge of complex variable theory,
the theory of ordinary differential equations, linear algebra, and functional
analysis. The authors have attempted to make the book sufficiently complete
and self-contained to make it accessible to a graduate student having no prior
knowledge of scattering or inverse scattering theory. The book may therefore
be suitable for a graduate textbook or as background reading in a seminar.