Most of the laws of physics are expressed in the form of differential equations; that is our legacy from Isaac Newton. The customary separation of the laws of nature from contingent boundary or initial conditions, which has become part of our physical intuition, is both based on and expressed in the properties of solutions of differential equations. Within these equations we make a further distinction: that between what in mechanics are called the equations of motion on the one hand and the specific forces and shapes on the other. The latter enter as given functions into the former. In most observations and experiments the "equations of motion," i. e. , the structure of the differential equations, are taken for granted and it is the form and the details of the forces that are under investigation. The method by which we learn what the shapes of objects and the forces between them are when they are too small, too large, too remote, or too inaccessi­ ble for direct experimentation, is to observe their detectable effects. The question then is how to infer these properties from observational data. For the theoreti­ cal physicist, the calculation of observable consequences from given differential equations with known or assumed forces and shapes or boundary conditions is the standard task of solving a "direct problem. " Comparison of the results with experiments confronts the theoretical predictions with nature.

Inhalt

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.