molecular physics, to the study of chemical reactions, with applications reach- ... Since nuclear, atomic and molecular physicists, and chemists and mathemati-.
Lecture Notes in Physics Edited by H. Araki, Kyoto,J. Ehlers, MLinchen,K. Hepp,ZL~rich R. Kippenhahn,MSnchen,D. Ruelle,Bures-sur-Yvette H.A. WeidenmLiller,Heidelberg,J. Wess, Karlsruheand J. Zittartz, K61n Managing Editor: W. Beiglb6ck
325 E. Br&ndas N. Elander (Eds.)
Resonances The Unifying Route Towards the Formulation of Dynamical Processes Foundations and Applications in Nuclear, Atomic and Molecular Physics Proceedings of a Symposium Held at Lertorpet, V&rmland, Sweden, August 19-26, 1987
# Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo Hong Kong
Editors Erkki Br&ndas Quantum Chemistry Group for Atomic, Molecular and Solid State Physics University of Uppsala, S-751 20 Uppsala, Sweden Nils Elander Manne Siegbahn Institute of Physics Frescativ&gen 24, S-104 05 Stockholm, Sweden
ISBN 3-540-50994-1 Springer-Verlag Berlin Heidelberg N e w Y o r k ISBN 0-38?-50994-1 Springer-Verlag N e w York Berlin Heidelberg
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, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1989 Printed in Germany Printing: Druckhaus Beltz, Hemsbach/Bergstr. Bookbinding: J. Sch&ffer GmbH & Co. KG., GrL~nstadt 2153/3140-543210 - Printed on acid-free paper
INTRODUCTION
Let any mountain be submerged gradually, and coral grow in the sea in which it is sinking, and there will be a ring of coral, and finally only a lagoon in the centre ... Coral islands are the last efforts of drowning continents to lift their heads above water. Regions of elevation and subsidence in the ocean may be traced by the state of coral reefs. The preceding quotation appears in the autobiography of Charles Darwin. The great scientist and universal genius, botanist, zoologist and geologist, Sir Charles Darwin gave birth to theories on most phenomena in nature through his exceptional intellectual and creative power. In a cited letter from Sir Charles Lyell to Sir John Herschel, the present, almost poetic, picture of pattern recognition, connected with Darwin's theory of coral island formation, is presented as an apparitional phenomenon. The quasi-stationary structure of a coral reef is depicted and comprehended by the supposed knowledge of the sinking mountain. In a general sense, this reasoning provides the flavour of pattern recognition as it appeared in the contentious theory of evolution associated with the origin of species.
Even if the above quoted passage refers to a macroscopic phase, and the L e r t o r p e t S y m p o s i u m o n Resonances - the unifying route towards the formulation of dynamical processes, held in the wilderness of V~rmland, Sweden, August 19-26, 1987, in contrast, focused on a microcosmic level, there are some subtle points in common, see further below. The subject of the symposium is to a large extent interdisciplinary and ranges from pure mathematics, via nuclear, atomic and molecular physics, to the study of chemical reactions, with applications reaching into the domains of biology and medicine. So, for instance, this approach involves a long list of topics including the correlation problem of the collision complex constituting the compound nucleus; shape resonances, such as those in a-decay; nuclear reactions and heavy ion resonances; Auger spectra; electron-atom scattering; molecular predissociations; wavepacket propagation; electron transfer; applications to van der Waals complexes; trapping of atoms and molecules on surfaces; far-infrared absorption in polar liquids; photon-counting in steady-state luminescence with possible implications for macromolecular phenomena. The surprising circumstance that scientists from so many different areas have so much in common rests on their veracity and ability to retrace the successive stages in the respective theoretical development to an equivalent or even identical origin. For instance, consider the picture used by a fusion oriented nuclear physicist who wants to describe how a helium nucleus is created by two colliding deuterons. This visualization will not differ much from the image painted by a theoretical chemist explaining the gaseous reactions of H2 and 02 to form H20. In both cases, the
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initial and final states of the participating particles are well defined. However, one does not possess detailed information concerning the connecting paths. The interpretation of these processes is usually in terms of shortlived unstable decaying intermediaries, called resonances. Although these ideas originated more than fifty years ago, the correct mathematical description was not forthcoming until the beginning of the seventies, and the exploitation by theoretical physicists and chemists followed shortly thereafter. The fundamental concept of an extended spectral classification, which subsequently provided the basis for the popular complex scaling method (CSM), had nevertheless been anticipated and utilized in connection with regularization routines. However, with the pioneering paper of Balslev and Combes, Commun. Math. Phys. 22, 280 (1971), the unexpected facets of a dilation analytic view gave support to a wealth of new applications. Since nuclear, atomic and molecular physicists, and chemists and mathematicians, all investigate various aspects of reaction paths and associated resonance phenomena, it was suggested that an interdisciplinary meeting of the L e r t o r p e t type would enhance the understanding and development of respective subject areas and would lead to improved ideas and fundamental knowledge concerning the elementary processes governing our natural environment. Returning to the quote at the beginning, one can draw a specific analogy between Darwin's geological theory and the present microscopic concept of a resonance. In the quantum mechanical formulation, the supposed relationship between motion and force is given by a generator. According to the postulates of quantum theory, this operator is self-adjoint, implying that its spectrum is always to be found on the real axis R. CSM, on the other hand, guarantees the existence of a generalized spectrum that is not necessarily real. Although this seems to be contradictory, it is not! Every structure or pattern that corresponds to such a generalized spectrum has a projection on R, which during a certain time interval defines the appropriate (projected) generator. In a sense one can say that the resonance is the sinking mountain that provides the knowledge of the dynamics of the quasi--stationary state, the coral island. The present book contains the invited talks, and accompanying discussions, that took place at L e r t o r p e t during a very intensive week in August, 1987. The number of participants was restricted to 35, the maximum capacity of the field station. In addition to scheduled lectures and extracurricular activities - organized sightseeing tours and study visits to local industries and municipalities - there was also ample time for spontaneous discussions, either privately or in small groups. To aid the reader of these proceedings, we have tried to structure the arrangement of contributions according to subject area. Although the actual lectures were chronologically scheduled with a certain thought of coherence, it does not follow
Introduction
v
that they occurred in exactly the same order as here. Nevertheless, we feel that these proceedings correctly reflect the sequence of events as they took place last year in the wilderness of V~rmland. The introductory article is concerned with the notion of a rigged hilbert space. This idea gives a well-known but important extension of traditional L2-theories. Even if this concept will not give a full characterization of the actual resonances, it will provide a proper imbedding of the theoretical models (see below) into the appropriate mathematical framework. The next two articles deal with a microlocal analysis of partial differential equations, the most modern and sophisticated mathematical theory of resonances, albeit of a physical nature and origin. Furthermore, various resonance formulations are appraised and compared. An interesting theorem on the equivalence between short range potentials and finite rank representations is outlined in the following contribution. Finally, the section on pure mathematics is concluded by one of the founding fathers of CSM; resonances with background potentials is the topic here. In an appendix the resonating group approach as well as the familiar back-rotation problem are given proper mathematical attention (see particularly the section on nuclear physics, where these questions are explicitly raised). The following three articles belong, like the introductory one, to the domain of mathematical physics, The almost periodic SchrSdinger equation is revisited in a very careful setting. In subsequent work, variational principles for non-self-adjoint operators are analyzed and applied to model problems. It was demonstrated for the first time that one can determine resonance states with such an accuracy that possible errors will be at most due to the numerical round-off of the computer. Expansion theorems involving bound states, resonances, etc. are at the center of interest in the next two papers beginning the section on selected applications in nuclear physics. It is surprising to note that the pioneers in this field made their achievements well before the appearance of CSM. The particular applications presented in the ensuing work seem to corroborate the equivalence theorem earlier referred to in connection with finite rank potentials. Moreover, the remarkable accuracy reached here led, as mentioned above, to a re-examination of the backrotation problem. The transition from nuclear to atomic and molecular physics is made smoothly in these proceedings. Here a different view is advocated; instead of transforming the evolution or propagator to a study of the analytic properties of the associated resolvent, one resorts directly to an explicit time dependent representation. The interplay between classical and quantum formulations is investigated as well as the possibility of treating relevant parts of the dynamics classically. The possibility of elaborating on genuine quantum effects, when necessary, is a very attractive alternative here. Further work pertains to a semi-classical formulation which, as is clearly demonstrated, allows for systematic corrections converging to the exact result.
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Returning to the view of the resonance as a Gamow wave or sinking mountain, the explicit boundary value problem has to be analyzed (see also appropriate papers in the nuclear physics section). Atomic and molecular applications are here presented in a multichannel version of CSM and/or optical potential models. The multichannel formulation is also presented in a potential model used to reproduce experimental data for scattering of He-atoms off a corrugated copper surface. Although not accounted for here, the somewhat controversial idea of a complex quasi-probability aroused comments. Since no satisfactory interpretation was offered as an alternative, we have decided to leave out this interesting discussion in these proceedings. In most applications of CSM one more or less automatically assumes complex symmetric representations. Even if this is not always the case, a fundamental theorem tells us that every matrix can be brought to complex symmetric form by a similarity transformation. A very simple construction is offered here as an additional concretizing proof. The form itself is shown to appear in certain extreme situations, i.e. organized forms called coherent-dissipative structures. Finally the complex scaling method is treated as an analytic extension of conventional theories such as those of the S-matrix and Hartree-Fock, as well as properly defined scalar products, etc. During the symposium it was generally felt that one could really benefit immensely from talking to scientists with motives and interests being fertilized in different fields. In the way science develoPs today, with increasing specialization, this is not at all a self-evident conclusion. It also needs to be said in this connection that the aim of interdisciplinary work is not to make mathematicians out of theoretically oriented physicists and chemists or to retrain a mathematician to be a physical chemist. Instead, we want to emphasize that the success of a program, like the present one, rests on the the premise that all who are sincerely participating should be deeply rooted in their own subject. This, of course, is easier said than done. However, the organizers of this workshop are convinced that with sufficiently many personal contacts, with the exchange of ideas, with a lack of fear of appearing stupid in another territory, with a willingness to appreciate foreign viewpoints, to listen to criticism, to debate' confusing terminology, and to freely share one's brainchild with colleagues on the other side of the fence, the present enterprise would lead to something more substantial than just another statement in support of the interdisciplinary belief. This symposium could not have been held without support from various loyal people and financial sources. First, we would like to thank all our students at that time: Christina Carlsund, who, besides running all kinds of errands also took a large number of photographs which will remind the participants of their good time at L e r t o r p e t in V~rmland; Erik Engdahl, who turned out to be an excellent
Introduction
vii
potato peeler, and, together with Mikael HSghede, also helped the participants with changes in their manuscripts, delivered to us on floppy disks written in TEX; the always stable and reliable Peter Krylstedt, who resolved every possible little problem that came his way, and also helped us to the bitter end with the editing of this book. Of course, this book would not have seen the light of day without the extraordinary effort of Lisa Mowat. With the help of her husband, Richard, she learnt TEX and typeset at least half of the contributions in this book. In this context we would also especially like to express our gratitude to the Manne Sigbahn Institute of Physics for its generosity in making available to us its computing facilities and for the assistance of its staff in the editorial work on this book. We could not have survived as well and happily without our two ladies, Susanne EngstrSm and Ann-Marie Karlstedt, who organized the kitchen and the wonderful lunches. Behind everything, and wanting everything, to the last detail, to run smoothly, but himself to be invisible, was D r A l l a n H e l l s t r a n d , the Managing Director of the A l b e r t a n d M a r i a B e r g s t r S r n F o u n d a t i o n , to whom we say thank you for the wonderful time you arranged for us all. To Prof. Ingmar BergstrSm, the grandson of Albert and Maria BergstrSm and his wife Britta, who gave us the idea to use L e r t o r p e t and then took time to guide us in the land of his childhood summers, we know that your efforts are well remembered and appreciated by all of us. Lektor Rolf Karlstedt was our local guru who seemed to know everyone in the area. Without his initiative and helpful contacts we would not have had the two nice ladies to help us, the industries to welcome, support and entertain us, and the Ferlin society to sing for us. Thank you again, Rolf. To Mr Finn Madison, managing director of the world's largest crisp bread factory we say thank you for the lunch and the tour of the plant. To Mr Svein Kalgraff, managing director of LesjSfors Industrifj~drar AB, we would like to express our deepest gratitude. Not only did he take us to a pleasant countryside inn at L£ngban, but he also arranged a guided tour of the old iron mill L~ngbanshyttan and the beautiful surrounding countryside. To Mr Uno Eriksson, manager of the FIMEK company, we are greatly indebted for providing the final symposium dinner. Typical Swedish Crayfish with all its accessories was served, and the non-educated part of the group was furthermore taught how to eat this crustacean. The dinner was a success and very much appreciated by the participants.
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The town of Filipstad is acknowledged with thanks for its practical support with expert advice of all kinds through its different agencies. We appreciated in particular the marvellous dinner at the Kalhyttans Herrg£rd. Kalhyttans Herrg£rd supplied us with the dinners at L E R T O R P E T . You really contributed to the success of the symposium by providing excellent propaganda for Swedish husmanskost at its best. Many thanks from all of us. The Swedish National Research Council (NFR) kindly supported the event in two ways. First, a considerable part of the funds which made the symposium possible came from an interdisciplinary program between Mathematics and Physics through the committee of Physics and Mathematics. Funds for this book and its editing were kindly supplied by the publishing committee of NFR. Finally, the Marianne and Marcus Wallenberg Foundation through a grant made it possible for us to carry out our plans at a late stage when we realised that our funds were not enough to support our wild plans. Their quick and informal support is gratefully acknowledged.
T h a n k You All !! Erkki Br~ndas
Quantum Chemistry Group for Research in Atomic Molecular and Solid State Physics Uppsala Sweden
Nils Elander
Manne Siegbahn Institute of Physics Stockholm Sweden
CONTENTS
INTRODUCTION
T O R I G G E D H I L B E R T SPACES (RttS)
Bengt Nagel 1. 2. 3. 4. 5. 6. 7.
Background and Introduction ............................................. E x a m p l e s of G e n e r a l i z e d Eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definition of G e l f a n d T r i p l e t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C o u n t a b l y - H i l b e r t Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . N u c l e a r Space (Special Case) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nuclear Spectral Theorem ................................................ References to Applications to Resonances and Virtual States . . . . . . . . . . . . . .
COMPARISON BETWEEN
1 4 5 6 7 8 9
DIFFERENT NOTIONS OF RESONANCES
Bernard Helffer 1. I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. A B r i e f R e v i e w on the A n a l y t i c D i s t o r t i o n or Dilation T e c h n i q u e s . . . . . . . . F r o m the Second Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F r o m t h e First Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Case a. T h e A n a l y t i c Dilation (or C o m p l e x Scaling) . . . . . . . . . . . . . . . . . . . . . . Case b. T h e E x t e r i o r Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Case c. T h e " R e g u l a r i z e d " E x t e r i o r Scaling (Hislop a n d Sigal) . . . . . . . . . . . Case d, A "Linearized" A n a l y t i c D i s t o r t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. C o m p a r i s o n t o t h e Definition of R e s o n a n c e s b y Helffer a n d S j S s t r a n d . . . . . 4. Final R e m a r k s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11 12 13 13 13 13 14 15 16 17
RESONANCES AND SEMICLASSICAL ANALYSIS
Johannes Sj~strand 1. 2. 3. 4.
Introduction ............................................................. A General Theory ....................................................... Results w h e n t h e Classical D y n a m i c s is Simple . . . . . . . . . . . . . . . . . . . . . . . . . . . E s t i m a t e s on t h e R e s o n a n c e s in M o r e G e n e r a l S i t u a t i o n s . . . . . . . . . . . . . . . . .
21 22 25 28
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Contents
RESONANCES WITH A BACKGROUND POTENTIAL
Erik Balslev 1. I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. E x p o n e n t i a l l y D e c a y i n g P e r t u r b a t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. B a c k g r o u n d P o t e n t i a l s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35 36 39
A p p e n d i x : T h e G a m o w F u n c t i o n A s s o c i a t e d w i t h an S - W a v e R e s o n a n c e . . . . 42
ON THE GENERAL INVERSION PROBLEM
Anders Melin 1. 2. 3. 4.
Introduction ............................................................. C o n s t r u c t i o n of I n t e r t w i n i n g O p e r a t o r s . . . . . . . . . . . : ...................... Trace Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T h e Miracle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47 48 52 54
WEYL'S THEORY STUDIES AND THE ONE-DIMENSIONAL ALMOST PERIODIC SCHRODINGER EQUATION
Ladislav Trlifaj 1. 2. 3. 4. 5. 6. 7.
Introduction ............................................................. Elementary Formulas .................................................... Periodic Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The BlochoWeyl Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inverse P r o b l e m s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Darboux Transformation ............................................ Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57 59 63 65 67 72 73
A GENERALIZATION OF ROUCHI~'S THEOREM WITH APPLICATION TO RESONANCES
Heinz Siedentop 1. 2. 3. 4.
Introduction ............................................................. T r a c e Ideals, D e t e r m i n a n t s , a n d All T h a t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A G e n e r a l i z a t i o n of R o u c h ~ ' s T h e o r e m for T r a c e Ideal O p e r a t o r s . . . . . . . . . A p p l i c a t i o n to R e s o n a n c e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77 78 80 82
Co nte nts
Xl
G E N E R A L I Z A T I O N OF MI~ILLER'S VARIATIONAL P R I N C I P L E Geert- Ulrich SSlter
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 2. An Exclusion Theorem for Operators with Hilbert-Schmidt Like Resolvents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 3. General Exclusion Theorems for Eigenvalues and Other Parts of the Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4. The Affiliated Birman-Schwinger-Rollnik Kernel . . . . . . . . . . . . . . . . . . . . . . . . . 91 5. Estimates for Trace Ideal Norms of w~ in Arbitrary Dimension . . . . . . . . . . . 92 6. The Modified Variational Principle of Miiller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 7. Rough Estimate for Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 7.1 Bounds for Resonances of Operators with Potentials with Compact Support in One Dimension . . . . . . . . . . . . 95 7.2 Bounds for Resonances of Operators with Dilation Analytic Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 7.3 Bounds for Resonances of Operators with Gaut] Potentials in One Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 8. S u m m a r y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Appendix : The Spectrum of an Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
102
R E S O N A N C E STATE EXPANSIONS IN N U C L E A R PHYSICS Tore Berggren
1. Ancestry: Gamow, Breit and Wigner, Siegert, Humblet and Rosenfeld .. 1.1 The Mittag-Leffier Side-Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Completeness: How to Prove It, How to Extend the Proof . . . . . . . . . . . . . . 2.1 Regularization Methods and Their Justification . . . . . . . . . . . . . . . . . . . . . 2.2 The Proof of Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The Extension of the Proof to Resonant States . . . . . . . . . . . . . . . . . . . . . . 3. The Off-spring: Resonance State Expansions Based on Completeness . . . 3.1 Some Properties of Bound, Resonant and Anti-Bound States . . . . . . . . 3.2 The Breit-Wigner Formula Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. The Next Generation: W h a t More Can We Do? . . . . . . . . . . . . . . . . . . . . . . . .
105 106 107 108 109 111 112 112 115 116
M I T T A G - L E F F L E R EXPANSIONS IN N U C L E A R PHYSICS Jens Bang . . . . . . . . . . . .
.....................................................
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ON THE ANALYTICAL CONTINUATION OF THE PARTIAL WAVE S-MATRIX USING COMPLEX SCALING TECHNIQUES
Magnus Rittby, Nils Elander and Erkki Briindas 1. I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. A n a l y t i c C o n t i n u a t i o n of t h e J o s t F u n c t i o n s a n d t h e S - M a t r i x . . . . . . . . . . . 3. A G e n e r a l i z a t i o n of L e v i n s o n ' s T h e o r e m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. D e r i v a t i o n of a G e n e r a l i z e d L e v i n s o n ' s T h e o r e m . . . . . . . . . . . . . . . . . . . . 4. A P a r t i a l W a v e S - M a t r i x E x p a n s i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. C o n n e c t i o n s w i t h t h e B r e i t - W i g n e r A n s a t z a n d t h e F a n g P a r a m e t e r s . . . . 6. C o n n e c t i o n s w i t h t h e I n v e r s e P r o b l e m a n d C o n c l u s i o n s . . . . . . . . . . . . . . . . .
129 131 134 135 140 144 146
A p p e n d i x : D e r i v a t i o n of E x p a n s i o n s for t h e S - M a t r i x . . . . . . . . . . . . . . . . . . . . .
148
CALCULATION OF RESONANT
WAVE FUNCTIONS IN NUCLEAR PHYSICS
Borbdla Gyarmati 1. 2. 3. 4.
Introduction ............................................................ O n e - P a r t i c l e R e s o n a n c e s in N u c l e a r P h y s i c s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Gamow Functions .................................................. T h e C o m p u t a t i o n of G a m o w F u n c t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 T h e D i r e c t N u m e r i c a l I n t e g r a t i o n : t h e P r o g r a m G A M O W . . . . . . . . . . . 4.2 A n A p p r o x i m a t i o n M e t h o d : T h e P o t e n t i a l Separable Expansion Method: The Code PSEUDO .................. 5. S u m m a r y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
THE USE OF GAMOW FUNCTIONS
153 154 155 163 164 165 174
IN NUCLEAR PROBLEMS
Tamas Vertse, Patrieia Curutehet and Roberto J. Liotta 1. 2. 3. 4.
Introduction ............................................................ Basic C o n c e p t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Normalization Procedures .............................................. A p p l i c a t i o n of t h e R e s o n a n c e S t a t e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 O n e - C h a n n e l Case: P o t e n t i a l S c a t t e r i n g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 T w o - C h a n n e l Case: I s o b a r i c A n a l o g u e R e s o n a n c e in t h e 2°spb(p, p) 2°spb R e a c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 G i a n t R e s o n a n c e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
179 181 182 185 185 186 189
xIII
Contents R E S O N A N C E S AS A N E I G E N V A L U E P R O B L E M
Lidia S. Ferreira 1. 2. 3. 4. 5.
Introduction ............................................................ G a m o w S t a t e s in the G e n e r a l C o n t e x t of R e s o n a n c e s . . . . . . . . . . . . . . . . . . . . Eigenvalue P r o b l e m for G a m o w Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G a m o w Vectors as a Basic Set of F u n c t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
201 202 205 208 212
A MODEL FOR STUDYING TIME DEPENDENT QUANTUM MECHANICAL PROCESSES AND ITS APPLICATION FOR QUASI-STATIONARY STATES
Jdnos Rgvai 1. 2. 3. 4. 5.
Introduction ............................................................ F o r m u l a t i o n of t h e M o d e l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S o l u t i o n of t h e T i m e E v o l h t i o n P r o b l e m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C a l c u l a t i o n of T r a n s i t i o n A m p l i t u d e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . N u m e r i c a l Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A p p e n d i x : F o r m u l a s for S t a t i o n a r y P r o b l e m s w i t h One- a n d T w o - T e r m Separable P o t e n t i a l s O n e - T e r m Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T w o - T e r m Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
TIME-DEPENDENT
215 216 217 220 225
230 231
DYNAMICS APPLIED TO ELECTRON TRANSFER
Erik Deumens and Yngve Ohrn 1. 2. 3. 4. 5. 6.
Introduction ............................................................ Theory ................................................................. Average Dynamics ...................................................... Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transition Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . N o n a d i a b a t i c Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
233 234 237 239 243 247
xlv SEMICLASSICAL DESCRIPTION
Contents OF RESONANCES
H. Jdrgen Korsch 1. I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Semiclassical B a c k g r o u n d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Free P r o p a g a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Reflection f r o m a Single T u r n i n g P o i n t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 R e f l e c t i o n f r o m a P o t e n t i a l B a r r i e r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 P a s s a g e T h r o u g h a C u r v e Crossing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. S h a p e - a n d O r b i t i n g - R e s o n a n c e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. R e s o n a n c e s for P u r e l y Repulsive P o t e n t i a l s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. C u r v e - C r o s s i n g : F e s h b a c h R e s o n a n c e s a n d P r e d i s s o c i a t i o n . . . . . . . . . . . . . . 6. R e s o n a n c e s in M u l t i p l e C u r v e - C r o s s i n g s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. N u m e r i c a l T e c h n i q u e s a n d G e n e r a l R e m a r k s . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements ........................................................
253 254 254 255 255 258 260 263 265 271 276 278
REGGE POLES AND ATOM-MOLECULE DIFFRACTION
Karl-Erik Thylwe 1. I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. C o m p l e x A n g u l a r M o m e n t u m T h e o r y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 P a r t i a l W a v e E x p a n s i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 S - M a t r i x P r o p e r t i e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 T r a n s f o r m a t i o n of t h e S c a t t e r i n g A m p l i t u d e . . . . . . . . . . . . . . . . . . . . . . . . . 3. D i f f r a c t i o n in He - N2 S c a t t e r i n g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 D i f f r a c t i o n M e c h a n i s m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 I s o t r o p i c P o t e n t i a l M o d e l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 A n i s o t r o p i c P o t e n t i a l M o d e l s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. S u m m a r y a n d C o n c l u s i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgment ..........................................................
281 283 284 285 290 296 297 298 299 306 308
E N E R G Y S H I F T S A N D W I D T H S IN A T O M I C AND MOLECULAR PHYSICS: MULTICHANNEL APPROACH
Roland Lefebvre 1. I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. M u l t i c h a n n e l C o m p l e x E n e r g y Q u a n t i z a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. C o m p l e x P o t e n t i a l s a n d C o m p l e x R o t a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. O p t i c a l P o t e n t i a l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
313 315 320 322
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3.2. C o m p l e x P a t h (or Generalized C o m p l e x R o t a t i o n ) . . . . . . . . . . . . . . . . . .
323
T W O S T U D Y C A S E S IN T H E C A L C U L A T I O N O F R E S O N A N C E S USING THE MULTICHANNEL SCHRODINGER EQUATION
Mario A. Natiello and Alejandro R. Engelmann Introduction .............................................................. T h e M u l t i c h a n n e l SchrSdinger E q u a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Algorithm ............................................................ Case 1: A M o l e c u l a r M o d e l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Case 2: T h e H e l i u m A t o m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concluding Remarks ...................................................... Acknowledgments .........................................................
329 330 332 334 338 343 344
MULTICHANNEL COMPLEX SCALED TITCHMARSH WEYL THEORY. A MODEL FOR DIATOMIC FRAGMENTATION
Erkki Br~ndas, Magnus Rittby and Nils Elander 1. I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. T h e o r y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . : ........................... 2.1 T i t c h m a r s h - W e y l ' s T h e o r y for a Set of Singular S e c o n d - O r d e r Differential E q u a t i o n s - A W a y of A n a l y z i n g t h e C o u p l e d E q u a t i o n s for a C o n t i n u u m P r o b l e m . . . . . . . . . . . . . . . . . . . 2.2 C o n n e c t i o n s w i t h S c a t t e r i n g F o r m u l a t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 T h e C o m p l e x D i l a t e d Analogue of t h e T i t c h r n a r s h - W e y l ' s T h e o r y for a Set of C o u p l e d Singular S e c o n d - O r d e r Differential E q u a t i o n s . . . 3. A p p l i c a t i o n t o a M o d e l of a D i a t o m i c C u r v e Crossing of t h e X 2 E + - B ~ z ~ + States in t h e M g H Radical . . . . . . . . . . . . . . . . . . . . . . 4. Possible A p p l i c a t i o n s t o P h o t o f r a g m e n t S p e c t r o s c o p y . . . . . . . . . . . . . . . . . . . 5. S u m m a r y a n d Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements ........................................................ A p p e n d i x : A d i a b a t i c and D i a b a t i c States of a D i a t o m i c Molecule . . . . . . . . . .
O N T H E WAY T O A M U L T I C O N F I G U R A T I O N O F R E S O N A N C E P H E N O M E N A IN A T O M S
345 347
347 357 359 362 371 374 376 376
TREATMENT
Nils Elander, Christina Carlsund, Peter Krylstedt and Peter Winkler 1. I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
384
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2. 3. 4.
5.
6. 7.
1.1. E x p e r i m e n t a l M o t i v a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. T h e o r e t i c a l A p p r o a c h e s to E l e c t r o n - A t o m S c a t t e r i n g P h e n o m e n a . . . Exterior Complex Dilation .............................................. Selfconsistent C o m p l e x D i l a t e d A p p r o a c h - P r e v i o u s W o r k . . . . . . . . . . . . . . F o r m u l a t i o n of t h e T h e o r y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. F o r m u l a t i o n of t h e E q u a t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. A Simple E x a m p l e : C o u p l i n g of T w o C o n f i g u r a t i o n s - H y d r o g e n i c T a r g e t . . . . . . . . . . . . . . . 4.3. T r a n s f o r m a t i o n t o a Set of C o u p l e d Differential E q u a t i o n s One Step Generalized Closed-Coupling Equations .................. 4.4. I t e r a t i v e G e n e r a l i z e d C l o s e d - C o u p l i n g E q u a t i o n s .................. 4.5. A W a y of Solving t h e N o n - l o c a l P r o b l e m . . . . . . . . . . . . . . . . . . . . . . . . . . . O n t h e E x t e r i o r C o m p l e x D i l a t i o n of t h e Close C o u p l i n g E q u a t i o n s . . . . . 5.1. A L o c a l E n e r g y D e p e n d e n t S t u d y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. A M o d e l P r o b l e m S t u d y U s i n g H y d r o g e n i c Basis . . . . . . . . . . . . . . . . . . . P a r t i a l - W a v e S - M a t r i x P o l e s a n d R e a l i t y - A M i t t a g - L e f f i e r E x p a n s i o n .. Future Perspectives ..................................................... 7.1. O n t h e P o s s i b i l i t y of P o l e E x p a n s i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. O n t h e A p p l i c a t i o n s to E l e c t r o n A t o m Collisions . . . . . . . . . . . . . . . . . . .
RESONANT
STATES IN THE MICROSCOPIC
384 387 390 392 394 394 395 397 398 401 403 405 414 415 422 423 424
CLUSTER MODEL
Andrds T. Kruppa 1. I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. T h e M i c r o s c o p i c C l u s t e r M o d e l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 T h e R e s o n a t i n g - G r o u p M o d e l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 T h e G e n e r a t o r - C o o r d i n a t e M e t h o d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. I m p o s i t i o n of G a m o w A s y m p t o t i c s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 R e a l i z a t i o n in t h e G e n e r a t o r - C o o r d i n a t e M e t h o d . . . . . . . . . . . . . . . . . . . . 3.1 A l g e b r a i c V e r s i o n of t h e R e s o n a t i n g - G r o u p M o d e l . . . . . . . . . . . . . . . . . . . 4. C o m p l e x Scaling in t h e C l u s t e r M o d e l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 T h e C o m p l e x Scaling . . . . . . . . . . . . : .................................. 4.2 T h e C l u s t e r M o d e l a n d t h e C o m p l e x Scaling . . . . . . . . . . . . . . . . . . . . . . . . 5. R e s o n a n c e s in SBe a n d S u m m a r y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A NOTE ON THE CLUSTER MODEL AND COMPLEX
433 434 434 437 440 440 442 445 445 446 448
SCALING
Erik Balslev . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
455
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C O M P L E X SCALING A P P L I E D TO T R A P P I N G OF ATOMS AND M O L E C U L E S ON SOLID SURFACES
Nimrod Moiseyev 1. 2. 3. 4. 5.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Complex Coordinate Method - a Brief Review . . . . . . . . . . . . . . . . . . . . . . Rotationally Mediated Selective Adsorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Trapping of Atoms at a Corrugated Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Complex Quasi-Probability for Atoms Trapped on Stepped Surfaces . . . . .
459 460 462 465 469
ON A T H E O R E M F O R C O M P L E X S Y M M E T R I C M A T R I C E S AND ITS R E L E V A N C E IN T H E S T U D Y OF DECAY P H E N O M E N A
Charles E. Reid and Erkki Br~ndas I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. P r o o f of the Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
476 477 480
C R E A T I O N OF LONG R A N G E O R D E R IN A M O R P H O U S C O N D E N S E D SYSTEMS
Erkki Brgndas and C. Aris Chatzidimitriou-Dreismann 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Coherent States, O D L R O and Large Eigenvalues of r(2)(g N2) . . . . . . . . . . . 3. The Complex Scaling M e t h o d (CSM), Similarity (Non-Unitary) Transformations and Complex Symmetric Forms . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The Complex Scaling Method, CSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Similarity Transformations and Complex Symmetric Forms . . . . . . . . . . 4. Subdynamics in the Light of CSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Emergence of Coherent-Dissipative Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 On Microscopic Irreversibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Spontaneous Creation of Coherent-Dissipative Structures in Amorphous Condensed Systems: Formal Derivations . . . . . . . . . . . . . . . . 5.3.A Jordan Blocks and the Reid-Br~ndas Corollary . . . . . . . . . . . . . . . . 5.3.B On the Second Order Density Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.C Density Matrix of the Canonical Ensemble and Complex Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
486 487 494 494 497 501 504 504 505 507 507 508 509
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5.3.D Coherent-Dissipative Structures on the Microscopic Level of Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Spontaneous Creation of Coherent-Dissipative S t r u c t u r e s F u r t h e r Derivations and Physical Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.A Spectroscopic Transitions and the Basis Set $ . . . . . . . . . . . . . . . . . 5.4.B On Fermionic Degrees of Freedom, and t he Units li, i + s} . . . . . 5.4.C On th e Spectral Resolution and th e Degeneracy Condition Ek = E . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.D On Geminals of E x t r e m e T y p e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.E On the Finite R ank of r (2) and th e Associated Dissipative P h e n o m e n a . . . . . . . . . . . . . . . . . . . . . . . . 5.4.F Minimal Size of the Coherent-Dissipative Structures Derivation and Physical I n t e r p r e t a t i o n . . . . . . . . . . . . . . . . . . . . . . . . 5.4.G F u r t h e r Remarks on the Physical I n t e r p r e t a t i o n of the Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Applications to Condensed M a t t e r Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . 6.1 On the Far-Infrared Absorption in Liquids . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.A Anomalous T e m p e r a t u r e Dependence of F I R Absorption Bands in Liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.B Coherent-Dissipative Structures and F I R Absorption . . . . . . . . . . 6.1.C S u p p l e m e n t a r y Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 On the S t e a d y - S t a t e Photoemission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.A D- F lu c t ua t i ons in the Luminiscence . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.B Coherent-Dissipative Structures and D-Fluctuations in the Luminiscence of Condensed Systems . . . . . . . . . . . . . . . . . . . . . . . . . .
512 514 514 515 516 516 517 517 520 522 523 523 525 529 531 531 533
T H E L E R T O R P E T S Y M P O S I U M V I E W ON A G E N E R A L I Z E D I N N E R PRODUC~I
Edited by Nils Elander and Erkki Br~ndas 1. I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Tore Berggren : T h e Regularization M e t h o d I . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Borb~la G y a r m a t i and Tamgs Vertse : T h e Regularization M e t h o d II . . . 4. Erkki Br~ndas and Nils E1ander : T h e Complex Scaling M e t h o d and the T u r n Over Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Nimrod Moiseyev : T h e C - P r o d u c t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. S u m m a r y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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SUBJECT INDEX .........................................................
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