Direct and Inverse Problems

This textbook can almost be viewed as a "how-to" manual for solving quantum inverse problems, that is, for deriving the potential from spectra or scattering data and also, as somewhat of a quantum "picture book" which should enhance the reader's quantum intuition. The formal exposition of inverse methods is paralleled by a discussion of the direct problem. Differential and finite-difference equations are presented side by side. The common features and (dis)advantages of a variety of solution methods are analyzed. To foster a better understanding, the physical meaning of the mathematical quantities are discussed explicitly. Wave confinement in continuum bound states, resonance and collective tunneling, energy shifts and the spectral and phase equivalence of various interactions are some of the physical problems covered. §Dieses Buch ist fast ein "Rezeptbuch" für die Lösung des quantenmechanischen inversen Problems, d.h. für die Ermittlung des Potentials aus Spektren oder Streudaten, sowie eine Art "Bilderbuch" der Quantenmechanik, welches das Gefühl des Lesers für die Quantenmechanik verbessern soll. Die formale Darstellung der inversen Methoden erfolgt parallel zu der des direkten Problems. Differential- und Differenzengleichungen werden Seite an Seite entwickelt. Eine Vielfalt von Lösungsmethoden wird präsentiert. Zur Vertiefung des Verständnisses werden die physikalischen Bedeutungen der mathematischen Größen im Detail diskutiert. 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.