Classical Nonintegrability, Quantum Chaos (Oberwolfach Seminars, 27) - Softcover

Knauf, Andreas; Sinai, Yakov G.; Knauf, A.; Sinai, Y.G.

Softcover

ISBN 10: 3764357088 ISBN 13: 9783764357085

Publisher: Birkhäuser, 1997

Synopsis

Our DMV Seminar on 'Classical Nonintegrability, Quantum Chaos' intended to introduce students and beginning researchers to the techniques applied in nonin­ tegrable classical and quantum dynamics. Several of these lectures are collected in this volume. The basic phenomenon of nonlinear dynamics is mixing in phase space, lead­ ing to a positive dynamical entropy and a loss of information about the initial state. The nonlinear motion in phase space gives rise to a linear action on phase space functions which in the case of iterated maps is given by a so-called transfer operator. Good mixing rates lead to a spectral gap for this operator. Similar to the use made of the Riemann zeta function in the investigation of the prime numbers, dynamical zeta functions are now being applied in nonlinear dynamics. In Chapter 2 V. Baladi first introduces dynamical zeta functions and transfer operators, illustrating and motivating these notions with a simple one-dimensional dynamical system. Then she presents a commented list of useful references, helping the newcomer to enter smoothly into this fast-developing field of research. Chapter 3 on irregular scattering and Chapter 4 on quantum chaos by A. Knauf deal with solutions of the Hamilton and the Schr6dinger equation. Scatter­ ing by a potential force tends to be irregular if three or more scattering centres are present, and a typical phenomenon is the occurrence of a Cantor set of bounded orbits. The presence of this set influences those scattering orbits which come close.

About this title

Review

This book includes several lectures given at the DMV Seminar "Classical Nonintegrability, Quantum Chaos". The aim of these lectures was to provide an introduction to the ideas and mathematical techniques of classical and quantum nonlinear dynamics. The lecture by Viviane Baladi presents a much needed overview of the current literature and includes an informal discussion of the pertinent problems and results. The chapters on irregular scattering and expanding maps illustrate techniques in nonlinear dynamics using the simplest nontrivial examples. The chapters on quantum chaos and Liouville surfaces stress a phase space geometry approach to semi-classical quantum theory. These lectures can serve as a basis for graduate seminars in mathematics and physics. -- Book Description

Edition details

• Publisher: Birkhäuser
• Publication date: 1997
• Language: English
• ISBN 10: 3764357088
• ISBN 13: 9783764357085
• Binding: Paperback
• Number of pages: 108