Inverse Schrödinger Scattering in Three Dimensions

ISBN-13: 9783642836732 / Angielski / Miękka / 2011 / 170 str.

Roger G. Newton

Opis

This book specifically explores the inverse Schrodinger scattering problem in three dimensions with a potential that is without any assumed symmetry properties. It covers most of the known methods for the exact solution of this problem, including detailed proofs of most of the needed theorems. The solutions of the three-dimensional inverse-scattering problem found during the last twenty years are presented together for the first time in coherent, self-contained, and detailed manner. This book combines the known results published previously in research papers only, corrects some errors, closes some blind alleys in previous developments, and also contains a number of new results."

Spis treści

I Use of the Scattering Solution.- 1. The Direct Scattering Problem.- 1.1 The Scattering Solution.- 1.2 Exceptional Points.- 1.3 Completeness.- 1.4 Asymptotics for Large |k|.- 1.5 Scattering Amplitude and S Matrix.- 1.6 Angular Momentum Projections.- 1.7 Proofs.- 1.8 Notes.- 2. The Inverse Problem.- 2.1 Introduction and Uniqueness.- 2.2 A First Approach.- 2.3 The Riemann-Hilbert Problem.- 2.3.1 No Bound States.- 2.3.2 Bound States.- 2.4 The x-dependence.- 2.4.1 Proofs.- 2.5 Variational Principles.- 2.6 The Jost Function.- 2.6.1 The Reduction Method.- 2.6.2 Solution Procedure.- 2.6.3 Angular Momentum Projections.- 2.6.4 Proofs.- 2.7 Perturbations and Stability.- 2.8 A Representation of the Potential.- 2.9 Miscellaneous Methods.- 2.10 Notes.- II Use of the Regular and Standing-Wave Solutions.- 3. The Regular Solution.- 3.1 How to Define a Regular Solution.- 3.2 Properties of the Regular Solution.- 3.2.1 Proof of Lemma 3.2.3.- 3.3 Completeness.- 3.4 Notes.- 4. The Inverse Problem.- 4.1 Generalized Gel’fand-Levitan Equations.- 4.2 An Example.- 4.3 Reduction to Central Potentials.- 4.4 Notes.- 5. Standing-Wave Solutions.- 5.1 The K-matrix.- 5.2 The $$\mathop \partial \limits^ - $$ -method.- 5.3 The Inverse Problem.- 5.4 Notes.- III Use of the Faddeev Solution.- 6. Faddeev’s Solution.- 6.1 The Green’s Function.- 6.2 The Integral Equation.- 6.2.1 Proofs.- 6.3 Exceptional Points.- 6.4 Large-r Asymptotics and Scattering.- 6.5 Notes.- 7. The Inverse Problem.- 7.1 The Scattering Amplitude as Input.- 7.2 A Gel’fand-Levitan Procedure.- 7.2.1 Completeness.- 7.2.2 The Povzner-Levitan Kernel.- 7.2.3 The Gel’fand-Levitan-Faddeev Equation.- 7.2.4 Proofs.- 7.3 A Marchenko Procedure.- 7.4 The $$ \mathop \partial \limits^ - $$ -approach.- 7.5 Notes.- References.- Index of Symbols.