Direct And Inverse Scattering On The Line

Richard Beals

This book deals with the theory of linear ordinary differentialoperators of arbitrary order. Unlike treatments that focus onspectral theory, this work centers on the construction ofspecial eigenfunctions (generalized Jost solutions) and on theinverse problem: the problem of reconstructing the operator fromminimal data associated to the special eigenfunctions. In thesecond order case this program includes spectral theory and isequivalent to quantum mechanical scattering theory; theessential analysis involves only the bounded eigenfunctions. Forhigher order operators, bounded eigenfunctions are againsufficient for spectral theory and quantum scattering theory,but they are far from sufficient for a successful inversetheory.

The authors give a complete and self-contained theory of theinverse problem for an ordinary differential operator of anyorder. The theory provides a linearization for the associatednonlinear evolution equations, including KdV and Boussinesq. Theauthors also discuss Darboux-Bäcklund transformations,related first-order systems and their evolutions, andapplications to spectral theory and quantum mechanicalscattering theory.

Among the book's most significant contributions are a newconstruction of normalized eigenfunctions and the first completetreatment of the self-adjoint inverse problem in order greaterthan two. In addition, the authors present the first analytictreatment of the corresponding flows, including a detaileddescription of the phase space for Boussinesq and otherequations.

The book is intended for mathematicians, physicists, andengineers in the area of soliton equations, as well as thoseinterested in the analytical aspects of inverse scattering or inthe 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 complexvariable theory, the theory of ordinary differential equations,linear algebra, and functional analysis. The authors haveattempted to make the book sufficiently complete andself-contained to make it accessible to a graduate studenthaving no prior knowledge of scattering or inverse scatteringtheory. The book may therefore be suitable for a graduatetextbook or as background reading in a seminar.

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