Classical Nonintegrability, Quantum Chaos

ISBN-13: 9783764357085 / Angielski / Miękka / 1997 / 102 str.

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

Opis

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."

Spis treści

1 Introduction.- 2 Dynamical Zeta Functions.- 2.1 Introduction and Motivation.- 2.1.1 Transfer Operators.- 2.1.2 Invariant Function Spaces.- 2.1.3 Quasicompactness.- 2.1.4 Weighted Dynamical Zeta Functions.- 2.2 Commented Bibliography.- 2.2.0 Foundations.- 2.2.1 Surveys.- 2.2.2 Applications.- 2.2.3 Subshifts of Finite Type and Axiom A.- 2.2.4 The Smooth Expanding Case.- 2.2.5 The Smooth Hyperbolic Case.- 2.2.6 The One-dimensional Case.- 2.2.7 The One-dimensional Case: Kneading Operator Approach.- 3 Irregular Scattering.- 3.1 Notions of Classical Potential Scattering.- 3.2 Centrally Symmetric Potentials.- 3.3 Scattering by Convex Obstacles.- 3.4 Symbolic Dynamics.- 3.5 Irregular Scattering by Potentials.- 3.6 Time Delay and the Differential Cross Section.- 4 Quantum Chaos.- 4.1 Husimi Functions.- 4.2 Pseudodifferential Operators.- 4.3 Fourier Integral Operators.- 4.4 The Schnirelman Theorem.- 4.5 Further Directions.- 5 Ergodicity and Mixing.- 6 Expanding Maps.- 7 Liouville Surfaces.- Participants.- Additional Talks.