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 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current versions, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1993 Softcover reprint of the hardcover 1st edition 1993 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Thomson Press (India) Ltd., New Delhi 41j3140jSPS-5 4 3 21 O-Printed on acid-free paper 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