An Introduction to Inverse Scattering and Inverse Spectral Methods

Foreword Preface Chapter 1: A Review of Basic Mathematical Tools, Lassi Paeivaerinta. Linear Operators on Hilbert Space Integral Operators and the Fredholm Alternative The Fourier Transform and the Hilbert Transform The Unique Continuation Principle (UCP) Unbounded Operators The Spectrum The Resolvent Kernel and the Fredholm Determinant A Particle in a Box Maxwell's Equations References Chapter 2: Multidimensional Inverse Scattering Theory, David Colton. Electromagnetic Scattering Problem Bessel Functions The Addition Formula Green's Formula Basic Properties of Far Field Patterns Spectral Theory of the Far Field Operator The Inverse Scattering Problem The Detection and Monitoring of Leukemia Regularization Closing Remarks References Chapter 3: Inverse Sturm-Liouville Problems, William Rundell. Introduction Preliminary Material The Liouville Transformation Asymptotic Expansions of the Eigenvalues and Eigenfunctions The Inverse Problem-A Historical Look A Completeness Result An Important Integral Operator Solving Hyperbolic Equations Uniqueness Proofs Constructive Algorithms Modification for Other Spectral Data Other Differential Equations Other Constructive Algorithms The Matrix Analogue Another Finite-Dimensional Algorithm Fourth-Order Problems References Chapter 4: Inverse Problems in Potential Scattering, Khosrow Chadan. Introduction Physical Background and Formulation of the Inverse Scattering Problem Scattering Theory for Partial Waves Gel'fand-Levitan Integral Equation Marchenko Equation Inverse Problem on the Line Nonlinear Evolution Equations Closing Remarks Appendix References Index.