Riassunto
Rapid progress in quantum theory brings us new important results which are often not immediately clear to all who need them. But fortunately, this is also followed by simplifications and unifications of our previous concepts. The inverse problem method ("The most beautiful idea of the XX-th century" - Zakharov et aI., 1980) has just both these aspects. It is rather astonishing that it took 50 years after the foundation of quantum mechanics for the creation of the "pictures" showing the direct connection of obser vables with interactions. Recently, illustrations of this type began to appear in the literature (e. g., how potentials are deformed with thc shift of one energy level or change of some resonance reduced width). Although they are transparent to those studying the quantum world and can be included within the necessary elements of quantum literacy, they are still largely unknown even to many specialists. For the first time, the most interesting of these pictures enriching our quantum intuition are col lected here and placed at your disposal. The readers of this monograph have the advantage of getting the latest information which became available after the publication of the Russian edition. It has been incor porated here in the simplest presentation possible. For example, new sections con cerning exactly solvable models, including the multi-channel, multi-dimensional ones and with time dependent potentials have been added. The first attempts in solving the three-body inverse problem are also mentioned.
Contenuti
I: One-dimensional, One-channel Systems.- 1 The principal Equations of Scattering Theory.- 1.1 General Remarks.- 1.2 Elements of the Direct and Inverse Problems.- 1.2.1 The Simplest Difference Schrödinger Equation.- 1.2.2 Potential Wells of Infinite Depth.- 1.2.3 The Direct Problem.- 1.2.4 The Inverse Problem.- 1.2.5 Scattering by a Potential of Finite Range.- 1.2.6 The Finite-Difference Analogue of the R-matrix Scattering Theory.- 1.2.7 Conditions of Orthonormality and Completeness of the Eigenfunctions of the Finite-Difference Schrödinger Operator on [O, ?].- 1.2.8 Relations Between the Scattering Parameters {E? and ??}.- 1.2.9 Reconstruction of the Potential on the Semi-axis 0 ?x Upper k Walf-plane.- 2.2.3 Potentials with S (k) with Two Poles in the Upper k Half-Plane.- 2.3 More General Models.- 2.3 Multi-term Degenerate Kernels of the Inverseproblem Equations.- 2.3.2 Models of One-Dimensional Motion on the Whole Axis.- 2.3.3 The Finite-Difference-Approach.- 2.3.4 The Rational Reflection Coefficient (no Bound States).- 2.3.5 The Finite-difference Approach.- 2.4 Potentials of the Finite-Range and Infinitely Deep Wells. R-matrix Models.- 2.5 Potentials Allowing Exact Solutions for Variable Angular Momenta.- 2.5.1 Potentials from Spectral Data at Fixed Energy and Variable l.- 2.5.2 Newton-Sabatier Potentials.- 2.5.3 The Generalized Crum-Krein Transformations.- 2.5.4 Lipperheide-Fidelday Potentials.- 2.6 Notes on the Literature.- 2.7 Exercises.- 3 Approximate Solutions.- 3.1 Convergence of the Approximations, Stability and Regularization of Solutions.- 3.2 Solutions Using Bargmann Potentials.- 3.4 Approximation of Datentials by Steps, at Discrete Points, and by Splines. The Role of the Upper Part of the Spectrum.- 3.5 Method of Multiple Solutions of the Direct Problem.- 3.6 Notes on the Literature.- 4 The Levinson Theorem.- 4.1 General Remarks.- 4.2 Simple Examples.- 4.3 The Coulomb Potential and Other Singular Interactions.- 4.4 Other Types of Interactions.- 4.4.1 Potentials Depending on Velocity.- 4.4.2 Potentials Depending on Energy.- 4.4.3 The Finite-Difference Schrödinger Equation.- 4.4.4 Motion Along the Axis.- 4.4.5 Nonlocal Potentials.- 4.5 Notes on the Literature.- II. Multi-channel, Multi-dimensional, Multi-particle Problems.- 5 Multi-channel Equations.- 5.1 General Remarks.- 5.2 The Formalism of Multi-channel Coupling.- 5.3 Finite-Difference Equations of Motion.- 5.4 Exactly Solvable Models.- 5.5 Notes on the Literature.- 5.6 Exercises.- 6 Multi-dimensional Problems.- 6.1 General Remarks.- 6.2 The Finite-Difference Formalism.- 6.3 Reduction of Multi-dimensional problems to Multichannel problems.- 6.4 The Multi-dimensional Inverse Problem.- 6.5 Separation of Variables in Noncentral Field.- 6.6 Exactly Solvable Models.- 6.7 Notes on the Literature.- 7 Multi-particle Systems.- 7.1 General Remarks.- 7.2 Asymptotic Hamiltonians and Boundary Conditions.- 7.3 Tunnelling Through Potential Barriers by Several Particles.- 7.4 Excitation of Collective Degrees of Freedom of Multi-particle Systems..- 7.4.1 Transformation of Amplitudes in Transition to the Reference Frame of the Target.- 7.5 The Method of Hyperspherical Functions (K-harmonics).- 7.6 The Levinson Theorem.- 7.7 Three-Particle Potentials.- 7.8 Notes on the Literature.- References.
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