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Chaos in classical and quantum mechanics. (English) Zbl 0727.70029
Interdisciplinary Applied Mathematics, 1. New York etc.: Springer-Verlag. xiii, 432 p. DM 68.00 (1990).
The monograph is an elementary introduction to the theory of dynamical systems, which do not follow simple, regular, and predictible patterns and which are now called chaotic dynamical systems.
The author is most interested in fields to which he has made extensive contributions. Hence, the questions that have a bearing on the connection between classical and quantum mechanics were singled out. In this way the very subtle problem of the existence of the chaotic features in quantum mechanics is frequently tackled from this point of view.
This book written in an informal style with arguments of geometric rather than algebraic nature offers a collection of ideas and examples rather than general concepts and mathematical theorems.
The monograph is divided in two main parts. The first part containing eleven chapters is devoted to the classical chaotic dynamical systems. From the content of this part we mention the study of periodic orbits, and the method of the surface section developed in the sixth aud seventh chapter, the connection between the soft chaos and the breakdown of the invariant tori due to the perturbation (the ninth chapter), the entropy in classical mechanics (the tenth chapter). The concept of the hard chaos illustrated by the study of the hydrogen atom inside a crystal, and called the anisotropic Kepler problem, is studied extensively in the eleventh chapter.
The second part of the monograph, devoted to the study of the quantum chaotic systems starts by recognizing in the thirteenth chapter that there is no chaos in the world of quantum mechanics. Therefore, the remaining chapters of this book constitute mainly an effort to organize what is known so far under some general headings rather than to present a coherent overall view.
The main topic studied successfully in the sixteenth, seventeenth and eighteenth chapter is that of the energy spectrum of classical chaotic systems and its connection by a trace formula with the quantum-mechanical energy levels. Some symptoms of chaos in quantum mechanics are explained in terms of classical orbits.
The nineteenth chapter is devoted to the motion on surface of constant negative curvature. The author emphasizes that many results of hyperbolic geometry are of central importance for the understanding of quantum chaos. The two topics of this last chapter (the twentieth chapter) have only recently been considered by the author in connection with the discussion of chaotic Hamiltonian systems. Finding a useful code for all classical trajectories seems to be the most important tool. The anisotropic Kepler problem is coded by binary sequences, and appears to be the simplest case of hard chaos. Codes for scattering seem to be easier to construct than for bound states. The concept of ultifractal sets is necessary in order to express the invariant measure defined in the code description in terms of the invariant measure defined in terms of the usual coordinates in phase space. Overall, the book can be highly recommended to a reader interested in learning about a wide range of topics in which the chaos is involved.

70K50 Bifurcations and instability for nonlinear problems in mechanics
81-02 Research exposition (monographs, survey articles) pertaining to quantum theory
70-02 Research exposition (monographs, survey articles) pertaining to mechanics of particles and systems
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
81Q50 Quantum chaos