Inverse Problems in Quantum Scattering Theory

I Some Results from Scattering Theory.- I.1 The Reduced Radial Schrödinger Equation.- I.2 The Regular Solution: S-Wave (l = 0).- I.3 The Jost Solution: S-Wave (l = 0).- I.4 The Jost Function and the Phase Shift.- I.5 Higher Waves.- I.6 Singular Potentials.- I.7 Comments and References.- II Bound States-Eigenfunction Expansions.- II.1 Bound States: The Levinson Theorem.- II.2 Integral Representation for the Jost Function.- II.3 Eigenfunction Expansion.- II.4 Miscellaneous Results.- II.5 Singular Potentials.- II.6 Comments and References.- III The Gel'fand-Levitan-Jost-Kohn Method.- III.1 The Povzner-Levitan Representation.- III.2 The Gel'fand-Levitan Integral Equation.- III.3 Krein's Equation.- III.4 Higher Waves.- III.5 More General Equations.- III.6 Concluding Summary of the Method.- III.7 Comments and References.- IV Applications of the Gel'fand-Levitan Equation.- IV.1 Introduction of New Bound States.- IV.2 Phase Equivalent Potentials.- IV.3 Bargmann Potentials.- IV.4 Transformations of the Schrödinger Equation.- IV.5 Comments and References.- V The Marchenko Method.- V.1 The Levin Representation.- V.2 The Marchenko Integral Equation.- V.3 Comments and References.- VI Examples.- VI.1 Bargmann Potentials.- VI.2 Singular Potentials.- VI.3 Comments and References.- VII Special Classes of Potentials.- VII.1 Yukawa Potentials and the Direct Problem.- VII.2 Yukawa Potentials and the Inverse Problem.- VII.3 Higher Waves-Coulomb Potential.- VII.4 Holomorphic Potentials.- VII.5 Comments and References.- VIII Nonlocal Separable Interactions.- VIII.1 The Direct Problem.- VIII.2 The Inverse Problem.- VIII.3 More General Interactions.- VIII.4 Applications.- VIII.5 Comments and References.- IX Miscellaneous Approaches to the Inverse Problems at Fixed l.- IX.1 Generalization to Other Central Potentials.- IX.2 Krein's Approach.- IX.3 Systems of Equations.- IX.4 Coupled Channels.- IX.5 Relativistic Problems.- IX.6 Discrete Forms of the Methods.- IX.7 Dispersion Relation Approach.- IX.8 Energy Dependent Potentials.- IX.9 Miscellaneous Results.- X Scattering Amplitudes from Elastic Cross Sections.- X.1 Introduction.- X.2 Constructive Methods.- X.3 Other Uniqueness Studies.- X.4 Local Results.- X.5 Uniqueness and Stability: A Reassessment.- X.6 Generalizations.- X.7 Comments and References.- XI Potentials from the Scattering Amplitude at Fixed Energy: General Equation and Mathematical Tools.- XI.1 Introduction.- XI.2 The Transformation Kernel.- XI.3 The Symmetric Kernel and the Integral Equation.- XI.4 The General Machinery.- XI.5 Further Study of the Integral Equation.- XI.6 Remarks on This Chapter.- XI.7 Connection Between the Problem at Fixed E and the Problem at Fixed l.- XII Potentials from the Scattering Amplitude at Fixed Energy: Matrix Methods.- XII.1 Introduction.- XII.2 A Method in Which the Index ? Runs Through Integers.- XII.3 Inversion of the Matrix M and Other Properties.- XII.4 Construction of cl from tan ?l.- XII.5 Construction of V(r)-Consistency of the Method.- XII.6 Generalized Matrix Methods.- XII.7 Miscellaneous Results.- XII.8 Interpolation Properties.- XII.9 Limitation of the Matrix Methods.- XIII Potentials from the Scattering Amplitude at Fixed Energy: Operator Methods.- XIII.1 Introduction.- XIII.2 Method for Potentials of the Yukawa Class.- XIII.3 Methods Using the Spectrum of the Schrödinger Operator.- XIII.4 Complete Solution.- XIII.5 Remarks on the Methods.- XIV The Three-Dimensional Inverse Problem.- XIV.1 Introduction.- XIV.2 New Outline of One-Dimensional Methods.- XIV. 3 Approach of the Three-Dimensional Problem Based on Faddeev's Green's Function.- XIV.4 Consistency and ?-Approaches.- XIV.5 Other Approaches in the Frequency Domain.- XIV.6 Time-Domain Inverse Scattering Theory.- XIV.7 Comments and References.- XV Miscellaneous Approaches to Inverse Problems at Fixed Energy.- XV.1 Methods Using Interpolation Properties.- XV.2 Methods Using Generalized Translation Operators.- XV.3 Remark on the Results Given in This Chapter.- XVI Approximate Methods.- XVI.1 Introduction.- XVI.2 Born Approximation.- XVI.3 The Semiclassical Approximation I.- XVI.4 Semiclassical Analysis II.- XVI.5 Semiclassical Analysis III.- XVI.6 From Approximate to Exact Methods.- XVI.7 Semiclassical Studies in Other Fields.- XVII Inverse Problems in One Dimension.- XVII.1 Introduction.- XVII.2 The Inverse Problem: Approaches Related with a Marchenko Equation.- XVII.3 The Inverse Problem: Other Approaches.- XVII.4 More General One-Dimensional Problems.- XVII.5 Comments and References.- XVII.A Appendix (Exercises for Readers).- XVIII Problems Connected with Discrete Spectra.- XVIII. 1 Introduction.- XVIII.2 Relations with Other Problems and Extensions.- XVIII.3 Inverse Problem in the Coupling Constant.- XVIII.4 Comments and References.- XIX Numerical Problem.- XIX.1 Introduction.- XIX.2 Numerical Methods for Local Potentials.- XIX.3 Numerical Methods for Inverse Spectral Problems.- XIX.4 Nonlocal Potentials.- Reference List.