D

OT46 Operator Theory: Advances and Applications Vol. 46

Editor: I. Gobberg Tel Aviv University Ramat Aviv, Israel

Editorial Office: School of Mathematical Sciences Tel Aviv University Ramat Aviv, Israel

Editorial Board:

A. Atzmon (Tel Aviv) J. A. Ball (Blacksburg) L. de Branges (West Lafayette) K. Clancey (Athens, USA) L. A. Coburn (Buffalo) R. G. Douglas (Stony Brook) H. Dym (Rehovot) A. Dynin (Columbus) P. A. Fillmore (Halifax) C. Foias (Bloomington) P. A. Fuhrmann (Beer Sheva) S. Goldberg (College Park) B. Gramsch (Mainz) J. A. Helton (La Jolla) D. Herrero (Tempe)

Honorary and Advisory Editorial Board:

P. R. Halmos (Santa Clara) T. Kato (Berkeley) P. D. Lax (New York) M. S. Livsic (Beer Sheva)

Birkhauser Verlag Basel· Boston· Berlin

M. A. Kaashoek (Amsterdam) T. Kailath (Stanford) H. G. Kaper (Argonne) S. T. Kuroda (Tokyo) P. Lancaster (Calgary) L. E. Lerer (Haifa) E. Meister (Darmstadt) B. Mityagin (Columbus) J. D. Pincus (Stony Brook) M. Rosenblum (Charlottesville) J. Rovnyak (Charlottesville) D. E. Sarason (Berkeley) H. Widom (Santa Cruz) D. Xia (Nashville)

S. G. Mikhlin (Leningrad) R. Phillips (Stanford) B. Sz.-Nagy (Szeged)

Order, Disorder and Chaos in Quantum Systems Proceedings of a conference held at Dubna, USSR on October 17-21,1989

1990

Edited by

P. Exner H. Neidhardt

Birkhauser Verlag Basel· Boston· Berlin

Editors' address:

Prof. P. Exner Prof. H. Neidhardt Laboratory of Theoretical Physics Joint Institute for Nuclear Research Head Post Office P.O. Box 79 Moscow USSR

Deutsche Bib60thek Cataloguing-in-Pub6cation Data

Order, disorder and chaos in quantum systems: proceedings of a conference held at Dubna, USSR, on October 17-21, 1989/ ed. by P. Exner ; H. Neidhardt. - Basel ; Boston ; Berlin : Birkhauser, 1990

(Operator theory ; Vol. 46) ISBN-13: 978-3-0348-7308-6 e-ISBN-13: 978-3-0348-7306-2 DOl: 10.1 007/978-3-0348-7306-2

NE: Exner, Pavel [Hrsg.]; GT

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use a fee is payable to »verwertungsgesellschaft Wort«, Munich.

© 1990 Birkhauser Verlag Basel Softcover reprint of the hardcover I st edition 1990 ISBN-13: 978-3-0348-7308-6

Dedicated to the memory of

Mark Grigorevich Krein

1907-1989

VII

PREFACE

The present volume collects the contributions to the conference "Order, disorder and chaos in quantum systems" which was held at Dubna last October. It is the third meeting in the series started three years ago in which we tried to put together mathematical

physicists from the member and non-member countries of JINR with their colleagues from soviet universities and institutes using this international centre as a convenient basis. As in the previous cases, new faces, subj ects and ideas appeared but the spirit remained the same, relaxed and inspirative.

Among this conference contributions, a majority should be listed in the "orderly" category. Being more specific, this means mostly various aspects of the theory of Schroedinger

operators that has been always a core of quantum mechanics. In spite of the fact that it is studied already for several decades, there are still many interesting problems to solve as some of the lectures collected below witness. At the same time, the theory extends to some new areas motivated by physical problems ; let us mention Schroedinger operators in complicated spatial domains appearing in some parts of solid-state physics or various models using the concept of contact interactions.

Our world is far from perfect and to keep a perfect

order is difficult not only in everyday life but also in most

physical systems. Theoreticians are used to take this fact into account introducing stochastic factors into their considerations.

This direction is represented by several lectures of this volume

though, frankly speaking, we hoped for more.

It has been known for long that even purely deterministic classical equations may yield highly irregular

VIII

solutions, however, only the last two decades gave boost to study of these phenomena, now commonly are essential for understanding of e. g. , turbulence. Naturally the chaotic behaviour can be found

called dynamical chaos, which such all-important effects as, question has arisen whether in quantum systems too. In

distinction to the classical case, unfortunately, there are some conceptual difficulties because even the definition of quantum chaos allows different approaches.

At the same time, this field offers many interesting problems. At the present stage, the investigation is concentrated mostly on analysis of simple quantum models where the chaotic behaviour could be manifested. Various relations to the topics mentioned above can be found, and we believe that in this way which does not rely entirely on numerical experiments one could achieve a deeper understanding of quantum chaotic phenomena.

We acknowledge with gratitude the access to facilities of Joint Institute for Nuclear Research and the support we got from its officials, particularly from Prof.A.N.Sissakian, in preparation of the conference. We want especially thank our colleagues Petr ~eba and Valentin Zagrebnov who helped us to organize the conference but could not from various reasons participate in editing this proceedings volume.

Dubna, February 1990

Pavel Exner Hagen Neidhardt

IX

TO THE MEMORY OF A GREAT MATHEMATICIAN

On the second day of the conference we learned about the death of Mark Grigorevich Krein. These sad news touched deeply each of us independently of his or her age, experience, scientific interests or nationality.

Mark Grigorevich was indisputably one of the greatest mathematicians of our century who enriched the science with many deep and beautiful concepts, methods, theorems and formulae. To name just a few of them, let us recall his contributions to functional analysis, harmonic analysis on locally compact groups or theory of self-adjoint extensions. One should mention also the eigenfunction expansions for ordinary differential operators, the indefinite-metrics spaces bearing now his name, as well as his results about stability of solutions to differential equations, inverse problems, non-selfadjoint operators and plenty of others.

In addition to his scientific merits, he was known as an outstanding teacher who managed to induce a true passion for mathematics to his students. A few older colleagues taking part in our conference were his disciples, and many younger ones were disciples of his disciples, and had also the opportunity to meet him in person. It makes their feeling of our loss even deeper.

We are convinced that only a historical perspective will allow to appreciate the great impact that the works of Professor Krein had on the development of modern mathematics in a full complexity. Reading carefully this book you can trace his ideas in a lot of the papers collected here. We deem therefore that to dedicate the present volume to Mark Grigorevich is the best way to honour his memory.

The editors

TABLE OF CONTENTS

Preface

To the memory of a great mathematician

1. Schroedinger operators : spectra, scattering and semiclassical behaviour

M.S.Birman, M.Z.Solomyak : Negative discrete spectrum of the Schroedinger operator with large coupling

VII

IX

constant : qualitative discussion 3

M.S.Birman : Discrete spectrum in the gaps of the continuous one in the large-coupling-constant limit 17

A.V.Sobolev : On the asymptotics of discrete spectrum for the schroedinger operator in electric and homogeneous magnetic fields 27

S.A.Vugalter, G.M.Zhislin : Asymptotics of the discrete spectrum of Hamiltonians of quantum systems with a homogeneous magnetic field 33

S.A.Vugalter : Asymptotics of eigenvalues for many-particle Hamiltonians at symmetry subspaces 55

V.lvrii, A.Kachalkina : Spectral asymptotics with highly accurate remainder estimates 61

P.Exner : Bound states and resonances in quantum wires 65

V.S.Buslaev : spectral properties of the operators H~ = -~xx+ p(x)~ + v(cx), p is periodic 85

S.V.Khryashchev : Discrete spectrum for a periodic Schroedinger operator perturbed by a decreasing potential 109

XI

~I

M.Novitskii : On a complete description of the principal discrete series of spectral invariants of the Hill operator 115

N.A.Chernyavskaya, L.A. Shuster : WKB-approximations from the perturbation theory viewpoint 119

H.Neidhardt : Hoving potentials and the completeness of wave operators. Part II : Propagating observables on scattering states 125

Yu.A.Kuperin, Yu.B.Melnikov : Representation of the three-body S-matrix in terms of effective amplitudes 149

E.A.Yarevsky : A three-body one-dimensional system with increasing interaction : wave function asymptotics 155

B.N.Zakhariev, V.S.Olkhovsky, V.M.Shilov : Surprises of quantum tunneling (something about the velocity of sub-barrier motion) 159

2. Point and contact interactions, self-adjoint extensions

W.Kirsch, F.Nitzschner : Lifshitz tails and non-Lifshitz tails for one-dimensional random point interactions 171

S.A.Cheremshantsev, K.A.Makarov Point interactions with an internal structure as limits of nonlocal point interactions 179

p.stovicek : Green's function for the Aharonov-Bohm effect with a non-Abelian gauge group 183

I.Yu.Popov : A model of zero-width slits and the real diffraction problem 195

I.Yu.Popov : Construction of an inelastic scatterer in nanoelectronics by the extension-theory methods 197

J.F.Brasche, W.Karwowski : On boundary theory for Schroedinger operators and stochastic processes 199

J.Dittrich, P.Exner, P.Seba : Dirac Hamiltonian with coulomb potential and contact intraction on a sphere 209

}(III

A.N.Kochubei: Self-adjoint extensions of Schroedinger operators with singular potentials 221

Yu.A.Kuperin, E.A.Yarevsky : Currents and the extension theory 229

3. Quantum chaos

P.Seba : Chaotic quantum billiards 237

F.M.lzrailev: Relevance of the localization to quasi-energy statistics in quantum chaotic systems 259

B.Milek, P.Seba : Singular continuous quasi-energy spectrum in the kicked rotator with separable perturbation : onset of quantum chaos ? 279

G.P.Berman, F.M.lzrailev : Relation between correlation functions and spectrum statistics in the region of quantum chaos 301

4. Other topics

P.B.Kurasov, B.S.Pavlov : Localization effects in non-homogeneous dielectrics 307

Yu.A.Kuperin : A self-adjoint dilation of the linear Boltzmann operator 315

G.A.Oerfel : Kato problem for functional-differential equations and difference Schroedinger operators 319

S.I.Petrukhnovsky : An existence theorem for some nonlinear nonlocal Schroedinger operators and the soliton­like dynamics for the corresponding dynamical systems 323

v.N.pivovarchik : The Sturm-Liouville problem with a potential linear in spectral parameter 327

M.A.Antonets, I.A.Shereshevsky : Stochastic model of tree growth 331

M.A.Antonets, I.A.Shereshevsky : Critical phenomenon in the stationary model of random growth 335

XIV

P.Bona : On a quantum-classical connection, hidden symmetries and a model of Josephson junction 339

V.R.struleckaya : Open multiquantum systems. Method of a generating functional 343

Yu.V.Egorov : Generalized functions and their applications 347

List of unpublished contributions 355

List of participants 357