KAM Theory and Semiclassical Approximations to Eigenfunctions

389 Pages·1993·31.72 MB·English

by  Professor Vladimir F. Lazutkin (auth.)

Ergebnisse der Mathematik
und ihrer Grenzgebiete
3. Folge . Band 24
A Series of Modern Surveys in Mathematics
Editorial Board
E. Bombieri, Princeton S. Feferman, Stanford
M. Gromov, Bures-sur-Yvette
H.W. Lenstra, Jr., Berkeley P.-L. Lions, Paris
R. Remmert (Managing Editor), Munster
W. Schmid, Cambridge, Mass. J-P. Serre, Paris
1. Tits, Paris
Vladimir F. Lazutkin
KAM Theory and
Semiclassical
Approximations
to Eigenfunctions
With an Addendum by A.1. Shnirelman
With 66 Figures
Springer-Verlag
Berlin Heidelberg New York
London Paris Tokyo
Hong Kong Barcelona
Budapest
Professor Vladimir F. Lazutkin
Department of Mathematical Physics
Institute of Physics
St. Petersburg State University
Ulyanov str. 1, kor. 1, 198904 Petrodvorets
St. Petersburg, Russia
Mathematics Subject Classification (1991):
35J10, 35P20, 35Q40, 58FXX, 58G25, 70HXX, 81Q05, 81Q20
ISBN-13: 978-3-642-76249-9 e-ISBN-13: 978-3-642-76247-5
DOl: 10.1007/978-3-642-76247-5
Library of Congress Cataloging-in-Publication Data. Lazutkin, V.F. (Vladimir Fedorovich)
KAM theory and semiclassical approximations to eigenfunctions j Vladimir F. Lazutkin; with
addendum by A.I. Shnirelman. p. cm.-(Ergebnisse der Mathematik und ihrer Grenzgebiete :
3. Folge, Bd. 24) Includes bibliographical references and index.
ISBN-13 :978-3-642-76249-9 I. Hamiltonian systems. 2.
Eigenfunctions. 3. Asymptotic distribution. 4. Schr6dinger operator. I. Title. II. Series.
QA614.83.L39 1993, 514'.74-dc20 93-17491
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Preface
More than ten years ago I wrote a short book "Convex billiards and Laplacian
eigenfunctions" which was issued in 1981 by Leningrad State University Press. It
contained a detailed account of results on caustics in convex domains in the plane
and their applications to the high-frequency asymptotics of eigenfunctions of the
Laplacian in those domains, both topics being of interest for physicists and
mathematicians. Issued in a very limited edition in Russian, that book was almost
unknown to the Western reader. Professor L.D. Faddeev proposed to publish the
English-translation in Springer-Verlag. In the process of preparing a new version,
I started to generalize the results to higher dimensions and applications to the
Schrodinger equation. The book was growing and the affair resulted in this volume.
I tried to give a comprehensive exposition of the topics involved, including all the
details. The main theme is KAM tori (in the COO setting) and eigenfunctions of the
Schrodinger operator, which "tend" to KAM tori of the corresponding classical
Hamiltonian system if II, the Planck constant, tends to zero. So I restrict myself
somewhat to what can be called the "quasiintegrable" case: the classical system
in question possesses KAM tori where it is quasiintegrable, the trajectories being
quasiperiodic. The behaviour oftrajectories in the complement seems to be chaotic.
Professor A.1. Shnirelman kindly agreed to write an Addendum to this book
devoted to eigenfunctions associated with chaotic regions, which, in my opinion,
makes for a more complete exposition of the subject.
I am grateful to Springer-Verlag for their great patience. I thank many people
(it is impossible to mention all of them here) who helped me directly or indirectly
during the preparation of the manuscript. Without their help the book could not
be finished.
Barcelona, 11 May, 1993 V.F. Lazutkin
Table of Contents
Introduction ......... . . . . . . . .
List of General Mathematical Notations 8
Part I. KAM Theory
Chapter I. Symplectic Dynamical Systems 15
§l. Symplectic Vector Spaces ........ . 15
§2. Symplectic Manifolds . . . . . . 33
§3. Symplectic Dynamical Systems 57
§4. Symplectic Gluing . . . . . . . 65
§5. Cross-sections ......... . 75
§6. Generalized Geodesic Flows . . 83
§7. Completely Integrable Hamiltonian Systems 91
§8. Systems in an Annulus 111
Notes to Chapter I ..... . 119
Chapter II. KAM Theorems 121
§9. The KAM Torus .......... 121
§10. KAM Set . . . . . . . . . . . . . . . 129
§11. The KAM Theorem in an Annulus 131
§12. Near a Torus . . . . . . . . . . . . . 133
§13. Near a Periodic Motion ...... 135
§14. Near the Boundary of Planar Convex Billiards 142
§15. The Robustness of a KAM Set . 150
Notes to Chapter II . . . . . . . . . . . . . . . . . . . . 159
Chapter III. Beyond the Tori . . . . . . . . . . . . . . . . . . . . . . . 160
§16. General Picture of Stochasticity Near KAM Tori. The Case of
More than Two Degrees of Freedom .............. 160
§17. Picture of Stochasticity Near KAM Tori in the Case of Two
Degrees of Freedom 169
Notes to Chapter III ............................ 186
VIII Table of Contents
Chapter IV. Proof of the Main Theorem ..... 188
§18. Two Reductions ........... . 188
§19. Machinery .............. . 191
§20. Description of the Iterative Process 198
§21. Reproduction of (20.1 i) and (20.2i). Convergence of Fi 200
§22. Estimates of '¥i+ 1 . .. . ..... 202
§23. Reproduction of (20.3i) . . . . . . . . . . . . . . . . . . . 204
§24. Reproduction of (20.4i) . . . . . . . . . . . . . . . . . . . 205
§25. Convergence of the Process and the Estimate of II <D - id II 206
§26. Derivatives of G at points of IRn x Iff ............ . 208
§27. The End of the Proof of Theorem 18.10 ........... . 209
§28. Deduction of the Theorem for Discrete Time from That of
Continuous Time 210
Notes to Chapter IV ......................... . 221
Part II. Eigenfunctions Asymptotics
Chapter V. Laplace-Beltrami-Schrodinger Operator and Quasimodes 225
§29. Basic Facts about Self-Adjoint Operators and Spectra 225
§30. Laplace-Beltrami-Schrodinger Operator ....... 226
§31. Particular Cases . . . . . . . . . . . . . . . . . . . . . . . 232
§32. Quasimodes ........ 235
§33. Degenerated Quasimodes 236
Notes to Chapter V . . . . . . . 240
Chapter VI. Maslov's Canonical Operator 242
§34. Assumptions .......... . 242
§35. The Local Canonical Operator 243
§36. The Commutation Rule .... 256
§37. Theory of Maslov's Indices .. 267
§38. A Global Formula for Maslov's Operator 284
Notes to Chapter VI ............... . 293
Chapter VII. Quasimodes Attached to a KAM Set ......... . 294
§39. The Canonical Maslov's Operator Associated with a KAM Set 294
§40. Quantum Conditions and the Set A 299
§41. Construction of Quasimodes 301
§42. Orthogonality . 305
Notes to Chapter VII ....... . 311
Table of Contents IX
Addendum (by A.I. Shnirelman). On the Asymptotic Properties of
Eigenfunctions in the Regions of Chaotic Motion . . . . . . . . . . . . . .. 313
Appendix I. Manifolds . . . . . . . . . . . . . 338
Appendix II. Derivatives of Superposition 358
Appendix III. The Stationary Phase Method 366
References ...................................... 375
Subject Index .................................... 381
Introduction
We shall study the asymptotic behaviour of eigenvalues and eigenfunctions of a
certain class of linear differential self-adjoint operators. A typical example is the
Schrodi~ger equation of quantum mechanics:
n
2
(0.1 ) - 2At/J + V(x)t/J = Et/J.
Here A = "L7=102/oxf is the Laplace operator in lR.",n is a small number, the
n
Planck constant (in CGSE units = 1.0544 x 10-27 ergs), the potential V is a
real-valued function which reflects the physical properties of the system, and E is
the spectral parameter (energy). One is looking for a function t/J: lRn --+ <C, a wave
function, which satisfies the equation (0.1) and behaves regularly at infinity. The
values of E such that the equation (0.1) has a nonzero square-integrable solution,
the eigenvalues, form the discrete spectrum {E of the Schrodinger operator.
k}
Corresponding nonzero and decreasing at infinity solutions of(O.l) and called eigen
functions, or bound states of the quantum system. (Here "nonzero" means that
the eigenfunction is not identically zero, but it is allowed to be zero on some
subsets of lRn.) Each chemical substance has an equation of the form (0.1) which
determines the structure of its molecule, and one can observe its eigenvalues as
dark thin strips in the real physical spectrum of the light passing through the
substance.
A similar problem is to find the frequencies of the sound vibrations in a
bounded domain Q c lR3. These frequencies {wk} can be determined as the square
roots of the eigenvalues of the wave operator - c2 A where A is the Laplace operator
in Q and c is the velocity of sound in Q, the corresponding wave functions t/Jk are
the nonzero solutions of the equation -c2At/Jk = WNk' satisfying certain homo
geneous conditions at the boundary of Q which reflect the physical conditions there.
A typical boundary condition is the Dirichlet one: t/J la = 0, which assumes that the
Q
density of the matter is very high outside Q.
The discrete spectrum of (0.1) is wholly determined by the potential V, as the
frequency spectrum of the domain Q is determined by its shape and boundary
conditions. How does one compute the spectrum? There exist very elaborate
variational methods which enable us to calculate the eigenvalues consecutively,
starting with the lowest one. Unfortunately these methods require a large amount
of calculation, even with the help of modern computers. They become practically
n
unrealizible if, in (0.1), is small, n ;?; 2, and the potential is sufficiently complicated.
The same is true for the problem of finding the high frequencies of a general
domain Q.
2 Introduction
For the reasons mentioned one is forced to use other approximate methods.
Ariadne's clue to find them is the idea that the quantum mechanics solutions turn
into the classical ones as h tends to zero. This means that one can construct
approximate solution to (0.1) using as a starting point, or as a skeleton, the solutions
to the corresponding equations of the classical mechanics case, Newton's ones. The
latter are ordinary differential equations, which may be written in the Hamiltonian
form for the variables x=(X1,X1,.··,xn) and P=(Pl,Pl, ... ,Pn):
oH(x,p) oH(x,p) .
(0.2) Xi= Pi = - ox. ,1;£ r ;£ n,
,
°Pi
where the Hamiltonian function is
(0.3) H(x,p)=-}lpI1 + V(x).
One expects to handle the equation (0.2) easier than the original partial differential
equation (0.1). The analogue of the classical counterpart to the problem of eigen
frequencies of a bounded domain Q is the billiards in Q, the mechanical system
which describes the motion of a material point (a billiard ball) in Q with elastic
reflections at the boundary. Because of their origin the said methods were called
semiclassical ones. They were used by physicists mostly for the one-dimensional
Schrodinger equations, and for those which could be reduced to the above
equations. Nevertheless, the class of problems solved in this way was large enough
to provide explanations to many physical phenomena, and the semiclassical
methods accompany quantum mechanics at all stages of its development starting
with the initial steps when N. Bohr used the planetary atom model to obtain the
formula for the spectrum of the hydrogen atom.
Serious obstacles arise when we attempt to extend the semiclassical methods
to a general multidimensional case. One may naturally suppose that the eigen
functions (bound states) have bounded motions in the corresponding classical system
as in their quantum counterparts. It means that there must be a definite way to
construct an approximation to an eigenfunction and eigenvalue which starts with
a suitable bounded invariant set in the phase space of the classical system, and
that, using this method, one can approximate all eigenfunctions if one considers
all such invariant subsets. This approach suggests the investigation of invariant
sets and their properties. So what do the invariant sets of a general multidimensional
mechanical system look like? The reader can get some idea of the appearance of
the partition of the phase space into invariant sets by looking at Fig. 1 which
presents some trajectories of the well known standard map, which is considered
to be a model for a general Hamiltonian system with two degrees of freedom.
Such maps arise as Poincare maps of cross sections in real mechanical systems.
All the features of genuine, general systems are presented in this phase portrait.
We see that some trajectories are organized in regular curves, these being
concentrated near periodic points. These are the so called KAM curves. Such
curves, in a Poincare section of a Hamiltonian system of differential equations,
are the traces of KAM tori which bear quasiperiodic motions. The abbreviation
KAM consists of the first letters of the names of the men who have discovered
Introduction 3
a
e
Fig. 1. This is the phase portrait of the standard Taylor-Chirikov-Greene mapping [(Example 8.8
and Eqs. (17.4)] for k = 0.9. Several trajectories are plotted on the unit square 0;;; q,;;; I, - 1/2;;; I;;; 1(2.
The following invariant sets are marked: (a) elliptic fixed point, (b) hyperbolic fixed point, (c) KAM
curves, (d) island of period 3, (e) stochastic layers, (f) elliptic periodic point of period 3, (g) hyperbolic
periodic point of period 3
and proved the existence of the invariant tori: A. N. Kolmogorov, V. I. Arnol'd,
and J. Moser. The first part of this book is devoted to the study of these tori.
The KAM curves intermit with filled zones which grow away from hyperbolic
periodic trajectories and look like stretched ovals joined in a sort of wreath. These
wreaths have nonzero width, and they bear a stochastic motion. Such wreaths are
called stochastic layers. They pervade the phase portrait densely as well as the
KAM curves and periodic trajectories. There does exist another type of invariant
set which is not seen in this picture; Cantori which are ruined KAM curves. The
reader can look at the portrait of a can torus in Fig. 44.
There is strong evidence that the characteristic feature of the whole picture is
its self-similarity: an arbitrarily chosen small piece contains another smaller piece
which resembles the initial picture provided one magnifies it with appropriate
coefficients in the appropriate directions. This is a typical phase portrait for general
Hamiltonian systems with two degrees of freedom. The two extreme degenerate
cases of this picture are firstly, an integrable one, when KAM curves form smooth