Complex analysis, microlocal calculus, and relativistic quantum theory: proceedings of the colloquium held at Les Houches, Centre de Physique, September 1979

Lecture Notes in

Physics

Edited by J. Ehlers, M0nchen, K. Hepp, Z~irich R. Kippenhahn, M0nchen, H. A. WeidenmLiller, Heidelberg and J. Zittartz, Kbln Managing Editor: W. Beiglbbck, Heidelberg

126 Complex Analysis, Microlocal Calculus and Relativistic Quantum Theory Proceedings of the Colloquium Held at Les Houches, Centre de Physique September 1979

Edited by D. lagolnitzer

Springer-Verlag Berlin Heidelberg New York 1980

Editor Daniel lagolnitzer S e r v i c e d e Physique Theorique, C E N Saclay BP 2, 91190 Gif-sur-Yvette/Frace

I S B N 3 - 5 4 0 - 0 9 9 9 6 - 4 Springer-Verlag Berlin H e i d e l b e r g N e w York ISBN 0 - 3 8 ? - 0 9 9 9 6 - 4 Springer-Verlag N e w York H e i d e l b e r g Berlin 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 the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin Heidelberg 1980 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2153/3140-543210

INTERNATIONAL COLLOQUI~I ON C~IPLEXANALYSIS, MICROLOCAL CALCULUS AND RELATIVISTIC QUANTU~4 THEORY Les Houches, Sept.3-13, 1979 Organization Committee : J.M. BONY (Maths, Orsay), J. BROS (Phys. Th~or., Saclay) D. IAGOLNITZER (Phys. Th@or., Saclay), J. LASCOUX (Phys. Th6or., Polytechnique), B. }IkLGRANGE (}~aths, Grenoble), F. PHAM (Maths, Nice), P. SCHAPIRA ~{aths,Paris Nord) Main organizer Participants ABRAHAM, D., Oxford, England ANDRONIKOV, E., Paris-Nord BALABANE, M., Paris-Nord BARLET, D., Nancy BENGEL, G. , M~nster, Germany BERG, B., Berlin, Germany BERGERE,. M. , Saclay BILLIONNET, C., Polytechnique, Paris BONY, J.M., Orsay BOUTET DE MONVEL, L., ENS-Paris BROS, J., Saclay CANDELPERGHER, A., Nice CHAZARAIN, J., N~ce CHUDNOVSKY, D., Saclay CHUDNOVSKY, G., Columbia Univ. USA CLERC, J.L., Nancy COLIN DE VERDIERE, Y., Grenoble DOUADY, A. , Orsay EPSTEIN, M., I.H.E.S. Paris ERMINE, J.L., Bordeaux FAVILLI, F., Pisa, Italy GAY, R., Bordeaux GERARD, R., Strasbourg GLASER, V., CERN, Geneva, Switzerland GRIGIS, L., Orsay GRUBB, G., Copenhagen, Denmark HATORI, T., Tokyo, Japan HONERKAMP, J. Freiburg, Germany HOUZEL, C., Paris-Nord IAGOLNITZER, D., Saclay ION, P.D.F., Heidelberg, Germany INTISSAR, R., Maroc JAEKEL, M.T. , ENS-Paris JIMBO, M., Kyoto, Japan KANTOR, J.M., Paris VII KAROWSKI, M. , Berlin, Germany KASHIWARA, M., Kyoto, Japan KATZ, A., Polytechnique, Paris KAWAI, T., Kyoto, Japan

LABESSE, J.P., Dijon LASCOUX, J.L., Polytechnique, Paris LOKAMCHAN, L., Nice LODAY, M., Strasbourg LAURENT, Y., Orsay LIEUTENANT, J.L. , Liege MAILLARD, J.M., ENS-Paris MAISONOBE, L., Nice MALGRANGE, B., Grenoble MARES RUBEN, G., Polytechnico, Mexico MEBKHOUT, Z., Orleans MIWA, T., Kyoto, Japan MONTEIRO, T. , Paris-Nord MORIMOTO, M., Tokyo, Japan MORI, Y., Okinawa, Japan MULASE, M. , Kyoto, Japan PESANTI, D., Orsay PHAM, F., Nice RADULESCU, N., Nice RAMIS, J.P. , Strasbourg ROMBALDI, J., Nice ROOS, A., France SABBAH, C., Paris SATO, M., Kyoto, Japan SCARPALEZOS, D., Paris SCHAPIRA, P., Paris-Nord SEBBAR, A., Bordeaux SJOSTRAND, J., Orsay SO~4ER, G. , Bielefeld, Germany STORA, R., CERN, Geneva, Switzerland TALEB, S., Paris TATARU-MIHAI, P. ,Max-Planck, Germany TREPEAU, J.M., Reims UNTERBERGER, A., Reims UNTERGERGER, J., Reims VAN DEN ESSEN, A. , Nijmegen, Netherlands VERDIER, J.L., ENS-Paris VOROS, A., Saclay XIA DAOXING, Shanghai, China

Organization Co~ittee of Centre de Physique des Houches : M.T. BEkL-~{ONOD (Phys. Solides, Orsay) D. THOULOUZE (CNRS, Grenoble)

FOREWORD

This volume presents the Proceedings of the Colloquium on "Complex Analysis, Microlocal Calculus and Relativistic Ouantum Theory", held at the Centre de Physique des Houches in September 1979. This Colloquium originated in the contacts developed during the seventies between two groups of (French and Japanese) mathematicians and a group of theoretical physicists who, with different motivations and approaches, were led to related, or even common, problems in the study of the singularity structure of functions (or distributions, hyperfunctions, microfunctions,...) of interest either in mathematics, in complex analysis and differential and microdifferential calculus, or in physics, in relativistic quantum theory. These contacts, including in particular previous meetings organized by F. Pham in Nice in 1973 and by H. Sato and collaborators in Kyoto in 1976, had proved to be useful. The present Colloquium, which was extended to related domains of common interest, has allowed the presentation of the important new developments and the exchanges that had appeared to be desirable or needed. Let us note, in this connection, that the separation between mathematicians and physicists is not always very neat: as will appear in these Proceedings, several of them in both groups have been led recently to contributions in either domain, the differences appearing mainly in the emphasis and in the main motivations and character of the various contributions. The topics treated have been classified in four parts. The two first parts are mainly mathematical, while the last two are more oriented towards physics. Some indications on the connections between various parts are given later. Part I presents the recent developments of microfunction theory and of the microlocal, or microdifferential, calculus (holonomic systems, second microlocalization) and related topics (essential support theory, ...). Other miscellaneous mathematical developments, on singularities of solutions of partial differential equations, pseudodifferential operators and generalizations, spectrum of operators, asymptotic expansions, monodromy, ... will be found in Part II. Part III is mainly devoted to the rigorous study of the general analytic and microanalytic structure of Green functions and of the S-matrix (i.e., of collision amplitudes), in axiomatic quantum field theory and in S-matrix theory, a domain in which appreciable progress has been made recently for multiparticle processes. Recent developments in the related study of Feynman integrals, and a few other topics, are also included. Finally, an important part of these Proceedings (Part IV) is devoted to the explicit determination of the S-mat=ix and, in some cases, of Green functions for

Vl

various models of field theory in two space-time dimensions, and to related physical and mathematical developments with emphasis.on those aspects that have, or should prove to have, a general character and give hopes of further developments. Approaches based on general principles and applying to theories "with soliton behaviour" are first presented (Sect.A). The more direct approach developed recently for quantization and solution of completely integrable systems is then introduced in Sect.B, where a general analysis of such systems, in connection with the isospectral deformation, is presented on the other hand. Finally, the recent developments on holonomic quantum fields, in connection with the isomonodromic deformation, and the corresponding solution of the Ising and other models by Sato-Miwa-Jimbo is presented in Sect. C. Complements on the Ising model are also included. While each part has its own unity, we emphasize however that the above division is to some extent arbitrary, both because the distinction between physics and mathematics is not always very neat and in view of the connections that will appear between the various parts. For instance, some texts of Part I (Iagolnitzer, Kashiwara-Kawa~, Van Den Essen) are directly relevant to the study of the S-matrix and of Feynman integrals. As another example, the study of the general structure of the S-matrix in Part III gives insight into some aspects of the two-dimensional models considered in Part IV. (Note, however, that possible extensions of specific aspects of these models to more dimensions seem to lead to structures that differ from the usual ones). Finally, a number of connections, not described here in detail, will appear between various texts of Parts I,II and IV. The Proceedings include short nontributions, which are a summary or an introduction to more complete works published elsewhere, and longer contributions, which either present original work or are review works (with some original aspects) in domains in which there was a need for an up-to-date and clear presentation of recent developments. On behalf of the Organization Committee, I would like to thank the directors of the Centre de Physique des Houches, and in particular Mrs. M.T. Beal-Monod, for their efficient help. I also wish to thank all participants and lecturers who contributed to the success of this Colloquium, and more particularly here all lecturers who, by their efforts in the preparation of their manuscripts, will contribute to the usefulness of these Proceedings. I am finally pleased to thank Mrs. E. Cotteverte for her efficient help in the final preparation of the manuscript.

D. IAGOLNITZER

CONTENTS

The l i s t of i n v i t e d lectures given during the Colloquium whose manuscripts have not been received is given at the end.

PART I

MICROFUNCTIONS,MICROLOCALCALCULUSAND RELATED TOPICS

D. I A G O L N I T Z E R - Essential Support Theory and u=O Theorems ......................... I M. KASHIWARA, T. KAWAI - The Theory of H o l o n o m i c Systems w i t h Regular Singularities and Its Relevance to Physical Problems ............. ............. 5 S e c o n d - M i c r o l o c a l i s a t i o n and A s y m p t o t i c Expansions ........................... 21 Y. L A U R E N T - D e u x i ~ m e M i c r o l o c a l i s a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Z.

MEBKHOUT

Sur le P r 0 b l ~ m e de H i l b e r t - R i e m a n n .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

-

P. S C H A P I R A - Conditions de P o s i t i v i t ~ dans une V a r i ~ t ~ Symplectique Complexe. A p p l i c a t i o n s ~ l'Etude des M i c r o f o n c t i o n s ......................... III A.

VAN DEN E S S E N

-

F u c h s i a n Systems of L i n e a r D i f f e r e n t i a l Equations

A s s o c i a t e d to N i l s s o n Class Functions and an A p p l i c a t i o n to F e y n m a n Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

PART I I J.M.

-

BONY-

MISCELLANEOUSMATHEMATICALDEVELOPMENTS Singularit~s des Solutions des Equations aux D ~ r i v ~ e s

Partielles non Lin~aires . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 J . C H A Z A R A I N - C o m p o r t e m e n t Semi Classique du Spectre d'un H a m i l t o n i e n Quantique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 G.V.

CHUDNOVSKY

-

Rational and Pad~ A p p r o x i m a t i o n s

to Solutions of Linear

D i f f e r e n t i a l Equations and the M o n o d r o m y Theory ............................. 136 B.

MALGRANGE

J.P,

RAMIS

-

-

M ~ t h o d e de la Phase Stationnaire et S o m m a t i o n de Borel ............ 170

Les S~ries k - Son~nables et Leurs A p p l i c a t i o n s ...................... 178

J. SJOSTRAND - R e f l e c t i o n of A n a l y t i c Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 A.

U N T E R B E R G E R - Les Op~rateurs M ~ t a d i f f ~ r e n t i e l s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

VIII

PART I I I

GENERALSTRUCTUREOF GREEN FUNCTIONS AND COLLISION AMPLITUDES

M.C.

-

J.

BERGERE

BROS

Asymptotic Behaviour of Feynman Integrals ......................... 242

Integral Relations in Complex Space and the Global Analytic and

-

Monodromic Structure of Green's Functions in Quantum Field Theory: Some General Ideas and Recent Results ....................................... 254 D. IAGOLNITZER - Microcausality, Macrocausality and the Physical-Region (Micro)Analytic S-Matrix .................................................... 263 G. SOMMER - Analytic 2-Particle Structure and Crossing Constraints ............... 296 D. XIA - On the Scattering Matrix with Indefinite Metric ......................... 306 On the Representation of the Local Current Algebra ...................... 311

PART IV (A)

TWO-DIMENSIONALMODELS AND RELATED DEVELOPMENTS

-

B, BERG - S-Matrix Theory of the Massive Thirring Model ..................... 3]6 M. KAROWSKI - Field Theories in |+l-Dimensions with Soliton Behaviour: Form Factors and Green's Functions ..................................... 344

(B)

D,V. CHUDNOVSKY - One and Multidimensional Completely Integrable Systems Arising from the Isospectral Deformation ....................... 352 J. HONERKAMP - Quantization of Exactly Integrable Field Theoretic Models 7 The Operator Transform Method ................................. 417

(c)

M.

SATO,

T.

MIWA,

M.

JIMBO

-

Aspects of Holonomic Quantum Fields -

Isomonodromic Deformation and Ising Model .............................. 429 D.B.

ABRAHAM

-

Recent Results for the Planar Ising Model .................... 492

Other Lectures Delivered During the Colloquium L. BOUTET DE MONVEL - Quantized Contact Manifolds Y. COLIN DE VERDIERE - Spectre Joint d'OPD qui Commutent H. EPSTEIN, V, GLASER - Unitarity and Analyticity in Axiomatic Quantum Field Theory R. GERARD - Th&orie des R~sidus Associ~s ~ une Connexion avec Singularit~s R~guli@res G. GRUBB - On the Spectrum of Pseudo-Differential Elliptic Boundary Value Problems C. HOUZEL - Th~or~me d'Image Directe pour les Syst~mes Micro-Diff~rentiels F.

PHAM

J.L.

-

Singularities of Phase Integrals and Picard-Lefsehetz Formula

VERDIER

-

Equations Diff~rentielles G~om~triques

A. VOROS - The Zeta Function of the Quartic Oscillator

PART I

MICROFUNCTIONS, MICROLOCALCALCULUS AND RELATED TOPICS

ESSENTIAL SUPPORT THEORY AND u=0 THEOF~MS D. IAGOLNITZER

DPh-T, CEN Saclay, BP n°2, 91190 Gif-s~-Yv#~te, France

The standard results of essential support theory Ill , or of hyperfunction theory [2], give no information on the essential support (= singular spectrum)of a product of distributions f',f" at u=0 points X (i.e. points where the essential supports of f' and f" contain opposite directions), even when the product itself is well defined in the neighborhood of X. The same remark applies also to products of bounded operators, such as the products of collision operators encountered in some applications in S-matrix theory (see the lecture of the author in Part III). The u=0 problem encountered there is crucial and has been at the origin of several works The approach to the u=0 problem developed in the framework of the theory of holonomic microfunctions by Kashiwara-Kawai is presented in [3] and references therein. We present here u=O results [4] obtained in the framework of essential support theory, on products of square integrable functions that satisfy a general regularity property R at X with respect to the relevant directions of their essential support. Analogous results [4a] also hold for products of bounded operators. Details in both cases will be given in [4c]. The basic facts on the essential support are recalled in Sect.l. Property R is probably linked with the second microlocalization introduced recently in microfunction theory. It is introduced in Sect.2 at the end of which a simple example is given in terms of analyticity properties. The announced u=0 theorem

is then pre-

sented in Sect.3. As in [3], the result is similar to the standard u¢0 one, except that limiting procedures that may efilarge the essential support have to be introduced. The latter are however different, the respective conditions of application of the results being themselves different in general.

|. ESSENTIAL SUPPORT Being given a tempered distribution f defined on the s p a c e ~ n of n real variables x=xi,...,Xn, the essential support of f is defined, at each point X, as a cone with apex at the origin in the s p a c e ~ n of the dual variables U=Ul,...,u n composed of the "singular directions" along which the generalized, or localized,

Fourier transform of f at X does not fall off exponentially in a well specified sense. Namaly, let F

be defined, for every y > 0, by

Y F (u;X)

=

f f(x)

e-iu'x-Y]ul(x-x) 2 dx

(I)

or, with an auxiliary real variable v, let : F(u;v,X)

=

(2)

f f(x) e -iu'x-v(x-x)2 dx

Then a direction Uo is by definition outside ESx(f) , or (X,@ o) @ ES(f) if there exist a neighbouring cone V(@ o) of Qo with apex at the origin, ~ > 0,Yo > 0, as also a polynomial P and q ~ 0 such that : IF(u;v,X) l

<

[P(lul) v -q] e -~v

(3)

in the region u E V(Go), 0 < v < YolUl. The important factor in these bounds is the factor e -~v ~ e -~Ylul , which expresses exponential fall off in the direction G

for all y > 0 sufficiently small o with a rate of fall off at least proportional to y . (Bounds of the form (3) without this factor are always satisfied). Whereas the exponential fall-off at y=0, i.e. of the usual Fourier transform f of f, corresponds to analyticity properties independent of the real point X (see e.g. [I]), the above notion of essential support characterizes by duality the real points where f is analytic or is the boundary value of an analytic function, and more generally possible decompositions of f into sums of boundary values of analytic functions from specified directions (which may depend on X) : see details in [I], where the notion of essential support and the results above are extended to general distributions defined in ~ n or on a real analytic manifold. The characterization of analytieity properties obtained [2] in hyperfunction theory in terms of the notion of

singal~spec~um

is similar to above, except that

the boundary values involved in the decompositions of f may ~ priori be hyperfunctions, even when f itself is a distribution. It is, however, proved in [5] that the two notions do coincide for distributions

(and coincide with Hormander's "analytic

wave front set").

2. REGULARITY PROPERTY R (for square integrable functions) The regularity property R is a condition on the way rates of exponential fall off of the generalized Fourier transform F

tend to zero in certain situations when Y directions of the essential support are approached. It asserts more precisely that certain uniform bounds are then satisfied (see below). Being given X and Q

o

~ ES(f), it follows from results of [I] that ~ in the

bounds (3) can always be chosen arbitrarily close to Max,'>0

{~';Qo ~ ESx(f)

' Vx s.t. (x-X) 2 < ~'}

(with appropriate choices of V(Q o) and Yo ~ 0).

,

Let us then consider a direction Uo that now belongs to the boundary of ESx(f) , and moreover to the boundary of o f X. The a b o v e r e s u l t

entails

U

ES (f), where N is some real neighborhood

xEN x that there exists

~ > 0 ( d e p e n d i n g o n l y on N) s u c h

that bounds of the form (3) be satisfied at X with this common u ~ f o ~

~ for all

directions Q outside

L) ESx(f). The choice of V(Q), Yo > O, ... depends on the xEN other hand on Q, and as a matter of fact yo(Q) necessarily tends to zero when the direction Q

o

is approached, since Q

E

o

ESx(f ). The main content of property R at

(X,Qo) , when it holds, is the condition that yo(Q) should not tend to zero faster than linearly with respect to the angle of Q with the boundary of

U ESx(f). xEN ~n order to introduce the precise statement of property R for sauare inte~rable

functions, we mention the following results that always hold in this case. First [4a], the bracket [P([uI)v -q] can always be replaced in the bounds (3) by a square integrable function d

of u whose norm is independent of v. This function may g priori V

depend on the direction considered outside ESx(f). on the other hand, there always exists, as easily seen, a uniform square integrable function d of u such that the bounds IF(u;v,X) l < d(u), without the exponential fall-off factor e -~v, be satisfied everywhere, in the whole region v ~ 0.

Property R (for square integrable functions) "Being given X and a direction Uo of ~(ESx(f)), property R is by definition satisfied by f at (X,Qo) if, being given any real neighborhood N of X, there exist a neighboring cone V(Q o) of Uo with apex at the origin, ~ > 0, X > 0 and a square integrable function d

V

of u, whose norm is independent of v, such that : [F(u;v,X) I

in the region u E V(@o), 0 ~ v < X

<

dv(U) e - ~ v

dist. {u,

U xEN

(4) ESx(f)}"

Example. Let f be, in the neighborhood of X, the boundary value of an analytic function f from the directions of an open (conveX) cone r in Imz-space (where z=x+iy is the complexified variable of x), in which case ESx(f) c C, where C is the closed dual cone of F. The above analyticity property means that there exist a neighborhood N of X and an open set B with profile F at the origin in y-space, such that f is analytic in {z=x+iy : xEN, yEB}. B has F as its profile at the origin if, being given any open cone F' with apex at the origin whose closure is contained (apart from the origin) in F, there exists O > 0 such that F'O{ly I < 0} ~ B ; p may, h o w e v e ~ shrink to zero in general when r' expands to F. Let ~ be a direction of the boundary ~F of F , if it exists, for which this is not the case, more precisely for which there exist 0(~) > 0 and a nei~hbonrinE cone V(~) of ~ with apex at the origin such that the set {y;Iy[
belongs

to the closure of B. If, moreover, this set is approached,

then

f is bounded in its analyticity domain when

property R is satisfied at (X,fio) for any Uo of $C

such that Qo.~=O. The converse is also essentially true.

3. U=0 THEOREM Theorem - "Let f', f" be square integrable functions such that property R is satisfied by f' and f" at (X,Q o) and (X,-Q o) respectively,

for any direction Uo

such that rio 6 ESx(f'), -rio C ESx(f"). Then : ESx(f'f" ) c {u;.~ u'n' u"n' X'n' X"n' u'n 6 ESx,(fl) , U"n 6 ESx,,(f") n n X' n

÷

X, X" n

÷

X, u' + u" ÷ u when n ÷ = o } n n

,

.

(5)"

The (easy) proof is based on the formula : F(u;v,X)

=

f

V p, v F'(u'; ~,X) (u-u'; ~,X) du'

(6)

where f=f'f". The result is a generalization of the particular case obtained when f',f" satisfy the properties of the example of Sect.2 : if F' N F" is empty (in which case X is a u=O point), but if ~F' N ~F" contains a direction ~ (or a set of such directions)

such that the properties described at the end are satisfied by f' and

f", then ESx(f) is contained in its dual half-space

(or in their intersection).

REFERENCES [I]

D. lagolnitzer,

in

S£ructuralAnalysis of Collision Amplitudes,

and D. Iagolnitzer, North-Holland, [2]

M. Sato, T. Kawai, M. Kashiwara,

Equations,

Amsterdam in

ed. by R. Balian

(1976), p.295, and references.

)~icrofunctio~ and Pseudo-diff~ential

Lecture Notes in Mathematics,

Springer-Verlag,

Heidelberg

(1973),

p.265. [3]

M. Kashiwara, T. Kawai, first text in this volume.

[4]

D. Iagolnitzer,

a) Comm. Math. Phys. 63, 49 (1978), b) Expos~ I, S~minaire

Goulaouic-Schwartz [5]

1978-79, c) in preparation.

J.M. Bony, in S~minaire Goulaouic-Schwartz

1976-77.

THE THEORY OF HOLONOMIC

SYSTEMS WITH REGULAR SINGULARITIES

AND ITS RELEVANCE

TO PHYSICAL

PROBLEMS

Masaki KASHIWARA D~partment

de Math~matiques,

Universit~

91405, Orsay,

de Paris-Sud

France

and Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606, Japan AND Takahiro Research

Institute

Kyoto University,

[Ii],

Sciences

Kyoto 606, Japan

The purpose of this report articles

KAWAI

for Mathematical

is to give a partial

[12] and indicate

their relevance

resume of our

to physical

So we have tried to make this report easy to understand, physicists,

by sometimes

sacrificing

We have also concentrated relate to physical

the generality

our attention

problems.

problems.

especially

for

of the statement.

on topics which,

we hope,

In this report we use the same notations

as in [12], In the theory of linear ordinary differential applications,

equations with regular

equations

singularities

and its

play a central

role.

Hence one might naturally wish to extend the theory to several variable~ case.

Then one natural

holonomic

systems,

choice would be to concentrate

partly because

theorem with ordinary

differential

because,

systems,

important

not general

in applications

Now, let

~

(e.g.

variety.

possibly with singularities.) locus

A

of

reg which is defined the form

T~X

N

equations

([4],

but holonomic [9],

be a holonomic

be its characteristic

they share the basic

[13].

~x-MOdule

(Hence

A

on p

A, we can find a suitable

contact

in a neighborhood

p

~(U)

U

of

for a non-singular

are particularly

[20] .... ). ~CT*X-T~X.

is a Lagrangian

For each point

on

finiteness

[8]) and partly

systems [15],

our attention

Let

A

variety

in the non-singular transformation

and brings

hypersurface

AregNU Y

in

to X.

We

choose

a coordinate

Y = (x I = 0} regular

system

and

~(p)

singularities

isomorphic

= (0;dXl~).

along

T~X

to the following :

where

x = (x I .... ,Xn)

U

=

is a column vector

identity matrix

of size

~(p)

is with ~

in a neighborhood

There

exists

N

unknown

N x N, Pl = XlDl'

an integer

of each power (When a matrix condition

(i),

We define abbreviation regular

A

A~j A

functions,

Pj = Dj

I

8x

~

such that the order

is said to be of order

of regular

at most

of a holonomic

singularities)

along

A

~(p):

is the

(j=2,...,n)

in

satisfies

with R.S.

by requiring

at each point

operators

the

0.)

~-Module

is said to be with R.S.

and

of each component

(~ >= 0) is less than m. of micro-differential operators

the notion

singularities

~)x-Module

m

is of

(j=l ..... n),

A. (j=l .... ,n) is an N x N matrix of micro-differential J which satisfies the following condition: (i)

so that

~

if and only if

~

0

with

~(~(p))(~)

Then by definition,

at

~x-MOdule

(pjI-Aj)U

near

(= the

that it is with

Areg.

A holonomic

if and only if so is

) LT,x_T x. See

systems

[12] Chap. with R.S.

I and Chap. V for more

and results

which

intrinsic

definition

show the naturality

of

of the above

definition. An important equations. sections namely,

class

of holonomic

A D-type equation of a regular

integrable

the sheaf of Nilsson

it is characterized Definition

i.

(=Deligne

type)

(~) from (~)

connection

class

in the sense

functions.

A holonomic

o~)x-Module

with singularities

Here and in what T~X to X. denotes

~

along

In terms

of Deligne of

[2],

~x-MOdules,

is said to be of D-type a hypersurface

the following

~ is with regular singularities f: ~ > ~ is bijective,

SS(~)

is that of the D-type

to the sheaf of meromorphic

as follows:

if and only if it satisfies

(3) (4)

~x-MOdules

corresponds

follows,

~

three

along

denotes

the characteristic

T~

Y = f-l(0),

conditions:

X. reg"

the canonical

variety

of the

projection O~)x-Module

Remark i. T~X

in

Since there may be some irreducible components other than SS(~),

is with R.S. the use of order.

~,

(3) does not imply immediately that (at least, at present)

the sheaf of linear differential operators of infinite

([12] Chap. V, Theorem 5.2.3.)

Example i. that

the condition

The proof for this fact requires

f

Let

f

be a holomorphic function defined on

X.

Suppose

is not identically zero on any connected components of

Then, for generic Remark 1.2.

~,

~X f~

is a holonomic system of D-type.

The precise condition which guarantees

~

X. ([ii]

to be generic

is also given there.) Example 2. Let

A4

Let

f

be the same as in Example I.

be a holonomic

along

T~regX

at each point

~/~*)

of

along

~

Denote

f-l(0)

by

Y.

~x-MOdule which is with regular singularities

Y

x

in

Yreg"

Then the localization

is a holonomic system of D-type along

Y.

([12] Chap. If, §3, Proposition 2.3.4.) The importance of the D-type equation lies in the fact that it is "generic" in the sense of Theorem 1 below. Defini_____tti_oon2n

A Lagrangian variety

a generic position at

p

holds in a neighborhood of Theorem i.

Let

hood of p in position at p D-type along SS(~)

C

)4

in

A

A

in

T*X-T~X

if and only if

is said to be in

A n-l(~(p))

= cXp

p.

be a holonomic

~x-Module defined in a neighbor-

T*X-T~X. Suppose that A d~f S u p p ~ is in a generic in A . Then there exist a holonomic system ~ of ~(A), a holonomic ~ - s u b - M o d u l e

T~X (**) and an injective

~

of

~

with

~)X,~(p)-linear homomorphism

¢: J~p --¢ ( ~ / ~ ) ~ ( p ) . If, in addition, any order of any sections of integer, then we do not need to divide ~p (*)

can be imbedded into Here

holonomic. (**)

~f

~

is not an half

by a sub-Module

~

~(p).

is equal to

([5], Theorem 3.1.)

Hence

~

~

~f-~,~-j=0

is isomorphic to

~

and it is known to be

for some integer

r.

, namel~

Remark

2.

The assumption

not too restrictive, A

that

because

A

is in a generic

a suitable

contact

position

at

p

transformation

is

brings

to this form. In the course

of the proof of Theorem

i, we also obtain

the same conditions

~

the follow-

ing Theorem

2.

~t//p

Then

Assume

is a f i n i t e l y

generated

on

and

A

as in Theorem

~:)X,~(p)-Module.

Furthermore

I.

we

have d~p

x,~(p) This

theorem

is-useful

amplitudes

3.

in [12]

As a matter (Chap.

holonomic order,

~x-MOdule

namely,

points

of fact,

p'

is in

(Cf.

the analytic

[13]

a much more

4.1.1):

with

C

striking

result

1 holds

system

T~X

structure

of

surfaces, which

§3.)

Theorem

a holonomic

SS(~)

~-l~(p)-T~x-¢Xp.

of Landau-Nakanishi

if we use micro-differential

there exist ~D

if

in investigating

region.

IV, Theorem

a-sub-Module

0

at cuspidal

are far from the physical Remark

p' = p

J~ p =

--X,p' ~ -i ~)@

Feynman

if

operators

~..

and a

is proved

for an arbitrary of infinite

of D-type,

its

~)X.~(p)-linear

homo-

morphism

¢: (&

@ ~) x Gx p

such that the associated

is

an

injective In

[12]. will the

fact, Most

method

amplitudes We space

of

sketch

of

the

some used

in

is one first germs

main

of

introduce

theorem

results ideas

[9]

of

to discuss

the

(~/~))~(p)

homomorphism.

imbedding

basic

~)x

homomorphism

~-linear this

('~x

--~

essential a vector

of microfunctions.

of

is the [12]

the

most

follow

proof.

the

We

ingredients C, is

this.

like

to mention

space

So,

principle

of our

which

the

result

from

hierarchical

space It

important

in

here

we

that for

Feynman

proof.

is an

analogue

of

holomorphic

of

the

functions defined on a cone with its apex at the origin, modulo the space of holomorphic functions defined on a neighborhood of the origin. (See (4.5.1) of [12] Chap. IV, §5 for the rigorous definition.)

This

space is introduced to make clear the action of micro-differential operators on holomorphic functions.

( C f . [6], [i] and [9].)

then prove that

is finite-dimensional

~p

V = Hom~(~p, P Homc(V,C).

is imbedded into

tOTS of

~,

let

C)

Let

{sj} Nj=I

be a base of

{¢~}~=i

V

be a system of genera-

and let

holomorphic function whose modulo class in

C

is

We can

and that

~j ,v ~v(si).

be a Then we can

J

prove that

~

functions on

can be extended to a multi-valued holomorphic X-~(A).

finite determination. holomorphic function

Furthermore,

thus extended functions are with

Since

is a hypersurface, we can choose a -I ~(A) = f (0). In this case,

f

on

~(A) X

such t h a t

the following result is proved by Deligne Let

~

functions

Qk

[2].

Then there exists Nilsson class

and single-valued holomorphic fucntion a k defined on m ~ = ~ ak~ k holds. k=l this result Wemcan obtain a much sharper result to the effect

such that

Using ~

~j,v"

~k

X-f-l(0)

that

denote one of

has the form

~ Qk,k with linear differential operators k=l (of course, possibly with infinite order). See [12] Chap. II, §2

Theorem 2.2.4 for the precise statement and the proof.

Here, instead of

repeating the proof, we present a heuristic, but instructive argument which, we hope, convince the reader that there must exist such an operator Qk" We know ([5], Theorem 2.7) that there exists a non-zero polynomial b(s)

which satisfies

(5)

b(s)fS %oI = p(s,x,Dx ) (fs+l ~ i )

for some

P(s)

= P(s,x,Dx) def

in

~)x[S], i.e., for some linear differ-

ential operator which is a polynomial in

s.

Then, a superficial

application of (5) would entail (6)

f-p 9,1 = P(-p)p...P(-l) 91 .

b(-s) s=l Since

aI

is single-valued on

X-f-l(0),

it has the form

10 oo

(7)

~ aI p~-oo

for holomorphic

,P fP , al, p = al, p (x).

functions

Hence,

combining

(6) and

(7), we might be finished by choosing

oo

~ al,-P p=l

as

QI"

Of course, of

+

al,pfP

[

p=O

this is a very formal reasoning,

may be zero and because coefficients

P(-p)..p .P(-1) II b ( - s ) s=l

Q1

the estimation

is not done.

Probably,

in the general

rather simple completely

transcendental.)

of the existence

3.

Let

so that it may become

it might be possible

Y

in [12] Chap.

from this type of the reasoning.

Once we get Theorem Theorem

even though

In f a c t , the proof given

different

is much more reader

f's.

case,

Still we hope

of the required

with respect

tangential

system

(i.e.,

to which

Y

for

II is

(Actually,

it will convince

it

the

Qk"

i, it is fairly easy to prove

be a submanifold

~x-MOdule

b(-p)

it will not be a very good

strategy to try to modify this formal reasoning rigorous

because

of the growth order of the

of

X.

Let

]~

the following be a holonomic

isnon:characteristic.

the system restricted

to

Y)

Then the

,~y

of

,I~

is

also with R.S. Intuitively i, because

speaking,

the restriction

[12] Chap. V,

this

§3 for a more precise

one should be a little careful non-characteristic Theorem

with respect

By a quantized

(*)

consequence

of Theorem

reasoning.

As a matter

in the reasoning, to the D-type

for

equation

Y

See

of fact,

may not be ~

used in

i.

ing result J'[

is an immediate

of a D-type equation must be with R.S.

contact

to the effect

with R.S.

transformation,

that,

along finite fiber (*)

Here "finite

a finite map for

fiber" means

p(~ : XxT*Y ~ Y

we can obtain a correspond-

if we integrate ~ : X --~

that T*Y

a holonomic

~x-MOdule

Y, then we get a

p~ol(supp~)n~el(u)-and

~

: XxT*Y --, Y

--~ U

T*X.

is

11 holonomic

£y-MOdule

~,~

For the application (Theorems condition

that

The additional ~

Theorem 4.

Let

the map

~ of

since the conditions

Let

is much the

the problems

which appear

in physics,

to

especially

Let

~)x-MOdule with R.S. system

7~y

of

Y

~

be is a

with R.S. be a holonomic

~ : X --~ Y for some

~x-MOdule

is projective,

N.

i.e.,

Then the integral

~-Module

of

with R.S.

Assume

~

is imbedded

into

7~

along fiber

~:

that y×pN~)

X--~ Y

with R.S.

Since the proof of these two theorems ourselves

Y

namely,

is usually not too difficult

Then the tangential

~

is a holonomic

VI,

results

on

condition needed there,

be a holonomic X.

~y-MOdule

Theorem 5.

the following

with Feynman amplitudes.

a sumbmanifold holonomic

problems,

is a ~x-Module,

in investigating

in connection

--~ Y

to physical

4 and 5) are more useful,

less restrictive. verify

with R.S.

with only referring

are complicated,

we content

the reader to [12] Chap. V, §3 and Chap.

§2. Using Theorem 5 we can improve several results

.... ) in that the holonomic

~-Module

of ours

whose existence

the papers quoted above is actually with R.S.

([7],[10],

is guaranteed

In particular,

in

we have

the following Theorem 6.

Analytically

fies a holonomic ~(D).

variety

( C f . [i0],

On the other hand, encounter

a serious

characteristic complete, functions.

Theorem

trouble,

(the so-called

u = 0

equations

with

in the extended

on

Theorem

problem)

R.S.

Landau

4 to physical

obtaining

SS(~y)

structure

(See [3],[15]

satis-

7.)

However,

information

the singularity

fD(p)

problems,

for it does not say anything

~y.

In fact, one unpleasant

an information. this problem

is contained

in applying

variety of

of course)

in analyzing

Feynman amplitude

system of linear differential

whose characteristic variety

renormalized

some

about the (desirably,

is crucially

of the S~matrix

we

important

and related

feature of analytic

S-matrix

theory

is closely tied with the lack of such and

[14] for this point.)

in several cases by finding

some information

We bypassed on

SS(~I)

12

for very special

ment in mathematics following

attained

satisfactory

Definition

3.

N X~ J~ = ~)X( [ifz(x)+ ).

]~i, i.e.,

Let

in [12] has enabled us to obtain the

results.

f

(Theorems

be a holomorphic

Wf is, by definition, the closure of f(x) ~ 0, ¢ = od logf(x)} in ~ × T*X

(ii)

W0

the dependence notation

Remark 4.

of

W0(f)

(0;x,~)E Wf}. W0

on

instead of

It is known that

Example 3.

Let

X

be

~2

W0

W0(f ) = T ~ 2 ~ 2 U T ~ I ~ 2

Theorem

7.

Let

by

Let

~

Y.

characteristic

f

{(o;x,~)~ ~×T*X; x~ X,

When it is necessary

to make

W 0.

is a Lagrangian variety.

{x~2

f

be

2 Xl(Xl-X2).

; Xl=0 }

(resp.,

Denote by

S1

{xE ~2, Xl_X~=0} )

UT*s2~2 U T ~ 0 } ~ 2

be a holomorphic

be a holonomic variety

X.

explicit, we sometimes use the

and let

(resp., $2) the hypersurface Then

f

7 and 8 below.)

function on

(i)

= {(x,~) E T * X ; def

The recent develop-

SS(~)

function on

X

and denote

system of D-type along is contained

in

Y.

f

-i

(0)

Then its

W0(f).

The proof of this theorem essentially uses the desingularization theorem of Hironaka.

The key-point

in the proof is that a D-type

equation remains to be a D-type equation after the desingularization procedure.

([ii],

§i.)

Next, by the aid of the result mentioned

in Example 2, we can

deduce the following result from Theorem 7. Theorem 8.

Let

M

be a real analytic manifold

ification.

Let

~

be a locally integrable

that there exist a holomorphic of

~X

function

which satisfy the following

(8)

f

(9]

~

(i0)

~,~ = ~)X/~ def

fl M

is not identically

f

on

conditions

holds on

X

M-f-l(0).

is a holonomic

(11) SS(/~) c T~X u ~ - l ( f - l ( o ) ) .

X

its complex-

~)x-MOdule.

M.

Assume

and a left Ideal ~

(8)~(12).

zero on any connected

is real-valued.

~ 0

and

function on

component of

X

and

13

(12)

7~

is with regular singularities

Then there exists a left Ideal

~'

T~

X, where Y=f-l(0). reg ~=)X which satisfies the

along

of

following conditions: (13)

~'~

(14)

~' : 7

(15)

~'d~f~X/~'

(16)

SS(~')

Remark 5.

= 0

holds

on

M.

X-f-leo).

on

is a holonomic

is contained in

~x-MOdule

with R.S.

W0(f).

As the following example shows,

~'

#j

in general.

Example 4. Let • be M, let x be f and choose the Heaviside function Y(x) as ~. If we choose ~ ~-f d as > , then conditions d (8)~(12) are all satisfied. In this case we must choose ~(x~-~) as~' The reader will see clearly that Theorem 8 provides us with a powerful tool in dealing with the restriction and the multiplication of holonomic hyperfunctions, if they are "tame" enough. Here the tameness means the quantitative aspect of ~, i.e., its local integrability or something like that. Such a tameness condition is used to prove that f+~ E > 0.

can be holomorphically extended to See [ii], §2 for details.

{~ e~;

Re ~>-E}

for some

Before ending this report, we mention some comparison theorems which justify our terminology "with regular singularities." Theorem 9.

Let

~

and

Af

be holonomic

~x-Modules with R.S.

Then

~X As an e a s y c o r o l l a r y Theorem 10. each point

Let x

of this

dd

in

X

theorem,

be a h o l o n o m i c and each i n t e g e r

~zfJ~ X(/~f, CgX,x) --~

we a l s o

have the

following

~)x-MOdule w i t h R.S. j,

the natural

~;~X(M,

Then,

for

homomorphism

~X,x )

is an isomorphism. Here (JVX x denotes the ring of formal power series around x,i.e, lim ~V,x/~k'A for the maximal ideal ~ of ~k

~X

14 As a matter of fact, we have the following more general result. Theorem

ii.

Let

~

and

~x Here

~X

denotes

~x(k)

the validity

~x-MOdule

statement

~.

([12]

(Cf. Ramis

of Theorem

Chap. VI,

case).

i.e.,

opera-

of the comparison ~

(Cf. Malgrange

i0 is also

§4, Theorem 6.4.1.)

[19], where the validity

the holonomic

of the

of Fuchsian

also the remark at the end of §4 of [12] Chap. VI.) the validity

operators,

theorem characterizes

theorem is used as the definition

does not imply that

Then

k.

of the comparison

x-MOdule with R.S. comparison

with R.S.

is the sheaf of micro-differential

We know that the converse

true for a holonomic Hence

~x-MOdules

the sheaf of formal micro-differential

o f o r d e r a t most

Remark 6.

be holonomic

(~,~) = ~ /4~ExC~t, ~x ~xJ)"

l i m SX / ~ . x ( k ) , where tors

~

systems.

theorem at the 0-th cohomology

is with R.S.

See

Note, however,

that

group

(even in the one-dimensional

[16].)

We do not give even the sketch of the proof of Theorem 9 here, because probably

it ls not very interesting

who is interested

in it is referred

mention one thing which may be helpful meaning

f jw~

be a holomorphic by

is a holonomic holomorphic

for the understanding

~f,

the localization

functions

of the form

singularities

in this case Theorem

and loose language Let

~

we

of the

~

Denote along

@fX

along

~ x f X ( X E ~) by ~ . f-l(0).

for a meromorphic consists

f-l(0).

9 can be expressed

JV~

of

function ~fk

(Cf. Mebkhout

with [17].)

in a rather symbolical

~x-MOdule

with R.S.

Then the solution

are identical whether we admit only pole-type

ties or essential

Then

of multi-valued

as follows:

be a holonomic ~

of

X.

@ F~" Then ~)X ~)X

f-l(0).

having essential

spaces of

function on

~Z)x-Module with R.S. and it consists

having poles along

Hence

The reader

However,

of Theorem 9.

Let Define

to physicists.

to [12] Chap. VI.

singularities

as its solutions.

singulari-

15 Appendix Utilizing this opportunity, we report that the recent progress in the theory of holonomic systems has enabled us to prove the conjecture in [13] (p.123) with a trivial modification. in [13] (pp.123~129) with prefix

C-

Hence the results stated

such as C-Theorem 4.6, i.e., the

results whose proof rely on the conjecture hold as they stand. modification,

i.e., the inclusion of Case I in the statement does not

require any essential change of the argument given in [13].) which replaces the conjecture is the following Theorem A.I. Let of

has the form of

Let

~9[ = ~ X / ~ p, where

T~X

(The

X

be a complex manifold and

be a holonomic

~

where

V1

p

a point in

T~X-T~L

~x-Module defined on a neighborhood

is a left Ideal of

VIUV2,

The result

t

~X"

and

Suppose that

= Supp /V~ def are Lagrangian submanifolds

V2

and they intersect along a submanifold

Vi2

of

V

T~X

with

codimv.Vl2 = 1 (j=l,2). Suppose further that TpVI2 = TpVI~ TpV 2 holds. J Assume that ~ is simple both on VI-VI2 and on V2-VI2. Then either one of the following two cases happens: Case I.

There exist simple holonomic

Case II.

respectively by Vj (j=l,2) such that /9~= ~ i ~ 2 holds on a neighborhood of p. In a neighborhood of p, / ~ is isomorphic to a holonomic gx-Module reduced.

~X/~ '

~x-HOdules

with the symbol Ideal

~j

(j=l,2) supported

~(~')

being

Proof. By Theorem 5.3.1 of [21], Chap. II, we can reduce the problem to the case where X is an open subset of ~2. Hence, by a quantized contact transformation, we may assume that ~=y=0}

and

V2 = { ( x , y ; ~ , q )

~ T~2;

V 1 = ((x,y;~,q) ~ T ~ 2 ;

x=y=0}.

Now, we know ( [ 1 2 ] Chap. V, §1, T h e o r e m 5 . 1 . 6 ) t h a t a coherent ~(0)-sub-Module ~_ of M such that ~ and that (A. 1)

~

= ~/~(-I) def

there exists = / ~ holds

satisfies the following conditions:

JV ~ = O. m

(A.2)

The support of any section of

Here and in what follows, JV

~

has codimension

denotes (f ~ OT~X(0);

0

in

V.

flv=0}, where

OT~x(k) denotes the sheaf of holomorphic functions on T~X which are homogeneous of degree k with respect to the fiber coordinate. ~n the

16

sequel we sometimes abbreviate ~T,X(k) to ~(k). It follows from the assumption of the simplicity of /~ that ~ is isomorphic to

~T,x(O)/JvI*~T,x(O)/Jv2 ftvj=O)

(j=l,2).

geometry.

on

V - V12, where

JV. = ( f e ~ r , x ( O ) ; J t h e f o l l o w i n g 1emma in a l g e b r a i c

L e t us now r e c a l l

We o m i t i t s

proof here.

Lemma A.2. Let CO, C 1 and C 2 be respectively the curves {(Zl,Z 2) ¢2; ZlZ2=O}, {(Zl,Z2) E ¢2., z2=O ) and {(Zl,Z2) ~ ~2., Zl=O). Let be anec0-Hodule such that where

~CD = ~ 2 / J j

with

is isomorphic to either

We :

~0}(~)

= 0

and

J.] = {f 6 ~ 2 ;

~C0

or

=

~IC0_{0}

~CI

f IC=0}j (j=l,2).

~

~C2

Then

~CI@ ~C2 .

find by this lemma that either

~F_~= ~ ( 0 ) / J V

or

(~(o)/JVl) ~ (~(o)IJvz). We first consider the case where

Let u and v be sections of respectively. Then we have

~

2 = (~(0)/JvI)

which give

(A.3)

~=

(A.4)

(xDx + 2yDy) u, (xDx + 2yDy)V E ~ .

(A.S)

D u x

6(o)u

~

~

+

and

@ (~(0)/Jv2).

ImodJvl

and

imodJv2,

6(O)v

xD v ~ ~ . y

Now the following result is proved in [18] (Theorem 3.2): Lemma A.3. Let P be a square matrix of micro-differential operators which are of order at most zero and defined in a neighborhood of p =(0;dy)~ T*~ 2. Then there exists an invertible matrix U of microdifferential operators of order at most 0 which satisfies the following:

(A. 6)

o 0 (U)

(A. 7)

(XDx+ 2yDy-P) = U ( X D x + 2 y D y - A ) U

and

= I -1,

where

A = o 0 ( P ) (P) •

By virtue of this result we may assume from the first that v satisfy

u

17

(A.8)

(xD x + 2yDy

for a constant

matrix

Now we prove Lemma A.4.

Let

Assume

u

that

- A)(u)

= 0

A.

the following u

be a column vector

for some

constant

length N

defined

matrix

A)u A.

Let

P(x,y,Dx,Dy)

in a n e i g h b o r h o o d

Then,

+ 2yDy,

we find

P.u = 0 J

Proof.

Let

be the s u b - M o d u l e

that

writing

~

Qu = 0. Q

P =

Then

~ P. jE £ J

QI'

the

of

functions.

"'"

r x N

Qu = 0

which

of

satisfies

[ • ,P] = jPj, w h e r e

~ = xD x

j. of

~(0) N

' Qr

which

be a system

matrix

holds

be a row v e c t o r

p = (0; dy)

with

for every

Let

denote

components.

unknown

= 0

Pu = 0.

such

N

satisfies (xD x + 2yDy

and let

of

with

QI'

consists

of

of g e n e r a t o r s

"'"

' Qr

by the definition.

Q of

as its row

On the other

hand,

we have

(A. 9)

([~ ,QI+QA)u

Hence

there

exists

Q( ~

- A) = ( ~

and

B E M(r,r;

ing

Q

holds,

with i.e.,

[ ~,Qj]

~Qu

R)Q. ~)

U-IQ,

Again

such

~(0))

such

by Lemma A.3

that

~

we m a y assume

[~ -Q] = BQ

= 0.

- QA

R = U(~ from

holds.

that

[ ~ ,Q] + QA = RQ,

there

exists

-B)U -I.

the first Hence,

U

~

Hence,

i.e.,

GL(r,~(0))

by replac-

that

Q(~

-A) = ( ~ - B ) Q

writing

Q =

~ Qj jE L

with

= jQD, we obtain

for every j.

j.

This

= BQj

QjA

implies

Furthermore,

for each p o l y n o m i a l

j, there

exists

that

Qj = 0

by the r e p e a t e d

(~Qjf(j+A))u holds

- Q(~-A)u

R ~ M(r,r;

jQj

many

=

a polynomial

=

holds

except

application

of

for f i n i t e l y (A.9),

we find

0

f(X).

On the other

f~(X) J

such

that

hand,

for each

integer

~8

fi(j+A ) = I

and

fi(k+A)

J

= 0

for

k # j

with

Qk ~ 0.

Therefore

J

Qiu = 0

holds

for e v e r y

j.

J

N o w let

P

be the r o w v e c t o r

there

exists

T6M(I,r;

with

[ ~ ,Rj]

= jRj, we have

obtain

in the s t a t e m e n t

such that

P = RQ

Pj = ~ R j _ k Q k.

of Lemma A.4.

holds. Since

exists

Then

Writing

R = ~ R. j ~ I J holds, we

Qk u = 0

P.u = 0. ]

Q.E.D.

N o w we r e s u m e there

~)

the p r o o f

P CM(2.2;

of the theorem.

~(0))

It follows

from

(A.5)

that

such that

I oII:l Dividing y.

P

by

xD x + 2yDy

By Lemma A.4 we m a y

hand,

since

contain Since

following

P

it is a linear

[~. ,xiDJD -j-k] x y

=i+j+2k

further

the o p e r a t o r

y,

c a n n o t be

A, we m a y assume

that

is of o r d e r combination

w i t h a,b,c

have and d

of

(XDx+2yDy)U

= au + bv

(XDx÷2yDy)V

= cu + dv

P

does not c o n t a i n

[ @ ,P] = -P. 0

x i D J D -j-k xy

On the other

and does not with

and since

P = 0. in

that

at m o s t

= ( i - j + 2 j + 2 k ) x i D J D -j-k x y

-i, we m u s t

equations

assume

i,j,k > 0. =

i-j+2j+2k

H e n c e we o b t a i n

the

~:

(A.IO)

Du

= 0

X

XV

=

0.

H e n c e we have

(A.II)

(yDy-a/2)u

= bv/2

and (A.12)

(yDy-1/2-d/2)v

Therefore

we h a v e

bv

= 0.

Similarly

= cu/2.

DxbV = ( 2 y D y - a ) D x U we o b t a i n

cu

= 0.

= 0 Thus

and

xbv

we f i n a l l y

= 0.

This

obtain

implies

19 (yDy-a/2)u

=

DxU

=

0

(A. 13) (yDy-i/2-d/2)v This implies

that

~

= xv = 0.

is the direct sum of

/ ( ~ (yDy-i/2-d/2)+~x), supported

by

V1

and

In the case where which gives

ImodJv.

implies that

~

~/(~

which are respectively

(yDy-a/2)+~Dx)

simple holonomic

and systems

V 2. ~

= ~(0)/Jv,

Then we have

satisfies

we take a section

~

=

the property

~(0)u

and

~

u

of

=~

= ~u.

This

stated in Case II of the

theorem.

Q.E.D.

References

[1]

Bony, J. M. and P. Schapira: Propagation des singularit4 analytiques pour les solutions des 4quations aux d~riv~es partielles. Ann. Inst. Fourier, Grenoble, 26, 81-140 (1976).

[z]

Deligne, P.: Equations differentielles ~ points singuliers r~guliers. Lecture Notes in Math. No.163, Berlin-HeidelbergNew York: Springer, 1970.

[3]

Iagolnitzer, D.: The 6__33, 49-96 (1978).

[4]

Kashiwara, M.: On the maximally overdetermined system of linear differential equations, I. Publ. RIMS, Kyoto Univ., i0, 563-579 (1975).

[5]

: On the holonomic systems of linear differential equations. II. Inventiones math., 49, 121-135 (1978).

[6]

Kashiwara, M. and T. Kawai: Micro-hyperbolic pseudo-differential operators I. J. Math. Soc. Japan, 27, 359-404 (1975).

u=0

structure

theorem.

--n

[7]

:

Micro-local

properties

of

Commun.

math.

Phys.,

s-

~ f.~.

Proc.

Japan Acad.,

j=l ~ 5!, 270-272

(1975).

[8]

: Finiteness theorem for holonomic systems of m i c r o differential equations. Proc. Japan Acad., 52, 341-343 (1976).

[9]

: Holonomic Feynman integrals.

[lO]

: Holonomic systems of linear differential equations and Feynman integrals. Publ. RIMS, Kyoto Univ., 12 Suppl., 131-140 (1977).

[11] with

[12] [13]

character and local monodromy structure of Commun. math. Phys., 54, 121-134 (1977).

: On t h e c h a r a c t e r i s t i c regular singularities.

variety of a holonomic system Adv. i n M a t h , 34, 1 6 3 - 1 8 4 ( 1 9 7 9 ) .

: On h o l o n o m i c s y s t e m s o f m i c r o - d i f f e r e n t i a l equations. III --Systems with regular singularities--. To a p p e a r . (RIMS Preprint No.293.) K a s h i w a r a , M., T. Kawai and T. O s h i m a : .A s t u d y integrals by micro-differential equations. P h y s . , 6_O0, 9 7 - 1 3 0 ( 1 9 7 8 ) .

o f Feynman Commun. m a t h .

20

[14]

Kashiwara, M., T. Kawai and H. P. Stapp: Micro-analyticity of the S-matrix and related functions. Commun. math. Phys., 6__66, 95-130 (1979).

[15]

Kawai, T. and H. P. Stapp: Discountinuity formula and Sato's conjecture. Publ. RIMS, Kyoto Univ., 12 Suppl., 155-232 (1977).

[16]

Malgrange, B.: Sur les points singuliers des 4quations diffe~entielles. Enseignement Math. 20, 147-176 (1974).

[17]

Mebkhout, Z.: Local cohomology of analytic spaces. Kyoto Univ., 12 Suppl., 247-256 (1977).

[18]

Oshima, T.: Singularities in contact geometry and degenerate pseudo-differential equations. J. Fac. Sci. Univ. Tokyo, IA, 2_!1, 43-83 (1974).

[19]

Ramis, J. P.: Variations sur le theme "GAGA". Lecture Notes in Math. No.694, pp.228-289, Berlin-Heidelberg-New York: Springer, 1978.

[20]

Sato, M.: Recent development in hyperfunction theory and its application to physics. Lecture Notes in Phys. No.39, pp. 36-48, Berlin-Heidelberg-New York: Springer, 1975.

[21]

Sato, M., T. Kawai and M. Kashiwara: Microfunctions and pseudodifferential equations. Lecture Notes in Math. No.287, pp. 265-529. Berlin-Heidelberg-New York: Springer, 1973.

Publ. RIMS,

SECOND-MICROLOCALIZATION

AND ASYMPTOTIC

EXPANSIONS

Masaki KASHIWARA D~partment de Math~matiques, Universit~ de Paris-Sud 91405, Orsay, France and Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606, Japan AND Takahiro KAWAI Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606, Japan

Abstract The asymptotic

expansions

of (holonomic)

microfunctions

is neatly

dealt with by the second micro-localization.

§I.

Introduction. Several years ago, Jeanquartier For a real-valued

analytic manifold

[5] proved the following:

real analytic

of dimensions

function

n, ~(t-f(x))

f(x)

defined on a real

has an asymptotic

expan

sion of the form

(1.1)

n-i

N

~

X

~

~

k +j

a

~=0 ~=i j=0 for distributions of the asymptotic function

~(x)

Recently,

(x)t

(log t) ~.

v'P'J a ,H,j(x),

expansion

as

t

has an asymtpotic

expansion

equations :

by definition,

technique,

Barbasch-Vogan

on a real semi-simple

and they studied some properties

term of the expansion.

satisfies,

Here the meaning

with compact support. by using a group-theoretic

[i] proved that any eigendistribution initial

tends to zero.

is given through the pairing with a C ~-

group

G

of the

Here we note that an eigendistribution

the following

system of linear differential

22

Pu = x(P)u

for each

P

in

LAU = 0

for each

A

in the Lie algebra of

(1.2)

where X

~

is the ring of G-biinvariant

is the character

sponding

of

}

LA

is the vector

to the inner automorphism

We can show that R.S. (*) and that later

and

6(t-f(x))

(§4.2) each microfunction sections.

Barbasch-Vogan

Hence

on

G

G,

corre-

a holonomic

system with

system with R.S.

As we show

system with

whose meaning will be clarified

in

and that of

in a neat and unified manner by employing

systems with R.S.

expansions

situation

i.i.

field on

on

G.

solution of a holonomic

the following

sider such asymptoti~

operators

the result of Jeanquartier

are explained

We now introduce

Definition

G

satisfies

expansion,

the theory of holonomic

a more general

of

(1.2) is also a holonomic

R.S. has an asymptotic subsequent

differential

G,

space

M (r)

so that we may con-

of distributions

mentioned

above

in

.

A distribution

u

if and only if there exist positive

on

~n

belongs

constants

e, m

to

M (r)

and

C

(r~m) such that

n (1.3) holds

I u(tx)~(x)dxl for every

~

These spaces asymptotic

< Ct

in

suplD~l

CO ({x~n;[xI<¢}).

M (r)

can be used as a scale in considering

expansion of a distribution

We say that

u

has an asymptotic

u ~

~0uj = if there exist a sequence j satisfy the following condition: (1.4)

For any

r~,

there exists

u

the

as follows:

expansion {uj}j> 0 =

NO ~

~ u. and we write j=0 J of distributions which

such that

No u- X u . ~ M

(r).

j=0 In particular, (1.5)

For any any

(~)

this condition rER,

implies

there exists

the following:

N 0Ez

such that

uj ~ M (r)

for

j ~ N 0.

See Kashiwara-Kawai [8], [9] for the theory of holonomic with regular singularities.

systems

23

In what follows we exclusively neous of degree

l+j

consider the case where

for some

u. is homoge] or, a little more generally, the

kEC,

n

case where case,

( ~ x D -~-j)Pu. = 0 holds for some p 6 ~ . ~=I v v ] n belongs to M(-Re~-J-~ +s) for any s > 0.

uj

In the latter

In order to exemplify the notion introduced here, we give the following ex@mple, where the j-th terms in the right hand side are homogeneous

of degree

2~+j

and

3~+~+j,

respectively.

(-~+½)

F (y2-x3)~ ~ jX 0 . ½ "= 4]j!F(-X+ +j)

(1.6)

+

~ j=0

/TF(~+I) 43j!r(~+j+~)cos~

The asymptotic

regarded as an asymptotic A

M (r)

expansion wifh respect to the origin

to try to micro-localize

{(x,~)ERnx/l~n;

x=0}.

Let

we introduce

1 2.

p = (0,/7i-~0)

the following

A distribution

u

There exist of

[0

-~

T~0}~ n =

be a point in

to

M (r) A,p M (r) A,p

A.

In

(rE~). if and only if

is satisfied:

×(x) ~ C~OR n)

and constants

i.e.,

spaces

belongs

{x=0}.

the notion as follows:

"

the following condition U

as the scale can be

be the conormal bundle of the origin,

this situation, Definition

3~+~+ s i n ~ x_3~+~)~(y).

x3JD 2j (-x+ Y

expansion done by using

Hence it is natural Let

x3JD2jlyl 2x Y

C

and

with m

X(0) 0, an open neighborhood such that n

(1.7)

l

l

holds for

T ~ 1

for every

The following Lemma 1.3.

(i)

(ii) r.

(*)

is If

If (1.7) holds for some

contained p

is not

See Hdrmander

~ 6C~(U).

lemma easily follows from the definition.

then (1.7) holds for any supp ~

sup ] D ~ ( ~ ) I

l~1=<m

X E C~0R n)

X g C0ORn)

such that

~(0) ~ 0

in the domain of definition in

W F ( u ) ! *) t h e n

u

with

belongs

of to

X(0) ~ 0,

and that

u. g(r) A,p

for every

[3] for the definition of the wave front set.

24

(iii)

If

u

belongs

M (r) and a(x) A,p of the origin, then au

in a neighborhood further (iv)

suppose Let

u

that

a(O)

If

constants

C, m

there

exist

such

holds

for

(v)

If

(vi) most

T > 1 u

Let 0.

(vii)

belongs P If

for

three

(c)

(~) u m~ M A,p

for

M

If we .

by

v

U

of

~0

u

belongs

its

Fourier

and

~

suplDa$[ $~C0(U

)

then

'

to

M (r) A,p"

The converse

are

at

equivalent:

p EA

p = (0"0)

For each distribution to M (r) A,p" of

u

there exists

(ii) is not true,

r

namely,

such that

u

the fact t h a t

u

to

M (r) for every r does not imply that WF(u) does not A,p p; For example, let n be 1 and consider the C~-function

contain

f(x)d~fv exp(-I/x2) xj, where

and a continuous

{xj}

combined

implies

However

WF(u)

is a sequence

that

function

tending

u(x) d~f f(x) g(x)

contains

some

p

We can also prove

in

1.4.

(x,@)

Let

the following

a(x,@)

= (0,@0)~A n x~ N

the following

conditions:

(1.9)

a(O,@)

= 0

(1.10)

[ a2a(x'@) ~xi~ j ]i,j

has

which Then

is not

C~

(iii) told (viii)

for any P

and r.

is contained

in

result.

(N ~ n).

n

0.

is in

be a c~-function

rank

to

g(x)

M (r) A,p A, because x. J

sing supp u.

of

to

and denote

conditions

for any

Theorem

M (r) "A,p"

M (r) then 3u/Bx k belongs to M~ I) A,p' be a classical pseudo-differential operator of order u belongs to M (r) then Pu belongs to M (r) A,p' A,p"

u 6 M (r) A,p

Remark.

belongs

distribution

every

(b)

belongs

au

to

defined

to

The following (a) u ~ M (r)

(viii) belongs

belongs

an o p e n n e i g h b o r h o o d

< cTr-Y n ¸

and

=

is a C~-function

that

I fv(T¢)~(¢)d~[

(1.8)

= O, t h e n

be a tempered

transform.

at

to

defined

Suppose

at

(0,@0)

that

in a neighborhood a(x,8)

satisfies

25

Denote by

p

the point

(0, /~-rdxa(0,@0)) E~C-IT~0}~n.

compactly supported distribution which belongs to function

l

w(@)

- lexp(-~

Then there exist constants

;

I w(t@)C#(@)d@l

C, m n

for

be a

Define a

%> 1

)u(x)dx.

and

~(>0)

such that

s p,O8~I

< CT =

holds

u

by

exp(-~a(x,e))u(x)dx

(i. Ii)

M (r) A,p"

Let

Ic ~ =

and f o r

every

~

O0

Co({e; le-e01 < E}).

in

It immediately follows from this result that a Fourier integral operator(~) A of order at most 0 such that the associated canonical transformation

~

preserves

this fact we can define

A

M A,p (r)

M (r) to M(r) In view of A,p A,~(p)" for every Lagrangian submanifold A of sends

/~rT~R n by using a suitable phase function a(x,@), even though we exclusively consider the case where A = /Z-IT~0}~n in this section. For example,

if

A = {(x',x";

~"=0}, then a distribution and only if there exist and E > 0 such that

Jz-f(~,,~,,) ~ x ~ n - ~ x £ _ - T O R Z ~ n - Z ) ;

u

belongs to

X GC;OR n)

with

M (r) A,p X(0)#0

x,=0,

(p=(0;~r~-l(~,0)) and constants

if C, m

r t l(lexp(-/7-1<~',x'>) (XU) (x',x")dx')~($',x")d$'dx" I

(1.12)

r-~ < C~

= holds for

T>I

l~'-~;l÷Ix"l

!

1<mSUp ID~, DxB,,,I

I~I Is and for every

~(~',x")

in

C~({(~',x")~IRZx]R n-~"

< ~}.

Now we want to investigate how these scaling spaces are related to micro-differential equations with regular singularities. Let us denote by T~0}En.

AE Then

the complexification ~A~(m)

has the zero of degree ~" E" A (,)

of

A = ~i-Tr?0~n , i.e., A E =

is, by definition, (j-m)

along

{P = ~. pj(x,D) ~ g; pj(x,~) J AE}. We abbreviate ~.E(0) to

It then follows from Lemma 1.3 (iii) that

See HSrmander

[3] for the definition.

26

(1.13)

Note also that, if a micro-differential then it has the form (1.14)

operator

P

belongs

to

~, A

O~l~lamA~(D) x a + Q,

where A (D) is homogeneous of degree I~I and Q belongs to A~(-I)" In order to see how %¢(m) is related to the asymptotic expansions, we consider a simple case where a distribution solution u(x) of the equation Pu = 0 has an asymptotic expansion of the form ~oUX+j(x)

where

j=

ux+j

is homogeneous

of degree

X+j.

It is clear

n

that

ux+j

belongs to

M(-Re~-J-2 ).

In this case, we find the

following relation: (1.15)

Pu

O£1~l~mA~(D)xauh (x)

If we define a homogeneous

then we obtain

(1.16) where by §5.)

another

L(P)fiX(~)

mod

differential operator

i~

L(P)(~,D¢)

by

equation

= O,

fiX(C) = ~I e x p ( - f Y i - < x , ¢ > ) u X ( x ) d x . ~A~,

M(-ReX-I-~ ).

is isomorphic to

I f we d e n o t e

~A~/ i~(-l).

the image o f

L

( C f . [8], Chap. I,

It follows from the definition that

~¢ consists of homoA geneous linear differential operators of degree 0 defined on A ¢. The following Theorem 1.5 shows the importance of the associated equation (1.16), especially because we will later prove that, if u is a solution of holonomic L-Module with R.S., then its asymptotic expansion is determined by its initial term. (See §4.2, Theorem 4.2.12 for the precise statement.) Theorem 1.5.

Let

X

manifold of T*X. Let ~X/~ is a holonomic

be a complex manifold and

A

~ be a left Ideal of ~X ~x-MOdule with R.S. Then

a Lagrangian subsuch that

~A~def

27

,~ :

L(P)v = 0

(P ~ -

is a holonomic system with R.S. on Proof.

~ ~,A)

T*A.

By a quantized contact transformation, we may assume the

following: (1.17)

X

is an open subset of

~l+n = {(t,x); t ~ ,

(1.18)

A

is the conormal bundle of

(1.19)

Supp/~ I, §6).

Yd~f{(t,x) 6X;

is in a generic position

x~n}. t=0}.

(in the sense of [8] Chap.

Then we know ([8] Chap. V, §i) that there exists a holonomic Module

~

with R.S. and a section

{12o)

fi of

1 ° G

~

~)X-

such that

= o

~9 X

=

and

(1.21)

1 @ fi corresponds to the section

u = (i mod ~ )

of

}{

by

the above isomorphism. We also know ([6]) the following: (1.22)

~[s](tSu)

is a coherent subholonomic

Here and in what follows, ~ [ s ]

denotes

~x-MOdule.

~[s] @ ~ X "

By using the

same reasoning as in [6] we can also show that ~ (1.23)

~[s](tSu)/~[s](tS+lu)

is a holonomic

O~y-MOdule with R.S.

Let us now denote by 4' the sub-Algebra of ~X generated by ~X' Dx. and tD t. Then, for any P in ~ ' , tSpt -s belongs to ~ [ s ] . J Moreover, any section of can be written in the form D~tsp.t -s

Lemma 1.6.

with

Let

p e~,.

P(s)

Here we note the following two lemmas.

be a section of

~[s]

of the form

28

j~O

DJtSp.t -s •

Proof. It

It

follows

suffices

case

If

j

t

for

to any

P(s) t su = 0

from

the

Show t h a t

integer

'

assumption

P.u J If

r.

holds

= 0 P.u J

then

j

that

(j>r) = 0

P.u = 0

holds for any

~oD~tSpJu entails

(j>r),

then

= 0

j

holds.

P u = 0 r we h a v e

in

this

r

0 =

~ D~tSp~u = j=0

~

J

~

s(s-l)-''(s-k+l)~S-k~j-k~~ ~t ~ju.

0~k~j~r

Looking at the coefficients tS-rPru = 0.

Lemma 1.7.

Hence we have

sr

~'u

by assigning Let

[s](tS+lu)

tSpu P

in the right hand side, we find

Pr u = 0.

We have an injective ~'u/t

Proof.

of

~y-linear

; ~ [s] (tSu)/~

to

Q.E.D.

homomorphism

[s] (tS+lu)

Pu (P E ~').

be an element of then we can find

tSpu = p(s)(tS+lu)

=

~'

If

tSpu

p(s)=~D~" t - Spj t s J

belongs to (Pj ~ o~' )

such that

~ D~tspj tu. J

Hence it follows from the preceding lemma that Pu = P o ( t U )

E~'(tu)

= t~'u.

Q.E.D. Now we resume the proof of Theorem 1.5. Let ~' denote the Ideal of ~ y which annihilates tSu mod ~ [ s ] ( t S + l u ) . Then it follows from (1.23) that ~y/~' is a holonomic On the other hand, Lemma 1.7 implies

~y-MOdule

w i t h R.S

~'u C t~}'u. Hence for any P in ~ ' , we can find Q in t ~ ' such that (P+Q)u = 0. Since o~' C ~A and L(P+Q) = L(P), the system ~ in Theorem 1.5 is stronger than the system ~y/~'. This means that is a holonomic O~y-MOdule with R.S. Q.E.D.

29 So far we have developed the theory of asymptotic expansions by the aid of scaling spaces M (r) (or its micro-localization M~r)).However, it would be much more desirable if we could find a suitable sheaf on which another suitable sheaf of rings of operators acts and in which the asymptotic expansion of (a class of) microfunctions can be considered. In the example (1.6), we can rewrite the right hand side

(1.24)

[° 21 ayl X

where

+

c

j=O J

Y

I eYF(~+I)

j

c'x3JD 2j

0 J

3~+}

[F(~+})cos~

(-x+

cj=Z(-X+ )/4Jj!r(-x+ +j)

Y

1

+ sin~ x

and

3,~+} 1

)j6(y),

cj = F(X+ )/4Jj'F(X+ +j).

In view of the growth order of c i and c~, it is easy to see that the infinite series of operators appearing in (1.24) do not preserve the local character but that it has "propagation velocity" of order 3 IxlZ. (Cf. Kashiwara-Kawai [7]). Having this in mind, ~e will introduce the sheaves ~A and ~A in the next section. We note that it has turned out that ~A coincides with the sheaf introduced by Laurent [14], [15] in different context.

§2.

The second-microlocalization

of operators.

As we mentioned at the end of the introduction, we want to find a sheaf of operators which is suitable for the manipulation of the asymptotic expansion of microfunctions. For this purpose we introduce the sheaves ~A and ~A starting from the sheaf ~ of simple holonomic A ~x-Module supported by a complex Lagrangian submanifold of T*X. Our procedure for finding the desired sheaves is similar to the way of constructing the sheaves of hyperfunctions and (micro-) differential operators starting from the sheaf of holomorphic functions For example,

the sheaf

infinite order) on = {(x,y) Een×¢n;

en x=Y }

~

of linear differential operators (of en is, by definition ~ ( ~ n ~ ~ n )' w h e r e and

~n ~n

denotes the sheaf of holomorphic

30

n-forms.

Note that

morphic to sheaf

~n"

fin Cn

is a dual sheaf of

@

cn

and locally iso-

We will follow this procedure in defining the

~oo

~A (Definition 2.2). We first prepare several notations. Let X be a complex manifold of dimension n and A a non-

singular Lagrangian submanifold of

T*X.

We denote by

y

the projec-

tion from T*X - T~X to P*X. In what follows, we will consider the problem locally in A, and hence we may assume that A = - I ~ with X = y (Aj.

~ be a simple holonomic $,z-MOflule with support fl and denote ~ t ~ ( ~ , ~X ) • Let JVr be ~@ ~* and A a def = T~(XxX) = {(x,y;~,~) ET*(X×X); x=y, ~+n=0}. Here we note the followlet

Let ~*

ing Lemma 2.1.

The system

~

is independent of the choice of

~.

Proof. Let ~ ' be another simple holonomic system with support A. Then there exists a constant X such that ~ ' is isomorphic to ~ " def = g(X) @& # ~ ' where denotes the sheaf of micro-differential g(X) operators of fractional order X+j (j~Z). Furthermore the isomorphism from ~ ' hand, y~"* = (~(-X)

to ~ " is unique up to constant factor. On the other ~(-X) @E ~ and hence y~" @ y~"* = (~(X) @E v~) ~

@gjk~)

= ~ *

=/~.

give rise to an isomorphism #~$ ~*. Since ((c~)*) -I depend on

~

Therefore from

= c-l~

~' *-1

~

and

~-I:

~,,~ ÷ ~ , ~

@ ~'* onto ~4~" @ ~ " * × c ~ ¢ , ~ does not

=

for

~.

Q.E.D.

After this observation we introduce the following Definition 2.2.

~7

d~f ~

a ( ~X×X @eX×X ~F)"

We can also consider the micro-localization of ~ A by the same procedure used in S-K-K [17] Chap. II to define the sheaf of micro (=pseudo)-differential~ operators starting from ~ 7n~) - In what follows, "A"Ax~*

denotes the comonoidal

center

(A×Aa-A a) U S* (AxA a)

Aa

i.e.,

A a

,

to P* (A×A a) Aa

S * (A×A a) Aa are denoted

and by

the ~

AxA a

with

The projection from "

projection

and

transform of

y

from

respectively.

S * (A×A a) Aa

to

31

Definition 2.3.

(i)

]~.n S*

g A def

(TT- ly~ ~) . (A×A a)

Aa oo

(ii)

~A

d~f Y iy, ~ A

"

Here and in what follows independent of the choice of

~ ~

are independent of the choice of as X.

~

when

A has the form

denotes

$~ @ g . ~ .

Since

~

is

, the definitions introduced above iV{. Hence we usually choose

T~X

~yl X

for a non-singular submanifold of

We now begin to explain how to obtain the concrete expression of the germ of ~ A" The corresponding result for ~ ~A is obtained by combining the results given below and the reasoning of S-K-K [17] Chap. II, §1.4, so we will state the result (Theorem 2.12), leaving the detailed arguments to the reader. See also Laurent [14]. We begin our discussion with proving several vanishing theorems needed for the concrete calculation of the relative cohomology groups introduced above. Proposition 2.4. and let

A

denote

each fiber of manifold.

Let

Y

be a non-singular hypersurface of

T~X.

U ÷ y(U)

Let

U

be an open subset of

is contractible and that

¥(U)

A

X = ~n such that

is a Stein

Then we have oo

(2.1)

Hj (U; g y [ x ) - 0

Proof.

Since

V d~f y(U)

(j~0) is Stein, there exists a fundamental system

of Stein open neighborhoods WkCX of V. Then W k n(X-Y) is also Stein. Denote by j the inclusion map from X-Y into X and let denote the sheaf j,j-i ~X" Denote ~ / @X by ~ . Then it follows from the definition that

(2.z)

oo x 2 -~-19; ~ y- 1 ~x" CyI

On the other hand, we have

(2.3)

HJ(Wk;JD)

= HJ(Wk-Y; O'X) -- 0

32 for

j ~i.

(2.4)

Hence we have

Hj(wk;~)

= 0

(j ~ i).

Therefore we find

(2.S) for

HJcu;C~I x) = !.~HJ(Wk;~O0" X) = 0 j ~I.

Q.E.D.

By a quantized contact transformation we can easily deduce the following corollary from Proposition 2.4. Corollary 2.5.

Let

a simple helonomic

A

be a Lagrangian submanifold of

~ - M o d u l e with support

A-T~X, we can find a neighborhood

W

of

A. p

T*X

and

Then for any

p

in

which satisfies the

following: For each open subset (2.6)

y(U)

U

of

W

satisfying the conditions

is a Stein manifold

and (2.7)

each fiber of

U + y(U)

is contractible,

we find HJ(u;J~ ~) = 0

for

j~0.

Next we deduce the following vanishing theorem from Corollary 2.5. In what follows, T~0}~n × (T~0}~n)a ~ T~0}~2n ~ ~2n

~{O}[¢n 0

~O}[~n

~ C =

and a point in

_, _2n ~{0}~

{o}I by

~2n"

X denotes ~n, V and ~ denotes

We denote a point in

(0,0;~,q).

(Theorem 2.6) denotes

~2n

Let us take a point

by p

(x,y) in

& a a v - T*X×x(XxX ) = {(0,0; ~,q); ~=rq~0}. By a linear transformation we may assume that p is (0,0; ~0,_~0) with ~0 = (0,...,0,i). Define

V~

by

{(0,0; ~,q) 6V;

$~+q~ ~ 0}

be a sufficiently small neighborhood of (2.6) and (2.7).

p

for

~=l,...,n

and let

U

which satisfies conditions

Then it follows from Corollary 2.5 that

HJ(UnVvI~ holds for

j~0

and

"''nVuk, ~ )

i £ v I <.-'< ~k £ n

is an open covering of (2.8)

= 0

U-(A a ~ V ) ,

H j~(U;~°°) = 0 A=

for

(0 £ k £ n).

Since

{U~Vz}

we obtain by a theorem of Leray

j > n

and n

H n ( U ; H ~) : ~ ( u n Aa

(2.9)

n

fAVz)I( [ ~'=(U~(t-~V~)). ~=I j =i ~#j

On the other hand, Theorem 1.2.2 of [8] Chap. I, §2 asserts that ~J holds for that

X×X, a v

( ~ X × X , ~ =) = 0

j < codimT,(XxX )(AaNV)

proj dim ~ .

It is easy to see

codimT,(Xxx)(Aaf~v) = 3n, proj dim H = 2n and

¢-Xxx,Aa (IV( E x x x ' A

--

"

Thus we obtain the following Theorem 2.6.

)~J (]f~) = 0 Aa

(j#n).

We next try to find a symbol sequence corresponding to an element ~-oo

in

"~A.p First

(peA). we n o t e t h a t n

n

2=1 holds by (2.9).

Let

P

c o r r e s p o n d i n g e l e m e n t in

j =r oo

be i n ~ A , p.

~j

Let

(~ ~P~(Dx'Dy))~(x'Y) 3 for sufficiently small

~ f ~ ( U n ~ V%) 2=1 which s a t i s f i e s c o n d i t i o n s ( 2 . 6 ) and ( 2 . 7 ) . we d e n o t e by 6 ( x , y ) t h e g e n e r a t o r o f A/'.

be a U

Here and in what f o l l o w s I t f o i l o w s from t h e

34

definition that

pj(~,~)

the following conditions

is homogeneous of degree

j

and satisfies

(2.10) and (2.11): n

(2.10)

e > 0

For each

and each compact subset

there exists a constant

C e,K

K

of

such that

sup[Pjl __< Ce,KeJ/j[ K

(j >

U ~(/AV.%) g=l

O)

holds. n

(2.11)

For each compact subset constant

RK

K

of

such that

U r~( r~ V.%), there exists a g=l

(j

suplpjl
<

O)

K holds. By the Laurent expansion, we have

(2.12)

n -c~g-I Pj(~,n) = a~zn PJ'c~(~).%=111(~.%+n.%)

It is then easy to see that, if we define pj,e(~) al, . . . , a £ _ ~g
~q(.%)(D)8belongs to

the first that

Pj(~,n)

1>0

[

aezn

pj

by

n -~£-i II (~g+q.%) , .%=1

d~(Un((3

Vk) ).

Hence we may assume f r o m

has the form

n

(2.13)

q~.%)(~,n)

-(~.%-i

(~) II (C.%+n.%) '~ g= i

Here we note that pj,~(~) is homogeneous of degree j+la[+n and defined on a neighborhood of ~0. Furthermore Cauchy's integral formula combined with (2.10) and (2.11) entails the following: There exists a neighborhood W of ~0 = (0,...,0,i) on which pj's are defined and satisfy the following conditions: (2.14)

For each such that

~ > 0

and

K > 0

there exists a constant

C

35 suplpj

a]

W

(2.15)

For each

'

K > 0

~JKI~I/j!

< C =

sup]pj ~] < (-j)!K]~]R -j ,

> o) =

there exists a constant

W

(j

g'~

=

K

R

K

such that

(j < 0).

On the other hand, we have the following formula: n

(2.16)

~=IE(Dx +Dy )

-1-a~

Hence, as a s e c t i o n of j~Z

xa n _1~ (x,y) ~(x,y) = ~ ~I(Dx~+DY~ ) "

n A/'°°(Ur~(raVe)),

n -a~-i pj,~(Dx) __121(Dx+I)y~.) 6(x,y)

is equal to n

~ j6Z ~.~" PJ '~ (Dx)X~ Z=I(Dx +Dy ) -1~ (x,y). Furthermore we have n

(Dxz+Dy~) k=l ~ (D+D~kJk) - 1~ ( x ' Y ) - k ~ (DXkyk+D ) - 1 6(x,y)e~(uN

(2.17)

(k~zVk))

and we find n

(2.18)

(x~-y~) II (Dv +D ) - l ~ ( x , y ) k=l

~k

Hence, as an element of

= 0.

Jk 0

~-co

H (U;~A),

n Wdef k~ l(Dxk+Dyk)-l~(x,y) satisfies the system of differential equations

{ Hence

w

(Dx +Dy )w = 0

(~=i,

. . . ,n)

(x~-y~)w = 0

(~=i,

... ,n).

can be identified with

~(x-y).

Therefore

w

can be

identified with the identity operator, and hence, J~ p~(Dx,Dy)~(x,y ) J

36

can be identified with

Now, if we define qj,i(g,x) g0

( ~ jEZ

qj,i(~,x)

is a holomorphic

and a polynomial

~-F p j ~(Dx)X ).

in

by

I~I =i ~

Pj_i_n,~(~)x ~, then

function defined on a neighborhood

x, where

~

is independent of

It is homogeneous of degree i in x and of degree j Further, they satisfy the following growth conditions: (2.19)

For each

e > 0

and each compact set

exists a constant

C

(2.20)

j ~ i.

For each

s > 0

and

in

of j.

~.

K

of

~ x Cnx' there

K

in

~ × Cn

such that

CE,K Ez+3 i! (j-i) [

s~ piqj'il ~ holds if

g,K

i

~

and each compact set

there

X'

exists a constant

R

g,K

suplqj i I < (i-~)! K

,

holds if

=

such that giRl- j

i!

~,K

i > j.

Now we introduce the following Definition 2.7.

The symbol sequence of

is, by definition,

P ~(~)

the doubly indexed sequence

(~CA

= T~0}~ n)

{qj,i(~,x)}i~+

which

j~Z satisfies the following conditions: (2.21)

qj i(~ x) ,

is a holomorphic

function defined on

~ x ~n

'

which

X

is homogeneous of degree j in ~ and a polynomial of degree i in x and which satisfies the growth conditions (2.19) and

(2.20). Using this definition, following form:

we can summalize our result in the

37 Theorem 2.8.

An operator

P

in

~A(fl)

determines a symbol sequence

{qj,i(~,x)}iE~+ which satisfies conditions (2.19) and (2.20) in a jam unique manner and each symbol sequence

{qj,i(~,x)}i~+

the above conditions determine an operator by defining it by multiplication by DX is considered. Remark 2.9.

P

satisfying the

jeZ belonging to

~A(fl)

~ qj i(Dx,X). Here qj,i(Dx,X) means that the j,1 ' x should be applied first and next the action by

We often express the symbol sequence of

{Pi,j(x,$)}igZ+, where

Pi,j

~co

P E ~A(~)

is homogeneous of degree

jEZ a homogeneous polynomial of degree is given in this form, we assign

i

in

x.

P E~A(~ )

j

in

as $

and

When the symbol sequence to it by setting

P =

Pi,j(X,Dx), namely, the (micro-)differentiation is applied first i,j and next comes the multiplicaiton by x. In this case {Pi,j(x,$)} is related to

{qi,j(~,x)}

in Theorem 2.8 by

1 ~ Pi,j(x,~) : [ ~ D~Dxqj+ic~i,i+ic~ ] (~,x) C~

and qj,i(~,x )

= [ (-l)c~![c~[D~D ~

xPi+l~l,j+l~l (x,~) •

C~

Furthermore {Pi,j(x,~)} satisfies the same growth conditions that {qj,i(~.x)} satisfies. Hence ~ is closed under the operation of taking the formal adjoint. Remark 2.10. ij ~ Pi,j(x'D) pj(x,~) consistent

that

Let

P = ~ pj(x,D) be in J in ~ A to P, by defining i s homogeneous o f d e g r e e

w i t h t h e embedding

g~l A ~

~X[ A.

Pi,j(x,~) i

restricted

of

to this

~A

coincides

subsheaf,

in

> ~

"

Ring s t r u c t u r e

Then we can assign

with that

x.

to be the part of This assignment

Note a l s o t h a t A"

of

~XI A

namely, the composition

when

it

the is

~ rm,n(X,Dx) m,n

is

38 of two operators

.~.Pi,j(X,Dx) l,j

and k,£

qk z(X'Dx) '

~

in

is given

by the following: (2.22)

rm,n(X, ~ ) = m=i+k_i~l~-T D~Pi,j(x'~)Dxqk,£(x'~)

Remark 2.11. operator

Theorem 2.8 establishes the correspondence between an ~~ ~A

in

An a n a l o g o u s an a f f i n e

result

variety

Let T~¢ n .

and a suitable

Y

of

Cn

be given

by

{x~¢n;

correspondence

sequences

c a n be o b t a i n e d Y

Denote

symbol sequence,

in the

(~d+l,...,~n) between

in the

by

~(a)

~".

}¢n.

A = T~¢ n

with

manner:

(Xl,...,Xd) Then t h e r e

(~ C A)

A = T~O

case where

following

X,d~ f

when

and t h e

= O} exists set

and

A =

a one-to-one

of doubly-indexed

{qi,j(x,~)}i~Z+ which satisfy the following condition

j~ (2.23) and growth conditions (2.19) and (2.20). (2.23)

qi , j(x,~)

is a holomorphic function defined' on

which is homogeneous polynomial

in

of degree

of degree

i

in

j

in

and a homogeneous

(x',[").

In a similar way we can find the characterization of operators ~'A" (Cf. S-K-K [17] Chap. II, §1.4.)

Theorem 2.12.

Let

~(U),

U

where

A

be

T~0}~n

and let

is an open subset of

exists a symbol sequence

i!jPi,j(X,Dx).

{Pi,j(x,~)}i,j~Z

conditions

Conversely, if

(2.24) ~ (2.28), then

P

be an operator in

T*A =~ ~n× x

following conditions (2.24) ~ (2.28) so that

in

[

~ × ~n( x ' , ~ " )

~

.

Then there

which satisfies the P

{Pi,j(x,~)}i,jEZ i,j~Pi,j (X,Dx)

can be expressed as satisfies the defines an operator

g~(~) .

(2.24)

Pi,j(x,~) is a holomorphic function defined on U and it is homogeneous of degree i in x and of degree j in ~.

(2.25)

For each

~ > 0

and each compact subset

K

of

U, there

39 exists

a constant

C C

s~plPi,jl holds (2.26)

for

s > 0

K

holds (2.27)

and each compact

a constant

supl

Pi,j

for

R

subset

K

of

U, there

such that

s,K

I < (i-j)'

i-j

=

i!

i ~ 0

and

j < i.

subset

K

For each compact RK

~,K ~j

j ~ i ~ 0.

For each exists

~

such that

E,K

" RE,K E

i

of

U, there

exists

of

U, there

exists

a constant

such that suplPi,jl

__< (-j)!RKJ

K

holds (2.28)

for

j < i < 0.

For each compact

subset

such that for each

K

E > 0

we can find another

a constant constant

C

RK E,K

so that suplpi,j I < (-i)! K = (j-i)! holds Remark

2.13.

by replacing we obtain (2.29)

for If

i < 0

and

is

T~ n

A

the condition

CE,K

~J-iR~i

j ~ i. with

Y = {x ~n',

Xl ..... Xd=0 }, then

(2.24) with the following

condition

(2.29),

the same result.

Pi,j(x,$)

is a holomorphic

homogeneous degree

j

of degree in

~.

i

in

function

defined

on

U

(Xl,...,Xd,~d+l,...,~n)

and it is and of

40

§3.

Construction

on which

_~ A

and

_~ A

act.

In this section we construct

sheaves

~A

and

~

on which

and

The f i r s t

~'A

the proof

act

respectively.

o f two t h e o r e m s w h i c h r e l a t e

tangential

sphere bundles

The r e s u l t s tion

of the sheaves

obtained

and those

there

Correspondence

on c o t a n g e n t i a l

are effectively

to study basic properties

§3.1.

subsection

between

of

~.

sheaves

is devoted to

t h e c o h o m o l o g y g r o u p s on sphere bundles.

used in the

and

second subsec-

~:.

on sphere bundles

and co-sphere

bundles. Let M

M

be a topological

with fiber dimension

Denote by

S (resp., 3*)

V*), namely, y*) D

denote

S*

T

(resp.,

onto

M).

is also denoted by Theorem denote

=

V-M

3.1.1.

Let

the complex

V

a real vector bundle

be the dual bundle associated

S* = (V*-M)/~ +.

(resp.f V*-M)

~)

class of

to

of

over

V.

with

V (resp.,

Denote by

y (resp.,

S (resp., S*).

(X,Vx,nx) 6(V-M)

the projection

The projection

Let

from

from

D

S

onto

onto

S*

~ (V*-M). M

~

be a complex

of sheaves

(3.1.1)

1RrK(S*;~)

S*.

= mr -1

M' = ~(K).

~(K)

of sheaves

~T,~-I~. Let

Let

K

on

onto

and let

be a closed and

K ° be the polar of

(K°;~)

S

K.

Then we have

[l-d].

Then we have

~RrK(S*;9 ) =IRrK(S*;IRr -1 (M') ( ~ ) )

-I Convexity means that each fiber Kx def K n ~ (x) is convex, ,-1 i.e., y (Kx) is convex. Proper convexity means that y*-l(Kx) (*)

does not contain

a line.

We

(resp.,

(resp.,

• (resp., ~).

convex (*) set of

Set

and

V*

(X,Vx;X,~x) E S ~ S * ; ~ 0}, where

properly

Proof.

and

is the equivalence

denote by from

from

((X,Vx,~x)

space Let

the sphere bundle

S = (V-M)/~ +

the projection

(Vx,~x)

d.

S)

41

and ]RF -i T

(K°;~")

]Rr

Since

= ]Rr(K°;

(M')

~

(,~) = ]R~,~-~r -1 ~T- i ( M ' )

with

"r

(Y)).

]RF -i (M')

(~')

holds, by replacing

M

(i~I')

M', we may assume from the first that

(5.1.2)

~(K) = M.

F u r t h e r m o r e , by c o n s i d e r i n g a f l a b b y r e s o I u t i o n of ~, t h e p r o b l e m t o t h e c a s e where ~- i s a f l a b b y s h e a f .

Let (5.1.3)

f

denote

~Im-l(s,_K)"

we c a n r e d u c e

Then we have

]RF(S*-K; ~ ) =

]RF(S*-K; ] R m ~ - l ~ )

= ]Rr( - I ( s , _ K ) ;

-1~)

= ]RF(S; ]Rf,f-l~).

On the other hand, we also have

(3.1.4) mr(s*;9) =]mr(z-Is,; =

-i~)

]Rr(~-Is; ~-15¢)

-- ]Rr is; 5~). H e n c e , by c o m b i n i n g ( 3 . 1 . 3 ) and ( 3 . 1 . 4 ) w i t h t h e l o n g e x a c t s e q u e n c e of relative c o h o m o l o g y g r o u p s , we o b t a i n t h e f o l l o w i n g t r i a n g l e : ]RFK (S*; ~ )

(S.l.S)

•, r ( s ; ~ )

;1RF(S; ] R f , f - l ~ :) .

42 On the other hand, of

~ f ( ~ )

the following

(cf. S-K-K

triangle

[17] p.270

follows

from the definition

Remark):

(3.1.6) P,I" (S ; ~ ) Therefore

is reduced

to calculate

purpose

we choose

subset

Un

of

-1 ~1~

.

where

it, we appeal a decreasing

S~

so that

of

to a limitting

R~f(~). procedure.

family of open and properly

K = r~ U n n~0

holds.

In

Let

f

For this convex

denote n

Note t h a t

fnl(x

,

U c = S*-U n

First of

fact, Un

to the calculation

(S~_Un)

(3.1.8)

set

>IRF (S; l R f , f - 1 ~ ) .

we find

Thus the problem order

-

x S Vx) = { ( x , V x , n x ) 6 S M

n

suppose

Un.

Then

~

;

(X,~x)

Unc and < V x , n x > =>

~

. that

(X,Vx)

fnl(x,Vx)

is not contained

in

U°n' t h e p o l a r

is a non-empty contractible

it follows

such that

Then,

for any

from the assumption that there exists f~l (X,Vx) < 0. Hence contains n

0}

in

f~l(X,Vx)

and any

t

such that

set.

In

(x,n 0)

in ( X , V x , - ~ 0) .

0 ~ t ~ I,

t~ - (l-t)~ 0 ~ f n l ( x , V x ). In fact,

if

(x, t~-(l-t)n 0)

=(x,tn-(l-t)n0+(l-t)n 0)

(3.1.9)

in this case.

]R ~ v i : f n ( 5 )

in

should be in

Thus we have seen that (X,Vx,-n 0)

were

Un, then U n because

fnl(x,Vx)

(x,t~) of its convexity.

is contractible

Hence we have

(X,Vx) = 0,

if

(X,Vx) ~ Un" °

to the point

'

43 Next we consider In this case, follows

Un

the case where

is contained

from the definition

if and only if

(X,qx)

is open,

fnl(x,Vx)

Therefore

we have

in

that

(X,Vx)

(X,Vx,nx)

is contained

is a closed

is contained

{(X,Vx)}°.

in

is contained

in

{(X,Vx)}°NU~.

annulus,

in

U°'n

On the other hand,

namely,

it

fnl(x,Vx) Since

homotopic

Un

to

S d-2

in

U °.

= ]RF(Sd-2 ÷ {point}; ~(X,Vx)) ~ ~)~,tf (~) n (X'Vx)

(S.l.lO)

= ~(X,Vx ) [l-d] ,

if

(x,v x)

n

Thus we find

(3.1.11)

]R~gfn(~)

= ~ [ 1 - d ] [Un

and hence

(3.1.12)

~F(S; ] R ~ X f ( ~ ) ) n

= ]RF(Un; ~) [ l - d ] ,

Since {Un}ng 0 is a decreasing family of proper convex sets whose limit set K is a closed and proper convex set each of whose fiber o {U n}n~0

not void, whose

is an increasing

limit set is

is flably.

K °.

Therefore

family

On the other hand, {H j (U n0", ~ ) } n ~ 0

of properly

convex

we have assumed

satisfies

the

if

sets

that

(ML)-condition.

Hence we have

(3.1.13)

~im

HJ+l-d(Un;~)

=

HJ+I-d(K°;~).

n

Combining

(3.1.7),

(3.1.12)

and

(3.1.13),

we finally

obtain

~rK(S*;~) = ~ r ( s ; m ~ C f ( ~ ) ) = ~r(K°;3~)

In application, special

structure.

of the cQhomology follows,

we usually It enables

groups

we consider

[l-d].

Q.E.D.

use Theorem

3.1.1 when

us to obtain more

in question,

the problem

concrete

as we show below.

in the following

~

has a expression

In what

geometric

situation

44 Let

N

manifold center

be a submanifold

M.

D e n o t e by

N, i . e . ,

~

of

codimension

:the

(real)

NM = (M-N) ~ SNM.

3.1.2.

Let

~ F S N M ( T - I ~ ). properly

(3.1.14)

Let

~

H{(S*; ~ ) U

ranges

ranges

over the set

CN(Z) n K ° = ~.

~T,~-I~

~,-~

M

•

let

M with

(SNM, S~M,N)

K

~

corre-

by

be a closed and

Then we have

H~+~-d(u;~),

~no

o v e r a s y s t e m o f open n e i g h b o r h o o d

Here

of

the projection

and define

and

S* = S~M.

= li~

of closed

subset

CN(Z)

denotes

of

U

of

such that

~(K)

and

Z

Z n N C~(K)

the normal cone of

Z

and

along

N.

Theorem 3.1.1 asserts

H{(S*;~) Hence

be

transform

We d e n o t e by

be a sheaf on

convex subset of

where

Proof.

7C

of a real analytic

monoidal

f r o m NM t o M. I n w h a t f o l l o w s , t h e t r i p l e t sponds to (S,S*,M) i n Theorem 3 . 1 . 1 . Theorem

d

it suffices

H j -i

= Hj + l - d (K°;~). T-I~(K)

to show that

~(K)

On the other hand,

(K°; ~;) = Ii__~H~(U;~). Z,U it follows

mr -1

(K°;$)

= m r -1

~(K)

T

from the definition

T

of

~

that

(W;z-17¢)' ~ (K)~K °

*%

where

W i s an open s u b s e t

obtain

the following

of

NM s u c h t h a t

W ~SNM = K°.

Hence we

triangle:

mT_ 1

(K° ; [~) ~(K)

(3.l.lS) lira'lRr(W;~c-l~2)W "

where

W

ranges

> 1.~ % I R r ( W - ( K ° ~ T , -1,rr ( K ) ) ; T - I " ~ )

over the set of neighborhoods

of

K°

in

'

such

45

that

W A SNM = K ° .

(3.1.16)

Then we have

linkl~r(W; - i ~ ) W"

: ~F(KO;

Now we note that a fiber of contractible

or

(SNM)x

we obtain the following

K°

according

over a point

as

x

is in

x

~(K)

in

N

is

or not.

Hence

triangle:

k(1

lRr ( N ; K )

(3.i.i7) N r (K ° ; -r- 1 J C )

> N r (N- ~r (K) ; ~ )

Next let us define a subset W-(K ° N z-l~(K))

-I~).

W

of

M-N

= -I(~o(N_~(K)),

by

because

[1 - dl •

T(W-SNM).

Then we have

K°-T-I~(K)

= T-I(N-~(K)

On the other hand, we easily see

]R~j..t ( ~ )

=

~]N[-d].

Therefore we obtain the following

triangle:

~F(W U(N-~(K)) ;jr6)

(3.1.18) mF (W- (K° c~ ~- 1~ (K)) ; Y(~) Comparing

(3.1.15),

following

triangle:

(3.1.16),

(3.1.17)

]RFT- 17r (K)

f

]RF (N; K ) Here we note that

and

(3.1.18), we find the

(K°; ~ )

-.<.

>

•r(N;YC)

the set of neighborhoods we obtain

> 1RF(N- ~ (K) ; Y-6) [ 1- d].

lim.mr((w-sNM ) U(N-~r(K));R;).

= Iim~$F(U;Y~) holds when U ranges over U" of N. Hence, by setting W'=(W-SNM)~(N-~(K)),

46

li__i%mru_~,z,(u; ~ ) U,W

lim~F (U; ~)

> li~F

U "

(U r~W'; ~)

U,W

,

Thus we finally obtain

(3.1,19) Since

RF

U-W'

ZnN = K

§3.2.

~-I~(K)

(K°; ~) = 11_~mrU w,(U;.T<]). U,W

ranges over the set of closed subs@t

and that

Z of

Definition and basic properties of sheaves In §2 we introduced sheaves

submanifold

A

of

U

such that

CN(Z) C K °, (3.1.19) is the required result. Q.E.D.

T*X.

~A

and

CA

and

and

C A.

for a Lagrangian

As we observed there, 8A

of second-microlocalization of operators. to find sheaves

~A

deserves the name

Hence it is natural to try

on which

~A

and

~A

act

respectively, as in the case of (micro-)differential operators. order to do this job, we hereafter assume that tion of a real analytic manifold purely imaginary locus imaginary)

A~

of

M

of dimension

A, i.e.,

Lagrangian submanifold of

A.

We denote

~

~gXY~

by

~.

n

sheaves

and

~x-MOdule

4

follows, we denote by Definition 3.2.1.

of microfunctions.

~A' we start with

(i)

~

Y~

is also a (purely

Throughout this sub-

and

the same procedure employed in defining the sheaf CM

and that the

CA

~M

with

the projection from

by using

of hyper-

In defining new

instead of

¸¸

~<

In parallel with the

discussion in §2, we will introduce sheaves

functions and the sheaf

In

is a complexifica-

AN/=TT*M,

~T*M.

section we choose and fix a simple holonomic support

X

Az~A*

~X" to

A.

In what

47

(ii)

~A def ~ n

Since we can easily prove that a

~oo

~A-MOdule.

T RA( - ~ ] R ) a .

~R

is a

~A-MOdule,

Using the same reasoning as in S-K-K [17] Chap. III,

§i, we can also verify that

~A

is an

gA-MOdule.

In order to study basic properties of several cohomology vanishing theorems. We first show that ~,

is also

A]R

~A

and

~A' we need

is purely n-codimensional with respect to

namely, we prove the following

Theorem 3.2.Z. Proof.

Let

~(j~)~

p

= 0

be a point in

for

A~.

j ~ n.

Since the problem is local in

T~X, we can find a real contact transformation which brings to

(vC-~T~M ' p0)

for a non-singular hypersurface

choose a local coordinate system Xn+/YTyn) .... Yn=0}

N

of

(A~,p)

M.

We may

z = (Zl,...,Zn) = (Xl+~C-lYl,...,

around ~(p0) e x so that M = {Yl ..... Yn=0}, N = {xl=Yl= and p0 = (0;¢C-~dXl) hold. We denote by Y the complex

hypersurface {Zl=0}. By a quantized contact transformation, we can bring ~ to a simple holonomic system whose support is T~X. Hence we may assume without loss of generality that

~

is

[17] Chap~ II, Definition 1.1.4 and Theorem 4.2.5.) 3.1.2 implies the following:

C~] X.

(S-K-K

Then Theorem

(3.2.1.) where

U

ranges over the set of open neighborhoods of

Z ranges over the set of closed subsets of following two conditions: (3.2.2)

Z ~ YC N

(3.2.3)

Cy(Z) ~ ( z = x + ~ y

Here we have identified system on UnZ

X.

~n TyX

~n

0

in

Z

with

X

by using the linear coordinate

Furthermore, by shrinking

is compact

and

Yl > 0) = 4.

U

and replacing

if necessary, we may assume without loss of generality

(3.2.4)

cn

which satisfy the

Z

with

48

Z C (z E c n ; Yl ~ IXl[)"

(3.2.5)

In what follows,

{z ~ n ;

If(z)]

compact set

We now r e c a l l Proposition Assume

for compact

subset

Z

of

Cn, ~

~ max]f] for each entire function f}. Z Z is holomorphically convex, if Z = Z the

3.2.3.

that both

following

Let Z1

Z1

and

result

([10]).

Z2

be compact

and Z2

denotes

We s a y t h a t holds.

subsets

are holomorphically

a

of

Cn.

convex.

Then we

have

H jZ1-Z2(¢n-z 2 ; (Yn) We will prove Let

~

the vanishing

denote

conditions

(3.2.2),

show that

Z

to

Z

belongs

satisfies

is equivalent

(3.2.6)

(Z-M) n Y

For each

(3.2.7)

~

(3.2.3), if

Z

by using subsets

(3.2.4)

belongs

(3.2.2)..

and

to

this result. Z

of

Cn

(3.2.5).

~.

We shall

First we show

Note that the condition

to

= 0.

implies

the following:

~ > 0, there

Z n{z E c n ;

Here and in what

of (3.2.1)

the condition

(3.2.2)

This condition

j ~ n.

the family of compact

satisfying that

for

= 0

exists

]y,] & ¢,

follows,

6

> 0

]Zl] ~ 6 )

z', x'

and

y'

such that

= 0. denote

(z2,...,Zn) ,

(x2,...,Xn) and (y2,...,yn) , respectively. Let z 0 be a point in Y-M. Define an open subset of

¢

by

~-{w=u+C~v~;

an entire

function

(3.2.s)

f ( z °) ~

v ~ c}.

f(z)

(z ~¢n)

We will which

R (c > 0) c show that there exists

satisfies

C

and

(3.2.9) Since

f(z)~R Rc

C

.

is a simply

connected

open subset

of

~, R c

is a Runge

49 domain.

Hence

there

exists

an entire

function

~(w)

(w ~ ~)

which

satisfies

(3.2.10)

~(f(zO)) g 0

Re

and (3.2.11)

Re 9(f(z))

Therefore

< 0

the existence

the condition

if

of such

f(z)

f(z).

f(z)

by

Z

We may assume without

satisfies loss of

(b > 0).

z I - /UTa(z2-x~) 2, where

which we shall specify First let us consider denote

that

that Im z 0 = (0,b,0,...,0)

Define

will entail

(3.2.2).

Now we try to find such generality

z gZ.

later.

Choose

the case where

Re(-cCTa(z2-x~)2 )

and

a

6

is a positive

as in (3.2.7)

IY'] ~ ~ = b/2.

Im(-/Z-la(z2-x~)2),

constant

for

Let

e = b/2. g

and

respectively.

we have g 2 = 4a2(x2 _x~)2 Y22 --< 4a2e2(x2-x~) 2 and

h = a(y~-(x2-x~)2 It then f o l l o w s (3.2.12)

h +

If there were

(3.2.13)

that 2 g < as 2 4a~2 = z

I

0 2

) < ae2-a(x2-x2 ) .

in

Z

such that

f(z) ~ Rc, then we should have

xl+g=0

Yl+h=t

with

t ~ c.

In view of (3.2.5),

we should

then conclude

h Then

50

(3.2.14) Combining

Igl

÷

h ~

(3.2.12)

IgJ

c

and

g

(3.2.14), we should obtain

2 ~¸ 2 >

c

as 2

-

4a~ and hence (3.2.15)

g2-4a~Zlgl+4ac~2-4a2~4

If we choose

c = ab 2, f(z 0) ~ R c.

c-2aE 2 = ab2-2a(b)

This implies that

(3.2.15)

of

< E})

c, f(Z n { l Y ' l

have fixed (= b/2).

a

Z

z c{zecn;Iz

zI I-~- - v ~ t l

> 0

is not satisfied.

Hence,

R c.

for this choice

Note that we need not

Next let us consider the case where there exists a constant

K

IY'] ~ such that

] £ K}.

In this case it follows from

holds

2 - ab 2 2

is compact,

<__ 0.

On the other hand, we have

is c o n t a i n e d in

so far.

Since

(3.2.16)

= (Igl-2aE2)2+4a~2(c-2a~2)

(3.2.5)

and

(3.2.7)

that

6 > --~-~ = Ha

for every positive

t.

+ ( K + [2x ~nl )

a (>0)

so that

< b2

£,

/2a

Hence if we choose

and C

~fa hold,

then

f(z)/a

sufficiently is

small

contained

condition

is

in

a > 0, R . c

in

f(z)/a

Rb2. is

Thus we h a v e

Then

contained

verified

it

is

in

that

obvious

Rb2. Z

that

Hence

satisfies

(3.2.3)

is

Z

satisfies

equivalent

to

the the

condition

following:

(3.2.3).

for f(z)

the

(3.2.2).

N e x t we s h o w t h a t condition

contained

The

51

For any

s > 0, there exists

~

such that S

(3.2.17)

Then it is easy to see that hence

(3.2.3).

Z

Z

satisfies

~0 dSf {Z ~ ~ ; Z = Z}

in (3.2.1).

satisfies

It is also clear that

and (3.2.5), if with

IZll ~ ~s}.

Z C{zeX; Yl ~ SlXll} u {z ~ X;

them•

Z

the condition satisfies

and

(3.2.4)

Therefore we may replace

when we consider the inductive

Next we shall show that,

for each

~ > 0

in 90' we can find a compact and holomorphically such that (3.2.18)

(3.2.17),

conditions

limit

and each

convex set

Z1 z2

0 ~ Z2

and (3.2.19)

Z1

Z 2 C{z ~¢n;

iz l < 4s} d~f Us"

If this is the case, we have (3.2.20)

link H31nu(U; Z 1 ,I] = link

Hj

Zl'~

=

%n )

c.

Zl~U~(z~ ) (u ~ z 2,

link 2 %n). Zl,~ H{ i- Z2 ( U n Z c;

Here z2C denotes {n-z 2. the excision theorem that

holds

for

%n )

j

Then it follows from Proposition

~ n.

Now we embark on the construction

of required

the set

{w=u+fi-Tv6~; v <

lul-B}

• .+Z2n).

Here

are sufficiently

~

and

B

and let

B < 2ac 2.

Z 2.

¢~(z) denote small positive

which we will specify later under the constraint (3.2.21)

3.2.3 and

Let

K denote

Zl-#TT~(z~+.. constants

52

We now define

Z2

may assume that

by Z1

~-I(K) ~ Z I.

Since

is contained in

Z1

is compact and since we

{z = x+ff-lY E ~n; 2y I =< [Xl]),

Z IN {IZl] ~ E } - U E is contained in Z 2 if we choose ~ and B sufficiently small. In view of (3.2.7) we can also conclude that Zln{[y']~}-U s

is contained in

Z 2.

Hence it suffices to show that

Z~ d~f Z1 n{]x'[~3~, [y'[~E, ]Zl]~E) - U is contained in Z 2. In order to see this, we have only to verify that Im ~(z) ~ IRe~(z) l-B holds on

Zi, namely,

(3.2.22)

yl+a(ly'I2-1x'I 2) ~ IXl+2~<x',y'>I-B

holds on if

Z½.

(3.2.23)

Since

Yl ~ ]x2I/2

ZI, (3.2.22) is satisfied

~ ( l y ' [ 2 - l x ' 1 2 + 2 l < x ' , y ' > l ) ~ -B

holds. Since ]x'[ ~ 3~ and from the constraint (3.2.21). Corollary 3.2.4. Proof.

holds on

[y'[ ~ c

The sheaf

~A

hold on

ZI, (3.2.23) follows Q.E.D.

is flabby.

Since the flabby dimension of ~

is equal to or at most

n,

this immediately follows from the theorem. By a

reasoning similar to the proof of Theorem 3.2.2, we can prove

Theorem 3.2.5.

There exists an injective

~x-homomorphlsm from

CMI A to ~A" More precisely, if we choose an ~x-linear injective homomorphism ~ from fl~ into CM, then there exists ~ such that the composition

~°~: ~4~÷ ~ A coincides with the composition of the

canonical homomorphism from

~

to A/L~

and that from 7~ ~

to

~A"

Probably the same argument will work for the proof of the following conjecture.

However, so far, we have been unable to prove

the conjecture completely.

(The point which we cannot prove at the

moment is the vanishing of

}5

= ~jdxj

vanishes.)

A( ~ lfl~

(j ~ n)

at

p

where

53 Conjecture 3.2.6.

(i)

~J,

AOT-I.~ IR) = 0

(ii)

0

> /~IR

(j~n)

> ~A

>0

is exact. In what follows, for a section supp s~ (f)

by

SS~"Af.

f

of the sheaf

~A' we denote

54

§4. Asymptotic expansion of solutions of holonomic systems with regular singularities. In this section we clarify the meaning of the asymptotic expansion of a microfunction solution of a holonomic system with aid of sheaves

~..

and

~A"

solution of a holonomic system with

R.S., we find a micro-differential satisfies.

equation of a special type which it

(Theorem 4.1.1 and Proposition 4.1.6 in ~4.1.)

~A

by the

Our procedure is as follows:

First, for a (microfunction)

employ

R.S. (*)

Next we

to bring each equation of that type to the simple and

canonical equation whose

~i-solution is easy to describe.

Using this

result we clarify the meaning of the asymptotic expansion of a microfunction solution of the equation in question with respect to a Lagrangian manifold A, namely, we grasp the notion of the asymptotic expansion of a microfunction by regarding it as a section of the sheaf

~4.1.

Holonomic systems with

R.S.

and indicial polynomials.

The purpose of this subsection is to prove Theorem 4.1.1 stated below.

The structure of the equation

(b(~)-P)u = 0

appearing there

will be investigated in the next sybsection by the aid of the sheaf ~A"

Before stating the theorem, we prepare some notations. Let

A

be a Lagrangian submanifold of

defining Ideal of i and by ~A Let ~A denote the sub-Ring of ~A(m) by ~A I X (m) (= is locally isomorphic to

T*X.

Denote by

IA

the

the Ideal {P ~ E X(1)'oI(P)IA = 0}. ~X generated by ~A and define

Ix(m) ~ A ). As we noted in §I, [A / [A(-I) ~ A ' which is a sub-Ring of the sheaf of

linear differential operators on

A.

Now let us consider a micro-differential operator which satisfies the conditions below. Here ~i and tively the first order and the zeroth order term of

(4.1.1)

~' ~ ~'A

(4.1.2)

d ~i ~ CX (= ~ j d x j ) J

~ = %1 + ~0 +... ~0 are respec%.

mod IA21

(*) R.S. is the abbreviation of "regular singularities". Kashiwa~a-Kawai [8], [9] for the definition.

See

58

(4.1.3)

1 ~ a2 ~i = ~0 - 2 j ~ 0

Here

denotes the sheaf of holomorphic

~I

easy to verify

that such

on

g

A.

always

~ A2 ~ X (-I)" ' For example, if d

1-forms

exists

A = r~ n

on

T~X.

It is then

and it is unique modulo

with

Y = {x~gn;xl=...=Xd=O},

then we may choose

}( ~ (x~D~+D~x~)) as ~. In what follows, w e J J J J j=l always use the letter {} to denote such an operator. Now we have the following Theorem R.S.

4.1.1.

Let

and let

there exist ator

A

u

be an arbitrary

a non-zero

P ~ ~A(-I)

(4.1.4)

deg b ~

(4.1.s)

(b(~)-p)u

follows

of order

Lemma

t=0}

tD t.

Let

and

Let

A~

the following

~

4.1.1,

denote

X

be

a non-zero

Ex-MOdule

A~.

p

If

of

b(s)

of

= ~4 , b ( @ ) ~ '

if it is the case,

Hence,

J,,'

~(m)~.

R.S.

@

which

~

denote is defined

be a coherent in

b(@)~

V, then there C a~(-l).

to show the existence

C ~ ' (-i) exists

and

@~'

an integer

of a co-

polynomial

b(@)

C ~'

hold.

r

such that

On the other hand,

-'b(@)~' = b ( O - J ) D t J d ~ ' = DtJ

by setting

V = {(t,x;z,g)E

,44 and a non-zero

then there

holds.

with

such that

First we note that it suffices

holds.

In

operators

and the projec-

Let

is contained

~(0)-sub-Module

b(@-j)~ ' (-j)

lemmas.

denotes

x ECn},

A, and let

Suppj~

polynomial

~'(-r)Cd~(-l)

~(m)

t=g=0, z # 0}.

Proof.

8~'

several

T*X-T~X

gl+n = { ( t , x ) ; t ~ g ,

of a point of

~,

respectively

herent fact,

Then oper-

X.

be a holonomic

(0)-sub-Module

such that

T*X.

conditions:

we prepare

8(0)-Module

A = {(t,x;z,g) ~ T ~ X ;

on a neighborhood exists

of

and a micro-differential

the sheaf of micro-differential

For an

onto

and

Theorem

denotes

m.

T*X }*X

4.1.2.

T'X;

submanifold

with

= 0.

at most

tion from

b(s)

Ix-NOdule

order P

g(m)

The symbols

satisfy

of a holonomic

Lagrangian

polynomial

which

In order to prove what

be a section

b(O-r+l).-'b(@)

C. ~ ' ( - j - l ) to be

b(@),

we obtain

In

56

b(O)~'

C

d~'(-r).

Now we is with

R.S.

prove

and

singularities M

Therefore

shall

since

along

M

0

/~

of

(M-exp(2~ - - ~ x ) ) N ~ V, there

~ of

~

as an

such that

sense

exist

exists

~'

z~.

~(-I) and

in

§i, of

V,

~

Since has

Corollary

[ii].

holds.

b.

regular

5.1.7.)

Then,

/~ Let

by the defini-

Since

End(/~) is finite-dimen/~j (M-X j ) Nj ~ j such that Nj. Hence we may assume from

and some integer

= 0

and that

Chap.V,

in the of

~ ~C

of such

sum of sub-Modules

X. e ~ x

the first that t~ere

C d~'(-r) is contained

([8]

~

is a direct

for some

along

Supp/~

is an endomorphism

sional,

b(O)~ existence

V~*X.

be the monodromy

tion,

the

~E ¢

holds. a coherent

@~,C~.

N~£+ /v~

such that has regular

~(0)-sub-Module

Let

~(0)-Module.

and Since

Ul,...,u m

such that

/~

be a system of generators

Then there exist

@ui = ZPijuj"

~

singularities

If we denote by

P.. ~ ~(0) (l~i,j ~m) z3 t the column vector (Ul,

u

...,Um) and by P the matrix (Pij), we have @u = Pu. We know ([II]) that there exists U E G L ( m ; ~(0)) and a matrix A(x) of functions in x

such that

@-P = U(@-A(x))U -I.

we may assume

that

nition

that

of

M

@u = A(x)u = 0

= 0.

the problem

integer

c

holds,

It-X] < c}

for any

b0(s)

and

by

x

c ~ (s-v)

and

G(x)(exp(2~/Ti-A(x))-exp(2~fTTX))

G(x)

and

ru

are commutative.

=b0(A(x))

ru

b0(O)r~ C ~(-1).

Lemma

Let

4.1.3

j4~

~x-SUb-Module

a coherent

with

A(x)

- exp(2~/z]-x))ru

x0, we can find an are contained Let

z~

~(0)-sub-Module

I b0(t) (t-A(x)) ~ dt. I .+1 (exp (2~/5Tt) -exp ( 2~ £5i-~) 2 ~ - ~ I~- ~ ,:~ 1

= b0(A(x))

holds,

and, moreover,

= 0. Q.E.D.

~v-Module

such that of

in

us now define

Hence we obtain

be a coherent of

u-lu,

from the defi-

= G(x)r(exp(2~/Z-l-A(x))-exp(2~/-~))ru

This e n t a i l s

herent

of

in the neighborhood.

Then

b0(@)

of

u

Since

(exp(2~vi-iA(x))

on a neighborhood

~=-c

A(x)

it follows

holds.

we have

so that all the eigenvalues

{tEe;

G(x)

by replacing

Then

Mu = exp(2~ - ~ A ( x ) ) u

(M-exp(2~/U-f%))ru Considering

Hence,

holds.

~ @~

~

and let

= ~

which

be a co-

~

holds.

is generated

Let

~

by

~.

be

Then we have the following:

(i)

~,(~(k)/~(k-l)) what ators

follows of order

= a~>k~/~k_1~-~

a~ k

denotes

at most

holds

for

k >> 0.

Here and in oper-

the sheaf of linear differential

k. o

(ii)

~k~ for

= {u~Y~f; k >> 0.

1 @u

is contained

in

~(k)

on

T'X}

holds

57

(m)

If a section

u

of

,~4 belongs

then there

exists

such that

Pu6d~m+k_ 1

to

d~(k)

a linear differential and

em(P)(p)

at a point operator

P

p

of

~*X,

of order

m

/ 0.

Proof. There e x i s t i n t e g e r s . rl,r2,N0,N1,N 2 sequence

(r 1 ~ r 2 )

and an exact

0 + /1~ ~ ~ "NO Po ~,N1 P1 ~-N2 N1

N2

such that ~ ( ~ N0) = ~ , P 0 ~ k )C (O~k_rl)N0, P l ( ~ k _ r 2 ) k > 0 and that

is exact

for

k ~ r 2.

Then the following

NO 8(0)

h

~(_rl )

is also exact.

Since

N1 P ~J/~

c~NI

k_rl

for

sequence

N2 [(-r2 )

is the cokernel

of

P0' we obtain

the exact

sequence

~ ( k ) + 6(k)

0 ÷ for each

k.

Hence

0 + ~(k)/~(k-1) is also exact. of holomorphic with respect (4.1.7)

£ ( k - r 1)

+

its symbol +

sequence

functions

on

that

~*X

groups

for

k >> 0

for

of coherent

sheaves

~,(~(k)/~,(k-l))

= ~-k

with a sufficiently

~ £9-(k), the sheaf

are homogeneous theorem

of degree

NO

for

~/~k-I large

k

sequence

on the vanishing

on a projective

÷ ~,(~(k) holds

(k-l)

Then the following

o

0 + ~,(~(k)/d~(k-l))

k ~ k0

which

by Serre's

Since ~ , ( ~ ( k ) ) ~ ~_k/~-k_l and (4.1.7), we obtain

(4.1.8)

6(k)/~

to the fiber coordinate.

o

(4.1.7)

NO

( & ( k ) / ~(k-1)) N0 + (C(k-rl)/ g(k-rl-l)) N 1

Here we recall

is exact

cohomology

NO

of

space. NI)

o

) ÷ ~,((9~(k-r I) k ~ 0, by comparing

. (4.1.6)

y k 0.

This proves

(i).

Next we

58

shall prove 1 @u

(ii)

belongs

such that

~4~

o

for

to

k ~ k 0.

o~(k)[~ X contains

~(~(j)/~(j-l)).

u.

Thus

proves

be a section

u.

If

j

k ~ k 0.

of

~

such that

Then there

is larger

than

exists

k, 1 @ u

j

1 @u

Hence

the induction

on

a

on

homogeneous

T~X

porves

to

degree

of

implies

that o ~ j _ l ~

that

~k

S

denote

r

Nullstellensatz

(ii)

implies Let

is contained

in

Then there

V

symbol

P~U~k+rv_l

~"

and

A

~x-MOdule

is zero

~

contains

contains

u.

This

with

a non-zero

P

which

(4.1.9)

= tPu,

order P & deg b

that there

exists

1 @u

on a neighborhood

Lemma

4.1.3 that there m

a.

Let

r

Then it follows

belongs

to

This proves

R.S.

operator for

(iii).

of

Assume A.

that

Let

u

b('s)

poly-

be the

from

~(k+rv-l)

v>>0. Q.E.D.

and let

Supp(~

@/~4)

be a section

of

and a linear differ-

the following:

loss of generality

of

~ @~[T~X"

a non-zero

holds

a homogeneous

a[s = 0.

be a linear differential is

where

P E ~ A"

We may assume without

be the section

exists

and

polynomial

satisfy

and

the set of points

be the same as in Lemma 4.1.2

on a neighborhood

ential operator

Proof.

P

PVu

exist

b(tDt)u

¢ 0

that

X, V

be a holonomic

Then there a(p)

and let

whose principal

4.1.4.

Let

~(k).

a

of order Hence

~i0-

such that

Hilbert's

where

j

let us prove

does not belong

nomial

Lemma

(4.1.8)

(ii).

Finally

order

u

J

in

A~.

Let

for some

of

polynomial p ~A,

exists

that

Then Lemma b(s)

where

A4 = d~u. 4.1.2

such that

@ =tD t.

Hence

a linear differential

Let

guarantees b(@)~E~(-l)@

it follows

operator

P

from

of

such that

(4.1.10)

~m(P) (p) ~ 0

(4.1.11)

Pb(@)u

By the condition

= Qu

with

(4.1.10)

Q ~m-l"

we may assume

that

P

has the form

m-i Dtm _ j !0 Aj (t ,X,Dx)DtJ, where

Aj

is of order at most

m-j.

Let

O. be written

in the form

59 m-1 j!oQj where

(t'X'Dx)DJt'

Qj

is of order

• -.(s+m-l),

we obtain

at most from

Then, s e t t i n g

m-l-j.

b(s) = b ( s ) s ( s + l )

(4.1.11)

m-I

b(@)u = tmDtb(@)u = t ( ~ (Ajb(@-m+l)+Qj)tm-lDJ)u. j=0 This proves Lemma

the required

4.1.5.

Let

X

and

be an arbitrary

holonomic

/lq.

exists

Then there

ferential

(4.1.12) where

A

a non-zero

P

which

order P < deg b

and

For each positive

to

be the Same as in Lemma

O~x-Module

with

R.S.

polynomial

satisfy

4.1.2.

and

b(s)

u

Let

/~

a section

and a linear

of

dif-

the following:

b ( t D t ) u = tPu,

Proof. X

operator

Q.E.D.

result.

X

defined

is an isomorphism (tN,x;z/NtN-l,~). subvariety Therefore

of

by

integer

fN(t,x)

N

we denote

= (tN,x).

by

fN

T'X,

and hence

CA(W )C { t = 0 } C T A ( T * X ) .

an integer

N

the map from

Then the associated

on T ' X - {t=0}, because fN,(t,x;T,C) Let W be Supp( ~ @/t q). Then W

there exist

tN-l~ N ~

P ~ ~A"

and a constant

fN*

= is a Lagrangian ([12],

C

map

Chap. X.)

such that

c(It~]+t~]) N,

or equivalently, (tT) N < ct(It~l+l~l)

N

Therefore

we have

(4.1.13)

(t~/N) N < c t N ( I t T / N ] + I C J ) N

on

-l(w n {t/o}) W'd~f(fN.)

The inequality

(4.1.13)

implies

In the sequel,

we fix such an

that

N

and set

f = fN"

Let

/4'

be

f*(M)

@0 and let R.S.

u

be the section

Further

satisfies exist

l@u

Supp( ~ @ /~')

all the conditions

a polynomial

tP(t,x,tDt,Dx)~ the primitive

b(s)

of

A4'.

Then

is contained

in

in the preceding

and

holds with

P ~

~A

degb>order

/~'

Hence

lemma. Hence,

Yq'

Therefore

such that P .

is also with

V~W'

b(tDt)~

if we "denote by

N-th root of i, we obtain

N-I . = ~1 .~ i t P ( ~ t , x , t D t , D x ) ~ " i=0 N-I . It is easy to see that N1 i=~0cltP(et,x,tDt,Dx)

can be written

form

tNQ(tN,x,tDt,Dx ), because

under

tion

t ~ et.

(4.1.14)

there

=

b(tDt)~

b(NtDt)u

Therefore

it is invariant

(4.1.14)

implies

in the

the transforma-

that

= tQ(t,x,NtDt,Dx)U

holds

outside

Hence

it follows

{t = 0}.

Here we note that

from Hilbert's

order Q =

Nullstellensatz

order F £ d e g b

.

that there exists

such that (4.1.15)

trb(N@)u

By multiplying b(O)u with

b(s)

= tr+IQ(t,x,N@,Dx)U.

the both

= @+l)...(s+r)b(Ns).

we can bring

Theorem

Y

Then we may assume

that

SuppA4

5.1.4.)

Proposition A

4.1.6.

of

obtain

~

([8]

Theorem

4.1.1

Let

A4

easily

exist

a system of generators

b(s)

and micro-differential

which

satisfy

the following

follows

Ul,...,u m three

A

with

([8] Chap.

of

~x-MOdule ~,

of

T*X.

with

transforto

T~X

1.6.4.) R.S., ~I, Theorem Q.E.D. 4.1.1. R.S.

and

Then there

a non-zero

Pij ~ EA (-I)

conditions:

V,

4.1.5.

from Theorem

submanifold

operators

and

~x-Module

from Lemma

be a holonomic

Lagrangian

position

position. follows

contact

I, §6, Corollary

is a holonomic

proposition

be an arbitrary

Chap.

Q.E.D.

the proof.

By a suitable

to a generic

X.

is in a generic

Therefore

The following

let

r Dt, we finally

by

This completes 4.1.1.

Supp ~

for a hypersurface because

of (4.1.15)

= t(O+2) -.,(@+r+l)Q(t,x,N~,Dx)U

Now let us prove mation,

sides

polynomial

(l~i,j ~m)

61

(4.1.16)

b ( ~ ) u i = ~ Pijuj

(4.1.17)

degb~

( 4 . i .18)

The difference

order P.. ij of any two different

roots of

b(s)

= 0

is

not an integer. Proof.

It follows

(0)-sub-Module say, r

from Theorem 4.1.1 that there exist a coherent ~

of

M

and a non-zero polynomial

b(s)

of degree,

which satisfy -

(4.1.19)

b(

C

and (4.i.20) We now show that we can choose additional

condition

(4.1.18).

~

and

b

so that they satisfy the

For this purpose,

we prove the follow-

ing : If (4.1.20),

b(s)

has the form

b(s)(s-X)

and satisfies

then there exists a coherent

~(0)-sub-Module

(4.1.19)

and

~'

which

satisfies (4.1.21)

(~-X)I](~-I) ~j' C 7 r ~' A+ l ~

We will prove this fact by showing

~' = ~ A Z + ( ~ - x - l ) ( ~ t is a required one. Furthermore contained

This

(-i).

that

(l)) ~'

is clearly

we see by an easy computation

in

(~ - ~ ) ~

}~+I~.

coherent that

E(0)-Module.

b(~-I)(~(i))

Hence we have

C ~'(-l)-

Therefore we obtain [~(~-l)~'¢b(~-i)~ This implies

A~

+b(~-l)(~(1))

C ~A

r+l~ . --

is

62

+2 ( _ 1 ) Z + ~ [ # + 1 ( ~

( ~ - l ) b ( ~ . - l ) J ~ ' C( ~-~)~[ Ar+l Z C ~ A Thus we obtain

§4.2.

the required

Transforming The purpose

cal one

equations

S-K-K

the nature

[17] Chap.

--~)~ -A fore the structure of tion and it enables

is different

II, §5).

acts on ~A ~A-solutions

us to clarify

sion of a microfunction the example

transformation

(4.1.5)

by using

of the reduction

the characteristic

Still

re-examine

is to find the canonical

of the form

in this subsection

cussed here changes question.

one.

does not act as a sheaf homomorphism and hence

(e.g.

Q.E.D.

to the canonical

Of this section

less to say, ~ A microfunctions, form discussed

result.

equations

micro-differential

Ar+l.2°. -~ ' (- 1).

_X)~C~

solution.

form of the ~.

on the sheaf of to the canonical

from that of the classi-

Actually variety

the reduction

of the equation

as a sheaf homomorphism. can be investigated

the meaning Before

(1.6) discussed

Need-

disin There-

by our reduc-

of the asymptotic

stating

expan-

our main theorem,

in ~i from the view point

we

of the

of the equations.

We first consider the following (y2-x3)S+ as its solution:

1

1

(yXDx+~yyDy-S)U

system of equations

which

admits

= 0

(4.2.1) (2yDx+3X2Dy)U Choosing

= 0

XDx+yDy+l

as

g, we obtain

the following

equation

(4.2.1). (4.2.2)

(8.-3s-3/2) ( ~ - 2 s - l ) u

For simplicity admits

we assume

an asymptotic oo

(4.2.3) where

u ~ uj

and

uj

expansion

s+i/2

is not an integer.

of the following

Then

u

form:

co

~ u.+ ~ v. j=0 J j=0 J '

respectively.

(4.2.4)

that

= x3D2yU/4.

vj By

=

are homogeneous (1.6)

of degree

in §I, we find

r(-s+I/2) j!F(-s+I/2+j)

~i 3_2,j xt~x Uyj u 0

2s+j

and

3s+I/2+j,

from

63

F(s+312) _v.1 3~2,j Uyj v0,

(4.2.5)

vj = j!F(s+3/2+j)^L#X

namely, (4.2.6)

u~

(x3/2Dy/2)s+i/2F(-s+i/2)I_s_i/2(x3/2Dy)Uo +(x3/2Dy/2)-s-I/2F(S+3/2)Is+i/2(x3/2Dy)VO,

where Iv(z ) is the v-th modified Bessel function. define v by (~ -5s/2-5/4)u, then we obtain

D with

A =

Further,

if we

v0

(z/2)s÷l/2r(-s+l/2)I_s_l/2(z),

B = (z/2)-s-1/2r(s÷3/2)Is÷l/2(z), c = (z/2)s+3/Zr(-s+l/2)z

and

'

-s-1/2 (

z)

D = (z/2)-s+i/2F(s+3/2)l 's+I/2(z)

where

z = x3/2D

Y

.

Then Lommel's

u0

formula for Bessel functions

entails

u

Hence this transformation

reduces the equation

(4.2.2) to

{

(~-2s-l)u 0 = 0

(4.2.9)

( ~ - 3 s - 3 / 2 ) v 0 = 0.

This example shows that, essentially in

our

sense

is

H e n c e we w a n t tions.

The

to

nothing have

but

the

a general

following

theorem

speaking,

transformation result

gives

us

on the

the asymptotic of

the

equation

transformation

a satisfactory

expansion

result

by

s5A"

of

euqa-

in

this

direction.

Theorem 4.2.1. t=0}.

Let

Assume that

A

be

T ~ l+n, where

A = (a v(x))iZ~,v~ N

Y = {(t,x)=(t,Xl,...,Xn)~ cl+n; and

Q = (Q~v(X,Dx, D t ) ) l ~ , v ~ N

64

satisfy m~

the follwoing

conditions ,(4.2.10) ~ (4.2.14)

for some integers

(I ! ~ ! N):

(4.2.10)

aUv(x)

(4.2.11)

a~v = 0

(4.2.12)

is holomorphic if

of the origin.

m -mv+l < 0.

If we denote by

k!4(x)

( l ! ~ _ < N)

matrix

X~(x)

is holomorphic

A, then

zero and

(4.2.13)

on a neighborhood

X~(0)-Xv(0)~£-{0}

the eigenvalues

of the

on a neighborhood

of

(I__<~, v_<_N).

Quve ~ A U - m v + 2 ( - l ) .

Then on a neighborhood

invertible matrix which s a t i s f i e s (4.2.14)

of

{(t,x;z,~);

R(X,Dx,D t )

t=x=0,

~=0)

there

exists

each o f whose e n t r y b e l o n g s to

(tDt-A(x)+Q(X,Dx,Dt))R(X,Dx,D

an

~)A

and

t) = R(X,Dx,D t) (tDt-A(x)).

Furthermore R(X,Dx,D t) can be expressed in the form ~RP(x,Dx)DtP, = P where RP(x,Dx) = 0 for p < 0, R 0 I, the identity matrix and order R p ~ a p Corollary Lagrangian

holds

4.2.2.

Let

submanifold

any two different

roots

micro-differential ~A(-I).

for

p > 0 b(s) of

operator

4.2.2. that

= tDt+I/2.

= 0

of degree

further

m

m

and

A

a

that the difference

is not an integer.

of order at most

in

(b(~)-P)u

By a suitable A = T ~ l+n

Furthermore,

operators

assume without

obtain

b(s)

Assume

-my).

Let

which belongs

P

of

be a

to

Then we have

tion we may assume

(m-l)

a = 2+max(m

be a polynomial T*X.

of

Proof of Corollary

ential

with

(S-K-K

quantized

with

by the division

[17] Chap.

loss of generality

theorem

II, §2, Theorem that

P

contact transformaEl+n; t=0} and

Y = {(t,x)

for micro-differ-

2.2.2),

is a polynomial

we may of degree

m-i t, i.e., P = =~0Pj(X'Dx'Dt ) ~ J. Then rewriting the equation J = 0 in the matrix form, we can apply Theorem 4.2.1 to

the required

result.

Q.E.D.

65 Proof of Theorem 4.2.1.

We try to find an invertible matrix

R

in the

following form: co

(4.2.16) where

R =

~ RP (X,Dx) DtP , p=0

RP(x,Dx)

order and

is a matrix of linea¥ differential

R 0 = I, the identity matrix.

this form, then its inverse can be constructed We expand

(4.2.17)

Q

Q =

in

in a similar manner.

differential

operators.

Note

order QJv_< m -mv+j+l. (4.2.14)

it suffices to define

sively by the following recursion pRP-[RP,A]

Let us now define an N2

(p~l)

succes-

=

in view of (4.2.18) and (4.2.19), of length

Rp

formula:

~ QJR k (p__>l). j+k=p j>l Here we note that order R P < ap (p > 0) holds with

RP

R

~ QJ(x,Dx)Dt j , j~l

In order to satisfy

(4.2.19)

of finite

also in this form, namely,

where QJ = (Q~v) is a m a t r i x of l i n e a r that (4.2.13) implies (4.2.18)

operators

If we can construct

by

N ×N

if

Rp

matrix

(av~(y))l
a = 2+max(mv-m

exists. A*(y) and

)

~'~

and a column vector

t(~Pl(x,y,D x) .....

rPN(x'y'Dx)' rPl(x'y'Dx) ..... rPl(x'y-Dx ) ..... ~PNN(X,y,Dx)), respectively. Then, instead of solving (4.2.19), we solve the following equation.

(4.2.20)

(p- (IN OA*!y)-A(x ) 0 IN))RP(x,y,Dx) (QJ(x,D x) @ IN)RP(x,y,Dx), j+k=p j__>l

where

A(x) ® IN

etc. denote

In fact, rewriting

the Kronecker product of the matrices.

rP~v'(x'y'D)6(x-y)A

we can find the required

Rp = (rP)

in the form

r p~v(x,Dx)6(x-y),

which satisfies

(4.2.19)

from the

solution RP of (4.2.20). If we denote I N @ A * ( y ) - A ( x ) @ IN by ~ , then the assumption (4.2.12) guarantees that (p-O~) -I exists for p = 1,2, ,'' . dummy variable

In order to facilitate Xn+ 1

and define an

the argument, we introduce a N2 x N 2

matrix

C

by

66

( ml (DXn+l)

.

IN @

)

0

".. 0

mN "

(Dxn+ 1

) -

-,-l+2n+l.

Then C i s i n v e r t i b l e on ~ = { ( t , x , y , X n + l ; ¢ , ~ , ~, ~ n+l ) ~ T • ~n+l/0}. Hence ~3(p) d~f C - I ( P - 0 I ) - I c is well-defined there. Furt h e r m o r e , by u s i n g the a s s u m p t i o n ( 4 . 2 . 1 1 ) , we can p r o v e t h a t ~(p) has t h e form b

(4.2.21)

~ C~(p) ~=0

where each satisfies

entry

C ~ (p)

of

CZ(p)

is of order at most

~

and it

N~(C~v(p);6) _< A/p ~+1. Here

N~

Kr6e

[2].

Defining

denotes

the formal

See Tahara sP

by

norm introduced

[18] Lemma

1.2.8

in Boutet

de Monvel

and

for the proof of this assertion

C-IR p, we can rewrite

(4.2.20)

into the following

form:

Sp :

~(p)

~ (C-I(Qj @IN)C)sk. j+k=p j~l

Thus we can w r i t e down Sp in terms o f ~(p) and C-I(Qj @ IN)C explicitly. Then, u s i n g the a s s u m p t i o n ( 4 . 2 . 1 3 ) and t h e d e c o m p o s i t i o n ( 4 . 2 . 2 1 ) , we can v e r i f y t h a t t h e i - t h homogeneous p a r t sP(x,y,~l ' g n + l ) o f t h e symbol s e q u e n c e o f Sp s a t i s f i e s t h e f o l l o w i n g : (4.2.22)

For each exist

e >0

and each compact

constants

CE, K

and

ME,K.

subset

K

of

~, there

such that

s u p l s ~ I < M .C p K c l p ! / i ! holds. Note also that (4.2.23)

s~ = 0, if

i < (b+2)p. oo

From the estimate

(4.2.22),

we easily

find that

R =

-p ~ RP(x,Dx)D t p=0

@7 Noo

belongs to

Remark 4.2.3. of

Q.E.D.

MN (=~A) . Let

A

be the same as in Theorem 4.2.1.

(4.2.23), the operator

[17] Chap. II, ~3), where

R

(and

R -I

X = £1+n.

also) acts on

A

Then in view

Cyl x

(S-K-K

On the other hand, it is easy to

verify that

m.x J (,,l,f o, Cyi x) --- <mx-J (dx. o, Oil x) N

holds for

(j : o,1)

N

J~0 = ~ X / ~x(tDt -A(x))" Hence the comparison theorem of N this type also holds for ~ = ~ x / ~ )N ( t D)t - A (' x ) +-Q ( X 'AD x ' D t

Remark 4.2.4. R

Because of the form (4.2.16) of the intertwining operator

constructed above, we find that

R

and its inverse

S

actually

belong to a smaller class of operators introduced by Laurent namely, R

and

S

belong to

2~

M N ( ~ VIA), where

[14],

V = {(t,x;~,~) ~ r~¢ l+n"

~;=0}. In order to apply Theorem 4.2.1 to the study of microfunction solution of the equation in question, we prepare the following Lemma 4.2.5. In what follows, A ~ denotes the purely imaginary locus of A, which is supposed to be Lagrangian. Lemma 4.2.5. ~A-solution (4.2.24)

Let A be the same as in Theorem 4.2.1. o f the following equation:

Let

U

be a

(tDt-X(x))u = 0

Then we have u = F(-X(x))(t+/---f0) X(x) ~(x) = 2~exp(-~cC~k(x)/2) (Dt//q--~)-k(x)-l~(t)~(x) Proof.

By applying

(Dr/d--T)-X(x)-I = exp (- (k (x) +l) log (Dt/ -/cY))

(4.2.24), we may replace (4.2.25)

(4.2.24) by

tu = 0

On the other hand, we have the following exact sequence: (4.2.26)

0 + ~-lOy

with ~ ( x ) ~ 3 n.

÷ Cy~Ix t

C~IX ÷ 0,

to

68

where

~

~

j

is the projection

T~n ~n+l

(4.2.26)

This

from

T~X

to

Y.

Since

(~-1 (~cn) = ~{JRnl((~cn )

implies

means

that

u = 6(t)

~(x)

holds

with

~(x) E

Q.E.D.

An"

The following theorem is an immediate consequence

of Theorem 4.2.1

and Lemma 4.2.5. Theorem 4.2.6. simplicity, U

Assume the same conditions

assume further that

be a microfunction

A(x)

solution of

as in Theorem 4.2.1.

is a constant matrix

(tDt-A+Q(X,Dx,Dt))U

has the following expression as a section of

(-Xl) (t+/~-r°).L1

= 0.

A.

For

Let

Then

U

~A:

91! x)

s

U = R

r(-~d)(t+/cY0) where

R~MN(~A)

A, Lj

has the form

fl

, kl,...,Xd

log(t+ -/710)

1

with

mj

tion in

kd

Ld

\gN(X)/,

are the mutually different eigenvalues

(log(t+/TT0))2/2!

being the multiplicity

of

... (log(t+ - ~ 0 ) ) m.-i 3 /(mj -i)!I m.-2 ... (log(t+/~i-0)) J /(mj-2)!

log(t+/710)

kj

and

~t(x)

is a hyperfunc-

x.

Remark 4.2.7. has the form

If

A(x)

is not supposed to be a constant matrix,

Rr ( - A ( x ) ) (t +/:-tO) A(x) ¢ (x)

of

U

69

with

}(x)6 ~ N

~n"

Corollary 4.2.8.

Assume the same conditions

Then for each microfunction = 0, there exist (~ = I,...,N)

u =

R e~

solution

A

u

(~ = I,...,N)

which satisfy

b(~)~

as in Corollary 4.2.2.

of the equation

(b(~)-P)u

and hyperfunctions

= 0

so that

N X R~ ~=I

holds as section of ~^. In particular, {(t,x)~ ~l+n;t=0}, then u has the form

if

A = T~{ l+n

with

Y =

I

d j -i 1 [ [ R (X,Dx,Dt) (t+ -/TT0) V(log(t+/zT0))J ~=i j=0 'J where

R ,j ~ ~ A '

b(l ) = 0

and

jv

is the multiplicity

of the root

Iv. These results show that the sheaf the asymptotic

~A

is fitted for the study of

expansion of a microfunction which satisfies

micro-differential equation. Note that can be formally rewritten in the form

a suitable

R(X,Dx,Dt)((f (x)(t+ - ~ 0 ) X )

co

~j (x) (t+/71-0) ~+j ,

j=0 if

A=Tv*~ l+n~

and

R

is the operator constructed

in Theorem 4.2.1.

In what follows, we call the expression

~ ~ ,~(R E ~ A) given ~= expanslo~ with respect to A

in Corollary 4.2.8 to be the asymptotic of the microfunction

solution

u

of the equation

(b(~)-P)u = 0.

Note

that Proposition 4.1.6 guarantees that such an equation always exists if u satisfies a holonomic system with R.S. Now, as far as we start from a single equation Corollary 4.2.8 is the best possible one of the sort. practical problems with holonomic

equations

systems.

systems on

A.

In such circumstances, For example,

in

as we mentioned

= YDxW = 0.

in §i,

is controlled again by

in the case of the example

(4.2.1) discussed at the beginning of this subsection, satisfy the following equati,ons beside (XDx/3+yDy/2-s)w

However,

of this type usually appear in connection

the top term of the asymptotic expansion holonomic

(b(~)-P)u = 0,

(4.2.9):

u0

and

v0

70 This situation is best formulated by regarding the original holonomic system as O~A-MOdule. In connection with such a formulation we first prepare some notions related to the Levi condition of the system. (Cf. Monteiro-Fernandes [16].) In what follows, A is a Lagrangian submanifold of T*X and ~A and JA(1) denote and {fe OT~X(1); f]k = 0}, respectively.

Definition 4.2.9.

(i)

~X(1);~I(P)]A= 0}

A d~f ~(J~/ (JAk-l+ JAk+l (-1)) (=

(ii)

{pc

~(JA(1)k/JA(1)k-l~T,X(1))).

Let d~ be a coherent 8x-Module and ~ its coherent ~(0)-subModule such that ~ = ~X ~ holds. Then Chi(d~) is, by definition, a subvariety of TA(T*X), the normal bundle along A, which is given by

Remark 4.2.10. The variety C h A ( ~ ) is independent of the choice of ~. We conjecture that it is an involutory subvariety of T*A---TA(T*X ) Remark 4.2.11. Denote by ~k the map from J k to JA(1)k/JA(1)k-I~]~T,X(1), which is induced from the isomorphism between Jk/(fik-l+Jk+l(-l)) and JA(1)k/JA(1)k-I~T,X(I ). Then, f o r a s y s t e m of the form

~X/fi,

ChA(~ ) = @A/(@k(/kA~S) k

)-

Now we begin the discussion on the structure of a holonomic system with R.S. regarded as a ~ - M o d u l e . In order to simplify the descrip tion, we will consider the case where X = gl+n = {(t,x);t e g, x e gn} and

A = {(t,x;T,~)e T'X; t=0, 6=0, T#0}.

x-Module with

R.S.

Take a coherent

Let

Jr{

be a holonomic

gA-sub-Module

J~

of

/~

such

that [XJJ/'=JI~. If we set J~(-l) = 6(-I)]~", then jY'/]~(-I) is a coherent ~k-Module. Here )t{k = ~k/ ~k(-l). By identifying ~k with the sheaf of homogeneous linear differential operators on A, we can regard J~/]~(-l) as a system of differential equations on A, i.e., x-space. If X = ~A u for a section is nothing but the following equations: L(P)u = 0

for

P £ ~A

such that

u

Pu = 0.

of ~v~, then

~//~-i)

71 As we have shown with

in ~I (Theorem

is a holonomic

system

R.S. Now it follows

~(0)-sub-Module say,

1.5), Jr/¢~(-l)

r

which

from Propositin

~

of J/4

satisfy

the following

that there polynomial

exist

a coherent

b(s)

of degree,

two conditions:

C ::+1(-1)~_~

(4.2.27)

b(tDt)~

(4.2.28)

The difference

of any roots

Take a system of generators having

of

Ul,...,u m

% column vector

4.1.6

and a non-zero

b(s)=0 of

~n

(tDt)~0Dx~...DxnUj,

a n range over all non-negative

is not a non-.zero integer.

M~

and denote by

as components,

integers

satisfying

u

where

the

~0

the constraint

n v~0~v~
and

l~j

£m.

it is easy to see that

Let u

N

be the length

satisfies

of the vector

u.

Then

the equation

tDtu = Au + Qu, where

A

is a constant

tial operators

that the eigenvalues consider

the system

sponding

vector whose

morphism

from

denote holds

~

of

of

~.

A

holds prove

N.

Q

is a matrix in Theorem

are the roots

components

of

f

J~

of )

generate

Thenitis

~

of linear 4.2.1.

differenHere we note

b(s)

= 0.

Let us now

and let

~

be the corre-

f

be the homo-

~4~.

Let

defined by assigning

and let

for the characteristic

is at most

and

the conditions

,4~ = ~ N / ~ N ( t D t _ A _ Q onto

the kernel

components

matrix

satisfying

be the

u

to

~(0)-Module

~.

Let

~'

generated

easy to see that

b(tDt)LCI~+iZ(-l)~

polynomial

of

The Artin-Rees

theorem

b(s)

A. Note that

can be modified

by

deg

to show that

for k => k~u _ ~if we take k 0 sufficiently large. We can easily that b ( t D ~ ) ~ K Z C --~ k+N+±~"~ (-I), and hence we obtain ok I

]~(tDt)(:ke:~')

C:A k+N`le

Then, by the same reasoning ators

. .. , v ~ )

v I .... ,v~

satisfies

of

(-I):~/~'

used above,

/:~AA~'

=:A

we can find a system

such that the row vector

of generv = (v 1 ,

72 tDtv = Av + Qv, where A is a constant matrix whose eigenvalues are the roots of b(s) = 0 (and hence those of b(s) = 0) and Q satisfies the conditions in Theorem 4.2.1.

Then, if we write

v = P(X,Dx,Dt)u

with

P E M(N,N; ~) which does not contain t, then we have (tDt-A-Q)P = p'(tDt-A-Q) with P'~M(N,N; ~). By comparing the coefficients of t, we can easily verify that P' = P. Hence we obtain

(tDt-A-Q)P

=

P(tDt-A-Q)

It follows from T h e o r e m 4 . 2 . 1 and

U(X,Dx,Dt) ~ G L ( N ; ~ )

that there exist

U(X,Dx,Dt)EGL(N;~A)

which satisfy

tDt-A- Q = U(tDt-A)U -1 and tDt_~_ ~ = ~ (tDt-A)U ~ ~-I . Hence we have

(tDt-A)(U-1pu) Set

= (~-Ipu)(tDt-A).

T = -'-U-Ipu=~TJ(x,Dx)D+j.~ J

(4.2.29) holds.

jT j = ATD-TDA Note that the difference

are not a non-zero = 0 x.

for

j ~ 0.

On the other hand,

U -I co

G j(x,Dx)Dt-"J

and

(4.2.30)

for

p0 = T

T and

some

and

Hence

of

A

U

and that of

(4.2.29) implies

is a linear differential have respectively

rj

operator

in

the form

.

~ H j(x,Dx)Dt J j =0

j =0 orderHJ < aj then we have

of any eigenvalues

integer by (4.2.28).

Therefore

oo

Since

Then

a.

Hence,

with

G9

=

H 0

= I

and

order G j

oo

if we write

P =

_

,

.

~ PJ(x,Dx)DtJ, j=0

order PJ ~ aj.

is an ~x-Module generated by u with the relation Pu = = ~ )~ is generated by 1 @ u with the same relation (tDt-A-Q)u 0, h A ~X

Hence,

~

if we take another generator

Wd~ f U - i(i @u)

of

~ A~ ~X

/~,

We

73 find (4.2.31)

(tDt-A)w = Tw = 0.

Hence

@ #(

generated by a generator identify

~A

~A-Module = ~pu

byusing

in

variety of

~A ~

u.

of

~.

Let

be a coherent

z~

4//,i/(-1)

in

C h A ( ~ ).

~-~4odule with

of ~

form a holonomic

is contained

~A-HOdule in

and let

(4.2.28)

and

with

R.S.

in (t,x)-variables.

~

A

enable us to t r a n s -

equations

theory.

we sometimes

However,

by its initial

properties

in terms of microfunctions

of and/or

in order to study the structure

need more precise

of the S-matrix.

information

For example,

use of "no sprout assumption"

theorem.

to

of the S-matrix.

is most neatly expressed

makes essential

R.S.

that any

expansions.

on the analyticity

support

with

It also asserts

It is now commonly accepted that the analyticity

microanalyticity

whose character-

ChA(~().

solution of such a system is determined

Discussion

factorization

theorem for

R.S.

the condition

system of micro-differential

equation

terms in its asymptotic

the S-matrix,

holds.

~'A NA ~ (W'/A/(-1))

=

a differential

essential

= ~

such that there exists a

This theorem implies t h a t o p e r a t o r s in

the S-matrix

the

~ A v ~/~'

Thus we obtain the following

which satisfies

is a holonomic

~ A gA,M

microfunction

~

[14]), we can easily show the characteristic

be a holonomic

b(s)

Here we

Let us take as ~/~(-i)

&-Module

Then we have

istic variety

§5.

(4.2.31).

Then by using the fact that

~(0)-sub-Module

non-zero polynomial b(~)HC/r/(-l).

is the

Remark 4.2.11 and an invertibility

(Laurent

is contained

Theorem 4.2.12.

(ii)

with the relation

with a sub-Module

generated by

Furthermore,

(i)

w

~

and (4.2.30) we can easily verify that

operators

/If

~ ~~ A ~ @ A~ , where

has the form

of

than the

Iagolnitzer-Stapp

[4]

in the study of pole-

This condition cannot be described in terms of

the micro-analyticity. The purpose of this section is to discuss how such a delicate

74 analyticity problem is related to the theory expounded so far, and how it is related to the holonomic character of the S-matrix. (See KawaiStapp [13] for the discussion on the holonomic character of the smatrix.) The theory of double-microlocalization with respect to an involutory manifold, which Laurent [14], [15] is now developing, is also very useful for this purpose. Now let A be r ~ n with y = {(Zl,Z,) ~ ~n; Zl=0 } and let V be {(z,~)E T ~ n ; C2 =...=~n =0)- Let L and W denote their purely imaginary loci. For a microfunction f, SS2f (Kashiwara-Laurent [i0]) and S~Lf are, by definition, subsets of TN(/~-IT*IRn) and TL(-¢C-IT*IRn), r~spectively. (Cf. the end of §3.2. Note that - ~ T ~ n and TL(-¢~T~n) can be identified by the Hamiltonian map H, because L is Lagrangian. Actually H (/~Tdxj) = -~TT~/~gj and H ( / ~ d g j ) = -~/~x. hold.) Using the coordinate system we denote a point in J n TW( - ~ T ~ R ) and a point in TL(/UTT~Rn) respectively by (x, ¢C-T~I; n n ¢C-Tj!2cj~/~j) and (x', -¢~l;/2-T(Cl~/~Xl + !2cj~/~j), where x,~ 1 J and cj are real. Note that the subset of TL(C~T~Rn) defined by w = 0 is identified with TW(-¢~T~iRn) iL. Let us now consider a microfunction f supported by L+ def {(x'~)~ ¢C-T(T~IRn-T~n~n); Xl--0' ~i >0' ~' = (~2 ..... ~n ) = 0). Then (5.1)

2 SSw(f) c { ( x , ~ I ;

c')~Tw(CC~T*IRn);

c'=0}

is equivant to the assertion that the d@fining function F(z) of the microfunction f is holomorphic on {zE ~n, izi 0), namely, f satisfies the no sprout assumption. (In the condition (5.1) c' = (c2,...,c n) is identified with Zcj~/~j.) Next consider the following condition

(5.2) ~L(f) C ((x',F-i-~l; c) ETL(-/-~T*~Rn); c=0}. Then, on the supposition that Conjecture 3.2.6 (ii) in §3.2 is valid, (5.2) is equivalent to the assertion that f belongs to M R ( = ~ R @ A 4 ) for a simple holonomic system ~ supported by A. In particular, the defining function F(z) of f can be analytically continued to the universal covering space of g-Y, where ~ is a neighborhood of 0 E~n. Hence the condition (5.2) is close to the holonomic character of f. However, if f is holonomic, we find another important property of f, namely, the finite determination property. This property is not implied by (5.2). The simplest example of f that satisfies

?5

condition

(5.2) but that is not holonomic is given by

Thus the sheaf

~A

(Xl+VCi-0)x2

supplies us with a link between the no sprout

assumption and the holonomic character of the function in question. Note also that SS2w(f) ~ SSL(f )

(5.3)

holds for a microfunction in Corollary 4.2.2.

f

which satisfies an equation dealt with

(See Remark 4.2.4.)

It is also noteworthy that

(5.3) is valid for an arbitrary holonomic microfunction necessarily supported by holonomic ~x-MOdule with R.S. so that

holds.

~,

L+).

f

(i.e., not

In fact, we know that, for each

we can find a holonomic

~x-Module

~reg

([8] Chap. V, §2, Theorem 5.2.1.

this colloquium.) Since fact implies (5.3).

~Xl A

See also our report [9] in ~~ 2~ is a subsheaf of ~ A and ~ V IA'this

We end this report by mentioning a fact which is a generalization of the no sprout assumption and will probably turn out to be useful in application. Let

f

be a hyperfunction defined on

trum is contained i n a p r o p e r l y with

convex cone

Rn

whose singularity

C={/~I-~v~n;

spec

~(g)>0}

~ being a real-calued real analytic homogeneous function such that

d~ never vanishes on V d~f { ~ n - { 0 } ; ~ (~)=0~" Let cone of C, i.e., { y ~ n ; > 0 ( ~ E C ) } . Suppose n ((x,~:-l~; c) ETv(CC-TT~]Rn); ~(~)=0, ~ c~/~j ~0}. j=l ~ function F(z) of f is holomorphic on { z ~ n IIm

z I <~(Re z),

Im z ~ F}, where

R n.

E

is a positive-valued function on

F be the~ dual that SS~f C Then the defining

References [i]

Barbasch, D. and D. A. Vogan: To appear.

The local structure of characters.

[2]

Boutet de Monvel, L. and P. Kr~e; Pseudo-dif~rential operators and Gevrey classes. Ann. Inst. Fourier, Grenoble, 17, 295-323 (1967).

[3]

Hormander, L. : Fourier integral operators. 79-183 (1971)

I.

Acta Math.,

127

76

[4]

Iagolnitzer, D. and H. P. Stapp: The pole-factorization theorem in S-matrix theory. Commun. math. Phys., 5_~7, 1-30 (1977).

[s]

Jeanquartier, P. : D~veloppement asymptotique de la distribution de Dirac attach~e ~ une fonction analytique. C. R. Acad. Sci. Paris, S~rie A, 271, i159-i161 (1970).

[6]

Kashiwara, M. : On the holonomic systems of linear differential equations. II. Inverntiones math., 4_99,121-135 (1978).

[7]

Kashiwara, M. and T. Kawai: ~licro-hyperbolic pseudo-differential operators I. J. Math. Soc. Japan, 27, 359-404 (1975).

[8]

: On holonomic systems of micro-differential equations. III - - Systems with regular s i n g u l a r i t i e s - To appear. (RIMS preprint No.293.)

[9]

: The theory of holonomic systems with regular singularities and its relevance to physical problems. This proceedings.

[I0] Kashiwara, M. and Y. Laurent: Th~or~mes d'annulation et double microlocalisation. To appear. [ii] Kashiwara, M. and T. Oshima: Systems of differential equations with regular singularities and their boundary value problems. Ann. of Math., 106, 145-200 (1977). [12] Kashiwara, M. and P. Schapira: Math., 142, 1-55 (1979).

Micro-hyperbolic systems.

Acta

[13] Kawai, T. and H. P. Stapp: Discontinuity formula and Sato's conjecture. Publ. RIMS,Kyoto Univ., 12 Suppl., 155-232 (1977). [14] Laurent, Y.: Double microlocalisation et probl~me de Cauchy dans le domaine complex. Proc. of the conference at St. Cast, 1979. [15]

:

The report in this proceedings.

[16] Monteiro-Fernandes, T. : Paris-Nord (1978).

Th~se de 3~me cycle ~ l'Universit~,

[17] Sato, M., T. Kawai and M. Kashiwara (referred to as S-K-K [17]): Microfunctions and pseudo-differential equations. Lecture Notes in Math. No.287, pp. 265-529, Berlin-HeidelbergNew York: Springer, 1973. [18] Tahara, H. : Fuchsian type equations and Fuchsian hyperbolic equations. To appear in Japanese J. Math.

.DEUXIEME rIICROLOCALISATION

YVES LAURENT Universit~ Paris-Sud, tlath~matiques 91405 ORSAY

Introduction : Dans [13] , Sato,KawaT et Kashiwara ont c o n s t r u i t sur le f i b r ~ cotangent T*

X ~ une v a r i ~ t ~ a n a l y t i q u e complexes.= X , le faisceau

m i c r o d i f f ~ r e n t i e l s ( q u ' i l s notent les op~rateurs d i f f ~ r e n t i e l s T*

sur

X

~X des op~rateurs

La premi6re p r o p r i ~ t ~ de ce faisceau est que

non c a r a c t ~ r i s t i q u e s dans une d i r e c t i o n de

X , sont i n v e r s i b l e s au voisinage de c e t t e d i r e c t i o n dans (~X " Nous construisons sur

le faisceau x*

~).

T*(T*

X)

(ou p l u t 6 t sur un q u o t i e n t de cet espace)

~52 des op~rateurs " 2 - m i c r o d i f f ~ r e n t i e l s " .

Nous dirons qu'un vecteur

est non m i c r o c a r a c t ~ r i s t i q u e pour un syst~me d'6quations m i c r o d i f f 6 r e n t i e l l e s

ce syst~me est i n v e r s i b l e au voisinage de f~rentielles. par Bony

[I]

x*

si

comme syst~me d'~quations 2 - m i c r o d i f -

On retrouve a i n s i la notion de vecteur m i c r o c a r a c t ~ r i s t i q u e d ~ f i n i e pour un op~rateur puis par Kashiwara-Schapira

En f a i t ,

[8]

pour un syst~me.

nous construirons toute une f a m i l l e de faisceaux semblables ~ .~2

qui permettrons de d 6 f i n i r pour un syst6me d'~quations m i c r o d i f f ~ r e n t i e l l e s

la no-

t i o n de "vecteur m i c r o c a r a c t 6 r i s t i q u e de type ( r , s ) " o~ (r,s); est un couple de rationnels. En p a r t i c u l i e r ,

pour

s = 1 et

t i o n de L~vi pour un syst~me de

r = ~ on retrouve la d ~ f i n i t i o n de la Condi-

T . Monteiro

[9]

Dans le cas g~n~ral nous d~montrons un th~or~me de Cauchy pour les syst~mes que nous r e l i o n s d'une p a r t au Probl~me de Cauchy pour les fonctions holomorphes ramifi~es ( g ~ n ~ r a l i s a n t Hamada -Leray-Wagschal

[4]

) et d ' a u t r e p a r t au polyg6ne

de Newton d'un op~rateur ou d'un syst~me ( g ~ n ~ r a l i s a n t au cas de ) l u s i e u r s v a r i a b l e s un th~or~me de Ramis [ i i ]

).

§1 . Op~rateurs m i c r o d i f f ~ r e n t i e l s de classe Soient

r

et

s

(r,s)

deux nombres r a t i o n n e l s t e l s que

: I _< s -< r -< +~

78 Si

U est un ouvert c6nique de

classe ( r , s ) sur

T*

~n

= ~n x ~n

U est une s~rie f o r m e l l e

une f o n c t i o n holomorphe sur

, par d ~ f i n i t i o n un symbole de

P(x,~) =

~ J~ j en

U , homoo6ne de degr6

P j ( x , ~ ) , 00

Pj(x,~)

est

~, t e l l e que :

A/ r : s < +~ ~K

compact de

U ,

3C > 0 , V~ > 0

~C

> 0

t.q.

pour

J z Jo

(i)

~j _> o

sup I P j ( x , ~ ) K

< C cJ Ic - c (j!)r

(ii)

Vj < 0

sup [ P j ( x , ~ ) K

~ CKJ[(-j)!] r

K compact de

U

3C > 0

t°q°

(i)

~j ~ 0

sup I P j ( x , ~ ) K

_< Cj

(ii)

Vj < 0

sup I P j ( x , ~ ) K

_< c - J [ ( - j ) !] s

B/ 1_< s < r < +~

C/ Si

r = +~

on suppose que

ne suppose aucune c o n d i t i o n pour Si

y =

(*)

avec

×(x)

1 (j!) r

Pj(x,~) z 0

e t si

s = +~

on

j < 0 .

est un diff~omorphisme de

§n

~ ( y , n ) = e dx'd~' P(x,~' + t M ( x , x ' ) n )

y = ×(x) , dx,d ~, = S -~,-[i . ~

et

on pose :

x = x' ~'= 0

M(x,x ~)

est la matrice d 6 f i n i e par

×(x) - X(X') : M ( x , x ' ) . ( x - x ' ) . On v o i t facilement que si ~(Y,n) est un symbole de classe n = tM(x,x)-I ~ > Si

P(x,~) est un symbole de classe (r,s)

X est une v a r i ~ t ~ a n a l y t i q u e complexe et

op~rateur m i c r o d i f f ~ r e n t i e l de classe de

X d'un symbole de classe

formule

sur l'image de

(,)

(r,s),

(r,s)

U par

(r,s)

sur

U ,

(x,~) +
U un ouvert de

T* X , un

est la donn~e dans chaque carte l o c a l e

tous ces symboles se transformant suivant la

dans les changements de cartes.

Les op~rateurs m i c r o d i f f ~ r e n t i e l s de classe

(r,s)

sur

T* X

forment un

faisceau que nous noterons J~x(r,s). ~rg~_: formels,

~X(~,I)

~X(~, ~)

est le faisceau

est le faisceau

fini

et enfin ~(1,1)

dre

infini

~X

est le faisceau ~

( s u i v a n t les d ~ f i n i t i o n s de

~X

des op~rateurs m i c r o d i f f 6 r e n t i e l s

des op~rateurs m i c r o d i f f ~ r e n t i e l s d ' o r d r e des op~rateurs m i c r o d i f f ~ r e n t i e l s d ' o r [13]

,

[5]

,

[7]

).

79 La formule

P.Q(x,~) =

1 Z n ~ T D~ P(x,~)D~ Q(x,~)

permet de munir

~x(r,s)

d'une s t r u c t u r e de faisceau d'anffeaux u n i t a i r e s . Pour not~

r = ~

et

P op~rateur de classe

(~,,s) , le symbole p r i n c i p a l de

~(P) , est le terme non nul de plus haut degr~ de

morphe homog~ne sur

P ,

P , c ' e s t une f o n c t i o n holo-

T* X .

On peut r ~ c r i r e

pour ces faisceaux le c h a p i t r e I I de

le r o l e des op~rateurs d'ordre f i n i

et~y(r,s)

pour

[13]

1 ~ r ~ s

,~z(~,s) jouant

c e l u i des op~rateurs

d'ordre i n f i n i . En p a r t i c u l i e r , ment si

~(P)

si

P ~ ~X(~,s),

il

est i n v e r s i b l e dans ~ ( ~ , s )

est i n v e r s i b l e et pour t o u t transformation canonique homog~ne

: T* X ÷ T* Y

, i l e x i s t e localement au-dessus de

ceaux d'anneaux ~ :

la r e s t r i c t i o n

~

A de la 1-forme canonique de

Si

Aa

feuilles

de

est l ' a n t i p o d a l e de

A × Aa

de

A

qui passent par

T*(X × X)

T* X ne s'annule pas) ou que

T* X). ,

A x Aa

donc est munie d'un f e u i l l e t a g e

lagrangienne de

invo-

A est r ~ g u l i ~ r e

l'ensemble des points oQ la l-forme s ' a n n u l e e s t

une sous-vari~t@ lagrangienne l i s s e de

T* X × X

A une sous-vari@t~ ( l i s s e )

T* X . Dans la s u i t e nous supposerons que

A est maximalement d~g~n~r~e ( i . e .

de

:

X une v a r i ~ t ~ analytique complexe et

l u t i v e homog~ne de (i.e.

~ un isomorphisme de f a i s -

~(r,s)~y(r,s).

§2 . Op@rateurs 2 - m i c r o d i f f ~ r e n t i e l s Soit

si et seule-

est une sous-vari~t~ i n v o l u t i v e

canonique. Soit

A × Aa ~ A

au voisinage de

~

T* X X × ,,

et

~

la r~union des

est une sous-vari~t~ TA ~

est un q u o t i e n t

T*(T*X)×T, XA . Soit

Nun

caract~ristiques l'id~al

~X-mOdule ( ~× = ~ X ( ~ , I ) ) simples sur

A

des symboles est r ~ d u i t et d @ f i n i t codimA (N'~X} .... = Ext ~X

S o i t .N*

ceau des formes d i f f ~ r e n t i e l l e s le p r o d u i t de

N et

N*

(cf

~ gauche coherent ~ un gOn~rateur et

.( N~ ~ X / I A

o0

I

est un ideal de ~X

).

® OX ~®~I

le dual de

N

holomorphes de degr~ maximum sur [13]

gendr~ par les ~l@ments de la forme

ch I I )

•

dont

Soit

L

(

~X est le f a i s X)

et

le sous-module de

f(u) ® v - u ® f(v)

pour

N ~ N* N ~N*

u ~ N , v c N*

enet

f c End(N) ~End(N*). Alors le q u o t i e n t support

~ . (Si

~

de

A est lagrangien

N 8 N*

par

~ = A × Aa

L

est un ~xX-mOdule holon6me de et

N = N ~ N*

). On pose

N(r,s) : ~<(r,s)e~ X N . Soient~ ~ : 7= (~-A)U TA~ + T ~ et y : TA~ canoniques ( ^ ~ e s t muni de la topologie de co~clat~ ( [13]

-

A =P *AKl e s et [7] )

projections et PA~

80

~(O,dim X) le faisceau des formes d i f f ~ r e n t i e l l e s est le fibr6 p r o j e c t i f ) , soit ~Xx X holomorphes de type (O,dim X) sur X x X . ~D~gD_~=!=

L~i~(r's~

~#~

(a application antipodale de TA ~ u~(r,s)

p = codimT* X A ) .

est le faisceau d6fini par :

~2~(r's) et

et

~(O,dim X) ( -1 ~(r,s))a© OX×X UX×X

JT:A-A

(r,s)J A

=

~

¥-Iy, (~(r,s) (r,s)J A ( = ~

] TA A_A) (N(r,s))).

R (ces d~finitions sont analogues a celles d e ~ et ~X de quelles on a remplac~ OX×X par N(r,s) et X par A ) . B~Cg~_:

On montre que

si

[13] et

[7]

dans les-

J # P

~T~( - i N ( r , s ) ) = 0 . A ~g§~gD_~=~=_: ~ ( r , s ) et ~ ~(r,s) sont munis de structures canoniques de faisceaux d'anneaux unitaires et on a un morphisme i n j e c t i f de faisceau d'anneaux : ~-l(~x(r,s)I A)÷ ~(r,s) Si A est ples sur A sont fi~es associ~es ~ que de la vari~t~

fix~e, les modules N a un g~n~rateur et ~ caract~ristiques s i m ~ localement isomorphes (par les transformations canoniques quantil ' i d e n t i t ~ ) donc, ~ isomorphisme local pros, ~ ( r , s ) ne d~pend A.

§3 . Symbole des op~rateurs 2-microdiff~rentiels Dans ce paragraphe, nous supposons que (Xl . . . . . Xn' Yl . . . . . yp, z I . . . . . Zq)

:

X est muni des coordonn~es locales

dans lesquelles

A ={(x,y,z,~,~,~)

E T*X/~=O,z=O}

Si A est une sous-vari~t~ involutive r~guli~re ou maximalement d~g~n~r~e de T*X , alors d'apr~s Oshima [12] , on peut toujours se ramener A c e cas par une transformation canonique homog~ne. Si

Z = {(x,y,z,x',y',z')

Nous prendrons N = CZIX×X ,TAA

~ X×X / y = y ' , z = z' = O}

N = ~X/~xDx I + ... + ~XDXn + ~ Z l +

alors A = TZ X x X.

... +~AZq

et alors

est muni des coordonn~es ( x , y , n , ~ , x * , ~ * ) .

Th~or~me . . . . . . . . . . . . 3.1. .. : Soit U un ouvert de T~ ~ , i l y a bijection entre les ~2~ (r,s) sur U et les s~ries formelles P = ~ sections de ~N (x,y,n,~,x , ,~ , ) o~

Pij

(x ,~*)

est une fonction holomorphe sur et

j

en

(n,~,x*)

U , homog#ne de ( i , j ) E Z=2 degr~ Pij avec les majorations suivantes :

i

en

81 VK compact de

U

~C > 0 Vc > 0 < 0

]C > 0

tels que (en posant

(i)

Vk < 0

(ii)

~k < 0

Vi

0

(iii)

(r = s)

Vk

0

~i < 0

sup [Pi,k+il K

(iii)'

(r > s) ~k

0

Vi < 0

sup IPi k+il .~ c k - i ( - i ) ! / ( k ! ) K

(iv)

(r = s )

Vk

0

Vim 0

sup IPi,k+i I ~ C k + i

(iV)'

(r > s)

Vk

0

Vi ~ 0

sup K

~9~9~_~:~: P = ~ Pij ~(r,s)

et

sup IPi,k+il ~ c - k - i ( ( - k ) ! ) s ( - i ) ! K

C-k c i ( ( _ k ) ! ) s / i !

sup IPi,k+il K

Si

Q = z Qij

k = j - i)

K

c

IPi,k+i I

~ Ck

r

1/(k!)ri!

ci I/(k[) r i!

O sont deux op@rateurs de ~ ( r , s )

Pet

, si

~ Cc~k c - i ( ' i ) ! / ( k ! ) r

R = PQ est le produit de

P et

de symboles Q dans l'anneau

de R est donn@ par

, le symbole R = Z Rij

Rk,P (x ,y ,q,~ ,x* ,~*) = ~= i+k-( ~ + y ~: j + ] - ( ~ + ~ + I x l )

i

L'injection canonique de ~-l(~(r,s)I A)

Dx, Dq~ Dy~ Pij Dx Dy ~ D~, Qkl

dans ~o~(r,s)

se traduit de la

mani@re suivante pour les symboles : Soit

P

un

op@rateur de ~ x ( r , s ) l

P = jE~Z P j ( x , y , z , ~ , n , ~ ) mog@ne de degr@ j

en

o~

Pj

A , P a un symbole

est une fonction d@finie au voisinage de

A , ho-

(~,n,~).

Pj se d~veloppe de mani@re unique en s@rie de Taylor sous la forme Pj(x,y,z,~,n,~) : p~B (x,y,n,~)zB~ ~ A]ors le symbole de ]'image de P dans ~N2~ ( r , s ) est d@fini par p = Z P.. avec P i j ( x , y , n , ~ , x * , ~ * ) = Z p~B(x,y,n,~)x*~ ~*B iEN,j~ la l~l+l~l:i §4 . [email protected]'ordre f i n i

- Polyg6ne de Newton d'un op@rateur :

Pla~ons nous tout d'abord dans le cadre du §3. i . e . A = {(x,y,z,~,n,~)/~ = 0 z = O} Soit ( r , s ) un couple de rationnels

tels que

i ~ s ~ r ~ +~

avec

S

K

+oo

.

82

i o n. . . 4.1. S o i t (i o, jo) ~ 52 ,~(r,s)[io,J o] est le sous.D. .~. f. i.n i t-2~ .... : faisceau de ~N ( r , s ) des op6rateurs P = ( i , j ) c ~ 2 P i j t e l s que :

A/ Si

1_< s < r_< +~

Pij -- 0 OU

si

i ~- i + ( j - i ) B/ Si

1 > ] i o + (Jo-io)

Pij ~ 0

b) Vk et pour

tiels

.

~1 i + ( j - i )

si

~(k)

t.q.P..

de classe

2~(r's)[i'J] (r,s)

> ~1 i ° + ( J o - i o )

~ 0

lJ

k o = ~1 l.o + ( J o - i o )

= ( i , j )U~

Si

I >-~ i o + ( J o - i o )

1 ~ s = r < +~

a)

~(r,s)

] -r i + ( j - l )

si

on a

! i + (j-i) S

i < ~(k)

est le faisceau des op~rateurs 2 - m i c r o d i f f ~ r e n -

(d'ordre f i n i ) .

~(r,s)[i,j]

on pose a ~ r ' S ) ( P )

et

~(ko) = i 0 .

P est une section de ~ ( r , s ) [ i o , J o ]

sous-faisceau

= k

et

si

P n ' e s t section d'aucun

s t r i c t e m e n t plus p e t i t que ~ ( r , s ) [ i o , J

o] ( ( r , s ) f i x 6 )

= Pio,Jo

Polyg6ne de Newton : Si Newton de {(i,k)

P est un op~rateur 2 - m i c r o d i f f ~ r e n t i e l

de symbole E Pij

' le polyg6ne de

P , NA(P) , est l'enveloppe convexe de la r~union des ensembles

c ~2/k ~ k o , i + k ~ i o + k o}

pour les couples

(io,ko)

t e l s que

Pio,io+ko ne s o i t pas identiquement nul. Si

P est un op6rateur m i c r o d i f f ~ r e n t i e l

polyg6ne de Newton de

P le long de

d 6 f i n i au voisinage de

A , le

A est le polyg6ne de Newton del'image

de

P

dans ~ ( 1 , 1 ) . ~]cgw~_~:~:

9

Les faisceaux ~ ( r , s )

, les symboles principaux et le p o l y -

g6ne de Newton sont d ~ f i n i s dans le cas p a r t i c u l i e r

ou

A ~ la forme du §3. mais

ces d 6 f i n i t i o n s sont i n v a r i a n t e s par transformation canonique de T* X et on peut donc consid~rer ~ ( r , s ) -

, aA(r's)

toute sous-vari6t~ i n v o l u t i v e

et le polyg6ne de Newton le long de

A r ~ g u l i ~ r e ou maximalement d~g~n~r~e de

A pour T* X .

88 ~[99~_~:~ A

Si

X = C et A = T{0}~ , le polyg6ne de Newton le long de

d@fini ci-dessus n ' e s t autre que le polyg6ne de Newton classique d'un op6rateur

diff~rentiel

(Ramis

[11]

~9~_~:

) ( au changement de v a r i a b l e 2 ~N = ~ (1,1)

Nous noterons

un op@rateur m i c r o d i f f @ r e n t i e l

oh(P )

et

i'

= i+k , j '

oh

O^ (1'1)

est le symbole qui est not@

= -k . Si

pros). P est

o^(o(P))

dans

[8] Les z@ros de sens de

[I]

et

OA(P) [8]

sont les d i r e c t i o n s microcaract@ristiques pour

tandis que les z@ros de

non fortement non m i c r o c a r a c t # r i s t i q u e s pour

OA(~'I)(P)

P au sens de

Etant donn~ un op~rateur m i c r o d i f f ~ r e n t i e l les fonctions les

Pi,j

olr'S)(P)

ou

pour

(i,j) r

et

du polyg6ne qui passent par nique de

[9]

P d ~ f i n i au voisinage de

A

,

est un sommet du polyg6ne de Newton sont des symbos

convenables

(i,j)

(r

et

s

@tant les pentes des c6t@s

e t donc sont i n v a r i a n t s par t r a n s f o r m a t i o n cano-

(les autres symboles o h ( r , S ) ( p )

T* X

P au

sont les d i r e c t i o n s

sont identiquement nuls)

Le th@or~me fondamental qui r e l i e les faisceaux ~ 2 ( r , s )

aux d i r e c t i o n s

m i c r o c a r a c t ~ r i s t i q u e s est le suivant : Th_~_o_r_~me__4_.5._:Si sur

U est un ouvert de

U ( i _< s _< r _< +~

et

s < +~ ), et si

T/~ A

, P une section de

o-(r,S)(p)

fr,s)

ne s'annule pas sur

U

A

il existe

Q ~ F(U,~(r,s))

tel que

P.Q = Q.P = 1 .

(Pour montrer ce th@or~me on u t i l i s e de Boutet de Monvel-Kree

[3]

On peut montrer pour ~.~(r,s) 3.1. de Boutet de Monvel

[2]

une norme f o r m e l l e semblable a c e l l e

). un th@or~me de f i n i t u d e analogue au th#or~me

, de ce th~or~me on d@duit d'une p a r t un th@or~me de

pr@paration et un th~or~me de d i v i s i o n

( c f [13]

th. 2.2.1. e t 2.2.2. ch I I )

et

d ' a u t r e p a r t le th~or@me suivant : Th~or~me 4.6. : - -

1,

.

.

.

.

Si

.

.

.

.

r

.

.

=,

et

s

sont r a t i o n n e l s ,

i _< s _< r < +~

et

s < +~

est un faisceau d'anneaux coherent e t noeth6rien, ~ f i b r e s noeth~riennes. 9 2. ~ ( r , s ) est p l a t sur -I(~XIA ) (~:T~÷A) (et sur

~-I(~x(~,S)IA). LC , 2oo G~P 3. C~N ( r , s ) est fid~lement p l a t sur ~ ( r , s ) .

, ~2(r,s)

84

§5 . Vecteurs microcaract@ristiques et Condition de Levi pour un systeme Probleme de Cauchy dans le complexe . D e f i n i t i o n 5.1. : Soient ou maximalement d@gener@e)de

A une sous-vari@t@ i n v o l u t i v e homogene ( r e g u l i e r e T* X et

c a r a c t e r i s t i q u e s simples sur Soit rationnels,

N un ~X-mOdule coherent & Un g@n@rateur

A

M un ~X-mOdule coherent d e f i n i au voisinage de 1 ~ s ~ r ~ +~

, s < +~

Tr : TA ~"÷

N*

Soient

= R HomJ~x(N,,~) ® Ox ~ ®~i

module coherent et

et

s

~-IM)

A

(Variete microcaract@ristique de type D@f~nit~on_5=2:_:

r

, on pose :

Ch2(r,s)(M) = support ( ~ ( r , s ) ® - 1 avec

A ; pour

(r,s)

M et [d]

de

M le long de A

N deux ~'X

).

modules coherents ,

, (d = codim support N) , M $ N*

TX X x X est une sous-vari@t@ lagrangienne de

est un

T* X x X

~<xX, on

pose : Ch2(r,s)(M,N) = Ch~

T*

Tx* X × X ~ T * X et on considerera (T'X). Si

n@r@e de sur

X x X( r ' s ) ( M £ N*) . Ch2( r,s)(M,N)

A est une sous~vari@t@ l i s s e i n v o l u t i v e T* X et si

N est un systeme a i

comme un sous-ensemble de

r@guliere ou maximalement d@g@-

gen@rateura caract@ristiques simples

A , on a une p r o j e c t i o n canonique ~ : T*( T*X)XT* XA + T~ A et on a pour tout M : Ch2(r,s)(M,N) = - 1 C h ~ ( r , s ) ( M ) ( c f [ 9 ] th. 4.18).

~X-mOdule coherent

On en d@duit facilement du theoreme 4.5. la p r o p o s i t i o n suivante : ~9P£§~£9_§=~_: ien

Si on i d e n t i f i e

H : T*(T*X) + T(T*X), Ch~ ( I , I ) ( M )

r i s t i q u e s pour

M le long de

T~

et

TA (T'X)

~ l ' a i d e du hamilto-

est l'ensemble des vecteursmicrocaracte-

A au sens de

[8]

, tandis que

l'ensemble des vecteurs non fortement non microcaract@ristiques A au sens de

[9]

La c o n d i t i o n de Levi ( [9] )

pour

Ch~(%1)(M) pour

est

M le long de

M s ' @ c r i t donc

C~2(~,1)(M) = Ch~(Z,Z)(M).

de

* ~ on d e f i n i t S o i t x * c TA A au p o i n t ~ comme s u i t s o = sup

{s/~

la s u i t e des "pentes c r i t i q u e s "

~ Ch~(s,1)(M)}

Sk+ I ~ sup{s/x* i CH~(S,Sk)(M)}

et par recurrence sur

k :

de

M le long

85 On d ~ f i n i t

ainsi une s u i t e croissante f i n i e i < s° < s I < ...

Si

M =~/~P

pour un op~rateur

A sont les valeurs le long

~

ou

~

< Sp <

de r a t i o n n e l s

:

Sp+ I = +~

P , les pentes c r i t i q u e s de

N le long de

parcourt les pentes du polyg6ne de Newton de

P

A Si

X = ~ , A = T {0}~

exceptionnelles de

s"

la s u i t e

de Ramis

On peut de m~me d ~ f i n i r

s o . . . . . Sp est la s u i t ~ des "valeurs

[11]

la s u i t e des pentes c r i t i q u e s d'un couple de syst~mes

(M,N). Remarquons e n f i n que si

A est une vari~t@ lagrangienne l i s s e , si

systOme holon6me de vari~tO c a r a c t ~ r i s t i q u e ( A identifi~

~ la section n u l l e de

A

, alors

T A ~ = T* A)

et

Ch~ ( I , I ) ( M )

M est un

c A

M est ~ points s i n g u l i e r s o

r ~ g u l i e r s au sens de Kashiwara-KawaY

[6]

si et seulement si

Ch~(~,I)(M) ~ A .

Nous allons maintenant ~ t u d i e r le probl~me de Cauchy dans le domaine complexe : D ~ f i n i t i o n 5.4. tique de type

(r,s)

* TyXxT* X(T'X)

:

Une sous-vari~t~

n Ch2(r,s)(M,N)

Th~or~me 5.5. : Soient voisinage de

(T*X)~ X Y dans

1 < s < r < +~ _

_

(M,N)

_

s < +~

~

X sera d i t e non m i c r o c a r a c t ~ r i s -

pour un couple de ~X-mOdules coherent (M,N)

Soient p : (T X)~xY ~T*Y d = dim X - dim Y .

et

Y de

•

est contenu dans la section n u l l e de

et

M et T* X et

Si

~: (T X)xxY ÷ T X

Si

~j ~ 0 My et

~

(r,s)

deux r a t i o n n e l s t e l s que

Y est non m i c r o c a r a c t ~ r i s t i q u e

de type

(r,s)

Ny

e ~. N)Ed] ~x

N.

X x* ~ (T*X)x X v e s t

C.h(N) n p - l ( p ( x * ) )

o~

les p r o j e c t i o n s canoniques

on a des isomorphismes :

N(r,s) = ~ r , s ) -

tel que

T*(T*X).

N deux ,~-modules coh~rents d ~ f i n i s au

~'IR= HOm~x(M,N(r,s))-~ =R HOm~x(M,~<~y)e_l ~ ( ~ s ) P ~Y avec

si :

Ch(M) n p - l ( p ( X * ) )

c {x*}

et

c { x * } l'isomorphisme precedent se r ~ d u i t ~ :

EXt~x(M,N(r'S))x,~ Ext~(My,N~r'S))p(x, ) d~signent les modules i n d u i t s par

M et

N sur

Y

[13] .

pour

86 Le th~or6me 5.5. s i g n i f i e que le probl~me de Cauchy est bien pos6 sur Y pour le couple (M,N ( r ' s ) ). II a ~t~ d~montr~ dans le cas r = s = 1 (faisceau ' ~ des op~rateurs d'ordre i n f i n i ) par Kashiwara-Schapira [8] ,dans le cas r = ~ , s = 1 (faisceau % des op#rateurs d'ordre f i n i ) T. Monteiro [10] aadapt# la d~monstration de Kashiwara-Schapira. Notre m~thode de d~monstration est diff~rente : Rappelons

Czlx

=

~ "x ~x/~x

( [13] )

que si

Z = { ( x , z ) ~ X/X= O}

ce qui permet CZI x

Czt X est d~fini par

+ ~D z

On a R HOm~x(M,N(r's) ) ~ R Hom~.x(M 8 N*

forme

,

([8]) pour

~(r,s) ) ' CXlXxX

de ramener ]e th~orame au cas p a r t i c u l i e r

Z hypersurface de

ou N est de ]a

X transverse ~ Y .

D'apr~s [13] , si ~= (T~X) ~z(YnZ)

et

d = codimxY

on a un triangle

:

c(r,s) zIx I~ -(r's~[2d]

2R CznYiZiX(r,s)[d]

÷ R 7T~

R FS(CZI X avec A= T X, ~

: (A=g) U A÷A , ~' A- % ÷ g 2R C Z : y f z i x ( r , s ) = =RFT*A ( ~ - l C z i x ( r , s ) ) [ d ]

d'o~ le triangle

et

:

B H°m~(M,~IX(r,s)llz H°m~(M,~F~(CzIx(r,s))[2d]

÷

2R ~ H°m~x(M,~Cz~yIZIX(r,s))[d]

, • est Nplat (sur r~-I(~xIA) , s(N = )CZIX) 2R =R HOm~x(M,B_ ~'.

donc 2R Cznyizjx(r,s))= = R ~'. (R_ Hom_I~x(~-IM, CZ~yIzIx(r- ,S)IT~A_A

?

2B

= R ~' (R Hom 2 r.s)(~N ( r ' s ) ® '~x ~-IM ' CZ~YIZIX(r's)JT*A-Z) = * ~N( ~-1~ Z D'apr~s l'hypoth#se du th#or~me, les supports Cznyjzix(r,s)2R

et

Ch~(r,s)(M)

de ~iN(r,s ) ' 2

section nulle donc: Horn ~X(M, =~. m'

2~

.,® - 1 M

"

(TznyZ) ×Z (T~ X)

de

sont d i s j o i n t s en dehors de la

~X

= 0 CZnYiZiX(r,s))

87 donc

R Hom~x (M,CzlX(r,s))IS ~ R Hom~x ( M , B r z ( C z I x ( r , s ) ) ) [ 2 d ]

.

Pour montrer le thOorOme 5 . 5 . , i l reste ~ montrer que HomeyX (M,R r z ( C z i x ( r , s ) ) ) x , [ 2 d ] si

x*

est un point tel que

= R Hom~y(My,Czny(r,s ) p(x*)

Ch(M) n p - l ( p ( X * ) ) =

{x*} ce qui se f a i t par un thOo-

rome de dualitO pour les ~'~(r,s)-modules analogue au th~or~me 3.5.6. ch. I I de [13]. Suivant [8] , on peut dOduire du thOorOme 5.5. un thOorOme sur le problOme de Cauchy dans les fonctions holomorphes ramifiOes :

nit :

Si Z est une hypersurface l i s s e de 1 O[ZIX ] = ~ Log~. Si

X • [ i , +~] ,

X et

X rationnel on pose

h° une Oquation de

(X) =

(X'I)IT X

Z

on dOfi-

et

I O[ZIX](X ) =~X(~) Log~o. I OEZIX](~ ) qui s'Ocrivent

est ]e faisceau des fonctions holomorphes ramifiOes autour de Z aj(x)(~o(x) - j + b(x) Log (x) avec localement j_>O ] a j ( x ) l < cj 1 pour x < +~ et a j ( x ) --- 0 si j _> Jo pour x = +~ ( j ! ) X- -- I Si

(Zi) i = l . . . . , r

sont des hypersurfaces de

comme le conoyau de l ' a p p l i c a t i o n r-l÷ Ox

~ 01 i=1 ZilX

Th~or~me_5:6: (Zi) i = l . . . . . r que

r 1 i=lZ O z i i x ( k )

:

( f l . . . . . f r - 1 )-~ ( f l ' f 2 - f l

Soit

X on d O f i n i t

Z

Y une sous-vari~tO de

des hypersurfaces de

. . . . . f r - 1 )" X, Z

une hypersurface de

Y ,

X transverses 2 ~ 2 et transverses ~ Y t e l l e s

Zi n Y = Z pour tout i . Soit

M

un ~X-mOdule cohOrent t e l que :

a) Ch M n p-I(T~Y) c U, T~.X 1

( p : (T'X)× x Y ÷ T'Y)

1

b) Y est non microcaract~ristique de type pour tout i . Alors pour tout

j

on a des isomorphismes :

(~,I)

pour

M sur

Tz.X\T~X ]

88 Ce th~or6me a ~t6 d~montr~ (sous des hypotheses plus r e s t r i c t i v e s ) p a r Hamada-Leray-Wagschal

[4]

Le th~or6me suivant donne une i n t e r p r # t a t i o n a n a l y t i q u e des pentes c r i t i q u e s d'un syst~me : Th~or~me_5:7~ : S o i t ment d~g#n~r~e de

T*X

tiques simples sur

A

Soit

une sous-vari~t~ i n v o l u t i v e r ~ g u l i ~ r e ou maximale-

N un "~-module coherent ~ un g#n~rateur, ~ caract~ris-

M un L~-module coherent et

c r i t i q u e s de ~(x*) = x

et

A

M le long de

x

A en t o u s l e s

un p o i n t de points

x*

A

de

t e l que les pentes (T~\A)tels

que

s o i e n t les m#mes.

Soient i ~ s o < s I < . . . pour t o u t ( r , s ) i l e x i s t e

ces pentes c r i t i q u e s . < Sp < Sp+I = +~ j ~ [O,...,p+l] t e l que (posant s_1=1 )

k Ex t~(M,N(r,s))~X = E t2tM'N(sj'sj-l))'~5~

Vk ~ 0

{,%

Soit caract~ristique

x ~ X et

M un ~v-module holon6me de v a r i 6 t ~ A

T{x}X U TxX

S o i t A = T{x}X - {x} . On suppose que pour t o u s l e s pentes c r i t i q u e s de

M le long de

Alors pour toute s o l u t i o n {x} , i l

de classe

(Les u l t r a d i s t r i b u t i o n s Z

~Nn

u

de

a~ ~ ( ~ ) ( y - x ) avec

T* A-A

les

M dans les hyperfonctions ~ support dans

e x i s t e une et une seule de ces pentes c r i t i q u e s

une u l t r a d i s t r i b u t i o n

points de

A sont les m#mes.

(sj)

sj

t e l l e que

et ne s o i t pas de classe

de classe (s)

a support dans

la~l ~C I~I ( I ~ I ' ) -s)

e t si

s = +~

(s)

{x}

u

pour

soit s > sj.

sont les s~ries

ce sont les d i s t r i b u -

tion~}. Dans le cas oQ X est de dimension 1, la c o n d i t i o n sur

M est toujours rem.

p l i e et on retrouve un th#or~me analogue au th~or6me i de Ramis [ i i ] Pour montrer le th#or~me 5.7.

.

, on se ram~ne &u cas de la dimension 1, grace

au th~or~me 5.5. §6 . Th~or6me de prolongement : ~2~_~z_:

Soient

deux r a t i o n n e l s avec Soit de

2

M et

N deux ~,X-mOdules coh~rents et

un ouvert de T* X ~ bord de classe

T* X . On suppose que la normale ~

Ch2(r,s)(M,N), alors

r

et

I ~ s ~ r < +~ , s < +~

~

en

R F~(R Hom ~ ( M , N ( r , s ) ) ) x * = X

CI / 0 .

au voisinage d'un p o i n t n ' a p p a r t i e n t pas

s

89 (Pour r = s = 1 ce th~or~me est d~montr# dans [7]

).

D~monstration : =RHOm~x(M,N(r,S))x,~ =RHOm~xxx(M ~ N*, C×:FXxx(r,S)(X*,x*) ce qui ram~ne au cas oO N est un module a un g#n~rateur a caract~ristiques simples sur une vari~t~ l i s s e A • I I s u f f i t alors de montrer que R r~(N(r,S))x* si

~*

est un ~(r,S)(x*,~*)-module_

est la normale ~ Q en x* . En e f f e t on a alors :

R r~ =R Hom~x(M,N(r,s))x*= =R Hom~ , X

=

,(Mx*, =R F~(N)x*),

= R Hom~NN(x,,~,)

:

(r,s)

((.~(r,s)

® ~-IM)(x*,~*),R?~(N)x, ) = 0

-1~

d'apr~s 1 'hypoth~se. [1]

J.M. BONY : Extension du th~or~me de Holmgren, Sem. Goulaouic-Schwarz

[2]

L. BOUTET de MONVEL : Op~rateurs pseudo-diff~rentiels analytiques.

1975-76, expos~ 17. Univ. Sci et M~d. Grenoble (75-76). [3]

L. BOUTET de MONVEL, P. KREE : Ann Inst. Fourier, Grenoble, 17,

[4]

T. HAMADA, J. LERAY, C. WAGSCHAL : Probl~me de Cauchy ramifi~, J. Math.

[5]

M. KASHIWAP~A : Cours sur les syst~mes d'~quations m i c r o d i f f ~ r e n t i e l l e s .

[6]

M. KASHIWARA, T. KAWAI : Systems with regular s i n g u l a r i t i e s

p. 295-323 (1967). Pures et Appl. 55, 297-352 (1976). Pr~publ. de l ' U n i v . Paris-Nord, C.S.P. 1978. (Preprint). [7]

M. KASHIWARA, P. SCHAPIRA : Microhyperbolic systems, Acta Math 142, 1-55, 1979.

[8]

M. KASHIWARA, P. SCHAPIRA : Probl~me de Cauchy dans le domaine complexe

[9]

T. MONTEIRO : Th~se de 3~ cycle ~ l ' U n i v e r s i t ~ Paris-Nord (1978).

Inv. Math 46, 17-38 (1978). [i0]

T. MONTEIRO : a r t i c l e a paraitre,

[Ii]

J.P. RAMIS : D6vissage Gevrey Ast~risque 59-60 (1978) p. 173-204.

[12]

T. OSHIMA : Singularities in contact geometry. J. Fac Sci. Univ. of Tokyo, Sec IA, 21, (1974), 43-83.

[13]

M. SATO, T. KAWAI, M. KASHIWARA : Hyperfunctions and pseudo-different i a l equations, Lect. Notes in Math. 287, Springer, 265-529.

SUR

LE

PROBL~ME par

HILBERT-RIEMANN

DE

Zoghman

MEBKHOUT

SOMMAI RE pages §.i

- Introduction

90

§.2 - P o s i t i o n du probl~me

92

§.3 - Syst~mes holonomes ( r a p p e l s )

94

§;4 - Enonc@ du th6or~me d ' e x i s t e n c e

97

§.5 - R ~ g u l a r i t ~ e t d u a l i t ~

98

§.6 - D~monstration ( i n d i c a t i o n s )

102

§.7 - A p p l i c a t i o n s

106

7 . 1 . D u a l i t ~ de P o i n c a r ~ - V e r d i e r

106

7 . 2 . Th~or6mes de b i d u a l i t ~

107

7.3. R~gularit~ et irr~gularit~

107

7 . 4 . Syst~me de Gauss-Manin

108 II0

Bibliographie

§.I

- INTRODUCTION

Nous pr~sentons dans cet expos~ le thgorgme d'existence de ([19]; chap.V) sur le probl~me de Hilbert-Riemann en dimension supgrieure pour les faisceaux analytiquement constructibles. Alors qu'en dimension un le problgme de Hilbert-Riemann ( =21~me problgme de Hilbert) a connu et conna~t encore une grande activit~ (voir par exemple les exposes de Sa~-Jimbo-Miwa dans ce m~me colloque), le premier r~sultat en dimension sup~rieure est d~ 5 P. Delisne [3] qui donne une description purement alg~brique de toutes les representations de dimension finie du groupe fondamental d'une

91

vari~tg alg~brique lisse sur le corps des nombres complexes. Pour ce faire, P. Deligne utilise une compactification d'Hironaka comme A. Grothendieck [4] qui avait d~j~ obtenu le cas de la representation triviale comme cas particulier du thgor~me de comparaison de Grothendieck [4]. Le module de d~monstration de Grothendieck a ~t~ fondamental pour ce genre de questions, parce qu'on utilise de fa~on syst~matique la rgsolution des singularitgs.

(voir [5],[9]). Ce bref article

[4] contient en germe plusieurs thgories dont la thgorie des systgmes g points singuliers r~guliers g plusieurs variables. II est bien connu que la cat~gorie de reprgsentations de dimension finie du groupe fondamental d'une vari~t~ analytique complexe lisse connexe est ~quivalente la catggorie des syst~mes locaux, c'est-~-dire des faisceaux localement constants de dimension finie, elle-m~me ~quivalente ~ la catggorie des connexions analytiques int~grables. C'est le thgorgme d'existence de Frobenius. Cependant, pour l'gtude des singularit~s, cette cat~gorie a des inconv~nients facheux : par exemple, l'image directe d'un systgme local par une application propre cesse en g~ngral d'@tre un syst~me local pour devenir un faisceau constructible. C'est la cat~gorie des faisceaux constructibles qui a l e s

bonnes proprigt~s de stabilit~ par images directes et r~ci-

proques. De ce point de vue, il devient naturel de reformuler le probl~me de HilbertRiemann pour les faisceaux constructibles qui auront pour analogues analytiques les systgmes holonomes. On d~montre [i9] que la cat~gorie d~riv~e des faisceaux constructible$ est ~quivalente ~ la cat~gorie d~rivge des syst~mes diff~rentiels lin~aires holonomes d'ordre infini. C'est I~ une g~n~ralisation du th~or~me d'existence de Frobenius dont la d~monstration nous fournira une r~ponse positive au probl~me de Hilbert-Riemann (local pour l'instant) pour les faisceaux constructibles. On doit donc s'attendre un dictionnaire entre ~nonc~s g~om~tri~ues concernant les faisceaux constructibles

02

et gnonc~es analytiques

concernant

ne, dans le dernier paragraphe, qui a gtg le point de dgpart Les dgmonstrations,

les systgmes diff~rentiels

On en exami-

quelques exemples dont la dualit~ de Poincarg-Verdier

([18] ; remarque 5.1).

qui ont des interm~diaires

vation par morphisme projectif de la r~gularit~, vent en d~tails dans [19]. A c e Hilbert-Riemann

lingaires.

int~ressants

tels que la conser-

ne sont qu'esquiss~es

titre, cet exposg est introductif

tout comme l'~taient

les exposes

mais se trou-

au problgme de

[20],[21] qui ~taient, eux, concen-

tr~s sur le th~or~me de dualit~ pour I ~ ~X-mOdules. L'auteur expose ses rgsultats et laisse le soin au lecteur de faire le lien ou de comparer s'il y a lieu avec d'autres

travaux. Nous essayons de presenter

les pro-

bl~mes de la fa§on la plus naturelle possible en nous appuyant sur la bibliographie.

§.2 - POSITION DU PROBLEME Soit (X , ~X) une varigt~ analytique complexe une hypersurface nique.

de X. Notons U le compl~mentaire

Soit, d'autre part, ~ u n e

tation du groupe fondamental vari~t~ alg~brique

et X = U

de fa~on un peu vague Question

(HR)I

lisse connexe de dimension n e t

monodromie

~I(U),

de Y e t

j : U~X

sur U, c'est-g-dire

l'inclusion

cano-

soit une represen-

soit un systgme local. Classiquement,

un de ses compartifi~s.

Y

U est une

On se pose alors deux questions

:

: est-ce que ~

est la monodromie

d'une ~quation diffgrentielle

sur

U? Question

(HR)2 : est-ce que ~ guli~re"

~U

O~"

d'une gquation diff~rentielle

"r~-

le long de Y ?

Par ~quation diff~rentielle, finition pour l'instant.

est la monodromie

on entend connexion intggrable

Pour r~pondre

dont on admettra la d~-

g (HR)I , il suffit de prendre

Mais cette connexion est "irr~guli~re"

en g~n~ral.

la connexion

Dans [3], P. Deligne

93

eonstruit une connexion "r~$uli~re" pour r~pondre g (HR) 2. Si U est alg~brique, alors cette connexion r~guli~re est alg~brisable. Comme nous l'avons vu dans l'introduction, il faut reformuler les questions pr~c~dentes pour les faisceaux constructibles. Soit ~

un faisceau constructible sur X,

c'est-~-dire tel qu'il existe une stratification (de Whitney) X = U X i pour laquelle la restriction

~ I X i g chaque strate est un systgme local. On se pose alors de fa§on

aussi vague deux questions : I

Question(HR) 1 : est-ce que

~

est solution d'un syst~me holonome ?

Question (HR~2 : est-ce que

~

est solution d'un syst~me holonome "r~gulier" ?

II convient de pr~ciser les rapports entre les questions (HR) et les questions (HR)' qui ne sont semblables qu'en apparence. Alors que la question (HR) 1 est facile, il n'en est plus de m~me de la question (HR) I. Une r~ponse locale g la question (HR)~ est n~cessaire pour r~pondre g la question (HR)~. Les questions (HR)' sont gtroitement li~es. Alors que le syst~me l o c a l , ~ est donn~ sur U dans les questions (HR), le faisceau constructible ~

est donn~ sur X. Pour r@cupgrer la r~ponse des questions

(HR) ~ partir des questions (HR)', il faut prendre un "prolongement" de ~ existe bien sQr un prolongement canonique, ~ savoir j ~ ,

g X. Ii

mais il en existe d'autres,

ce qui introduit un certain arbitraire. En fait, m~me en dimension un, il faut consid~rer les solutions supgrieures pour une bonne comprehension du problgme. Par exemple, u_~nsyst~me diff~rentiel peut donner naissance g u n tout comme on verra qu'.~ faisceau ~ p e u t

complexe de solution

donner naissance ~ un complexe de syst~mes

diff~rentiels. Pour toutes ces raisons, il apparalt clairement que le cadre naturel est les categories d~riv~es qui sont d~sormais fondamentales m~me pour passer d'une solution locale ~ une solution globale. Nous renvoyons le lecteur non averti aux notes originales de Verdier (Lectures Notes in Math. n°569) qui restent la meilleure r~f~rence pour la question. Pour la commodit~ du lecteur, nous allons reprendre

94

quelques notions sur les syst~mes diff~rentiels maintenant bien connues.

§.3 - SYSTEMES HOLONOMES

(RAPPELS)

Soit toujours (X, ~X) une vari~tg analytique complexe lisse de dimension n. Le faisceau des op~rateurs diff~rentiels d'anneaux unitaires non commutatifs. faisceau

~X

d'ordre fini est un faisceau cohgrent

Ii contient comme sous-faiseeau d'anneaux le

~X" En coordonn~es locales (x I ..... Xn) , une section P(X,~x) de ~

est un

pol~nSme de la forme dS

P(X'~x) =

I~I~ ~ m

o~ a (x) sont des fonctions holomorphes

;

as(x)

-dx~

~ = (~l,...,~n) E N n ; Isl = s I + ... +~n "

Le f a i s c e a u ~ X op~re g gauche sur le faisceau

~ X qui devient un syst~me diffgren-

tiel, ~ savoir le syst~me de "De Rham". Plus ggn~ralement, d'ordre fini est un ~ X - m O d u l e

un syst~me diff~rentiel

~ gauche coherent, c'est-g-dire admettant localement

une presentation finie. Une connexion int~grable est simplement un systgme diff~rentiel qui soit libre de type fini en tant que

BE-mOdule.

soit de type fini. Si on a un vrai syst~me diff~rentiel,

Ii suffit d'ailleurs qu'il c'est-~-dire une matrice

P =P(x,~--~) de p lignes et q eolonnes d'op~rateurs diff~rentiels, conoyau ~ d e

l'application lin~aire ~ q

P ~P

on lui associe le

d~finie par P. C'est un syst~me dif-

f~rentiel par construction. Pour notre but, le faisceau ~ X op~rateurs diffgrentiels

est insuffisant.

II faut introduire le ~

des

d'ordre infini. C'est un faisceau non coherent d'anneaux

unitaires non commutatifs.ll

contient comme sous-faisceau d'anneaux ~ X

il est fid~lement plat [22]. En coordonn~es

locales x = (Xl, ....,Xn) , une section

d

cale P (X,~x) est une s~rie de la forme P

d

=

sur lequel

I

Isl= 0

a S (x) S!

d~ dx ~

lo-

9S

avec la condition I

tim [%(x)LT~7= 0 uniform~ment sur tout compact. Le faisceau ~

op~re g gauche sur

~'X" On d~finit un syst~me diffgrentiel d'oro o

dre infini comme un ~X-mOdule g gauche ~

admettant une presentation locale de la

forme mais o3 la matrice P e s t d'ordre fini. On dit admissible dans [22]. Ceei ramgne l'~rude des ~ - m o d u l e s

admissibles g l'~tude d e s e x - m o d u l e s

coh~rents. Mais un pro-

bl~me fondamental et difficile est de d~cider qu'~tant donng un ~)X-mOdule, s'il est admissible ou pas (voir le th~orgme 4.1). La vari~t~ caract~ristique d'un op~rateur diff~rentiel dtordre fini est l'hypersurface du fibr~ cotangent T*X d~fini par son symbole principal, Pour d~finir la vari~t~ caraet~ristique d'un systgme diff~rentiel, une m~thode consiste ~ introduire le faisceau des op~rateurs micro-diff~rentiels

~

d'ordre fini. C'est un faisceau

coherent d'anneaux unitaires non commutatifs sur T*X. Si J ~ e s t

un syst~me diff~ren-

tiel, la vari~t~ caract~ristique S~(¢J~) est le sous-ensemble de T*X d~fini par support

~X o3 ~ d6signe la projection de T*X sur X. Le faisceau d'anneaux ~ - I ~ x

~X contient c o m e sous-faisceau

sur lequel il est fid~lement plat [22]. Le thfior~me fondamental

([22] thfior~me) affirme que la vari~tfi caractfiristique S~(~) d'un syst~me diff65.3.2 rentiel non n u l ~ est un sous-espace analytique involutif de T*X. En particulier, on a l'infigalit6 fondamentale dim X = n

~

dim S~(~)

Par dfifinition, un syst~me h o l o n o m e ~ e s t

~

2n = dim T*X .

un syst~me tel que dim S~(¢~) = n .

Par exemple, une connexion intfisrable est un syst~me holonome dont la vari~t6 caract~ristique est la section nulle de T*X. De ce point de vue, la notion de syst~me

g6

holonome assouplit considgrablement la notion de connexion int~grable en admettant des sinsularit~s dams la vari~t~ caract~ristique en toute codimension. La vari~t~ caract~ristique d'un systgme d'ordre infini est d~finie de la m~me fagot grace l'admissibilit~. On a la notion de syst~me holonome d'ordre infini.

Soit un ~X-mOdule ~ gauche ~ .

On construit son complexe de De Rham D R y )

(voir par exemple [16] page 103) : I n 0 ax ~ ........... ~0~,®~ x ~o e"X e"x Si ¢ ~ e s t une connexion int~grable, le lemme de Poincar~ affirme que le complexe

o

DR(M)

~,.~,-~vO~

est cohomologiquement trivial en degr~s positifs.

Le th~or~me d'existence de Cauchy affirme que les sections horizontales de ~ , Ker(V) qui est la cohomologie en degr~s z~ro sont un syst~me local de vectoriels complexes de dimension finie. Si la vari~t~ caract~ristique d'un systgme holonome a des singularit~s, le lermne de Poincar~ et le th~or~me d'existence de Cauchy cessent d'avoir lieu. On les remplace par le th~or~me de Kashiwara [8] qui affirme que les solutions holomorphes d'un syst~me holonome forment des faisceaux constructibles. Pour formuler de fagot precise le th~orgme de Kashiwara, il y a int~r~t tout de suite travailler dams les categories d~riv~es.

On d~montre (voir par exemple [16] page

104) que le complexe DR(¢~) est isomorphe dams D(¢X) , catggories dgriv~es de la cat~ gorie des faisceaux d'espaces vectoriels complexes, ~ D'autre part, le complexe ~ h o m ~ ( ~ , sont autres que les ~ X t ~ x ( ~

~X)

hom~)X(~X,~).

dont les faisceaux de cohomologie ne

, ~X) est par d~finition le complexe des solutions

holomorphes. La raison est simple, On peut voir facilement que le premier faisceau s'identifie au faisceau des solutions holomorphes du systgme dams une presentation locale. Alors le th~orgme de Kashiwara [8] s'~nonce comme suit :

97

Th~or~me 3.].- Le complexe m ~ _ q ~ ,

~x) est (analytiquement) constructihle si

est un syst~me holonome.

Si ~ e s t

une connexion int~grable, le th~or~me 3.1 se r~duit au th~or~me d'exis-

tence de Cauchy conjugu~ au lemme de Poincarg. Nous noterons D(¢X) c la cat~gorie d~riv~e des complexes born~s de CX-modules ~ cohomologie constructible. De m~me nous noterons D(~X) h (resp D ( ~ ) h )

la cat~gorie des complexes de ~X-mOdules

de ~-modules) g gauche horn, s ~ cbhomologie ~x-holonome

(resp ~)~-holonome). II

r~sulte facilement du th~or~me 3.1 que le foncteur ~ h o m ~ * ~hom

(resp

, ~X)

(resp

( * , ~ X )) envoie D(~X) h ( resp D~)X) h ) dans D(¢X)c .

§ . 4 - ENONCE DU THEOREME D'EXISTENCE

Soit ~ u n

complexe de D(¢X). La structure de ~-module ~ gauche de ~xinduit

une structure de complexe de D(~X~ sur ~ h O m C X ( ~ , ~X).

Th~or~me 4.].- Le complexe tRhOmCX (~ ,

~X) appartient g D(~X) h si ~

appartient

D(£X)c •

De fa~on un peu plus parlante, le th~or~me 4. ] affirme que les faisceaux ~XtCx(~ i , ex ) sont des syst~mes diff~rentiels holonomes d'ordre infini. Des exempies simples (par exemple pour ~

~gal au faisceau caract~ristique d'une sous-vari~t~

Y lisse de X) montrent que fRh__.~¢X(~ , (~X) n'appartient pas ~ D(~)X) h g cause de la presence des sinsularit~s essentielles. D'oN la n~cessit~ du faisceau ~DX. Par le th~or~me 3.1 et le th~orgme 4.1, on dispose de deux foncteurs IF : D(~X~ h

)_ D(~X)

98

C

:

D(¢X) e

~ D(~X) h

~(#) : mh°mCx(~' ~x) Proposition 4.~.- Les foncteurs ~ et ~ sont inverses l'un de l'autre et ~tablissent donc une ~quivalence de cat~gorie entre D(~)X~h et D(¢X) c.

La d~monstration de cette proposition utilise d'une part le thgor~me de bidualit~ locale de nature analytique (th~or~me 2.1, chap. Ill ; [19]) qu'on peut r~sumer en disant que le syst~me de De Rham " ~ X "

est dualisant dans D(~X~

, d'autre part le h

th~orgme de bidualit~ locale de nature g~om~trique d~ ~ Verdier [23] que l'on peut r~sumer en disant que le faisceau constant "C X'' est dualisant dans D(CX)c.

Le th~o-

r~me 4.1 et la proposition 4.2 donnent une r~ponse globale tr~s precise globale g la !

question (HR) I du paragraphe 2. La d~monstration du th~or~me 4.1 apporte une r~ponse locale mais precise ~ laquestion (HR)2 comme nous allons le voir. L'auteur espgre utiliser la mgthode de la descente cohomologique comme pour le th~orgme de dualit~ v

([19] ; chap.IV) pour rgpondre $1obalement ~ (HR)2. On peut donc s'attendre, en vertu de l'@quivalence des eat@$ories D(¢X)

et D(~X~ c

, ~ un dictionnaire entre @nonc@s h

g~om~triques et ~nonc~s analytiques. Nous verrons de tels exemples dans le paragraphe 7. II n'est pas ~tonnant que les d~monstrations combinent les m~thodes analytiques introduites dans [16],[17] pour traiter le cas du faisceau caract~ristique d'un sous-espace analytique Y de X ayant des sinsularit~s arbitraires et les m~thodes ggom~triques utilisant les d~vissages de faisceaux constructibles et la r~solution des singularit~$. Nous allons rappeler rapidement dans les paragraphes 5 et 6 ces diff~rentes mgthodes. §.5

Soit~un

complexe de D ( ~ X ~

- REGULARITE ET DUALITE

et Y un sous-espace analytique complexe de X d~-

99

fini par un Ideal ~y. Consid~rons le complexe de cohomologie locale alg~brique introduit par Grothendieck dans [4] : fRF[y](~)

En f a i t ~ F [ y ] ( ~

D~f k ,~) = ~ lira h O m ~ X ( ~ X / ~y

est un objet de D(~X). II faut pr~ciser la structure de ~ -

modules. Le complexefRF[y](~)

est un sous-complexe detRry(~),cohomologie

locale

ordinaire, o~ op~re le faisceau ~ X parce que les op~rateurs diff~rentiels sont locaux et diminuent le support. Pour voir que ~ X opgre sur[RF[y](~),

il suffit de v~rifier

qu'un champ de vecteurs le conserve. Ce qui est fait dans ([16] ; page 98). On rappelle alors ('[19], proposition 3.1, chap. Ill) que les conditions a), b) et c) suivantes sont ~quivalentes : a) L'homomorphisme canonique dans D(CX) est un isomorphisme DRy_[[y] (~))

> DR~R_~Fy(~))

b) L'homomorphisme canonique dans D(CX) est un isomorphisme '

c) L'homomorphisme canonique dans D ( ~

Dans le cadre des ~X-mOdules,

est un isomorphisme :

les conditions a), b) et c) sont ~tablies pour

la ~remi~re fois dans [17] pour le syst~me de De Rham [la condition a) est le th~or~me 3.1, la condition b) est le th~or~me I.I et la condition c) est le th~or~me 1.2, [17]]. Pour un syst~me g~n~ral, la d~monstration de l'~quivalence des conditions a), b) et c) est toute pareille. Naturellement, les conditions a), b) et c) ont une Iongue histoire dans d'autres situations. En dimension un, la condition a) est due Picard (les formes de deuxi~me esp~ce calculent l'homologie d'une surface de Riemann)

100

En dimension sup~rieure et Y un diviseur g croisements normaux, elle eat due Atiyah-Hodge (Integrals of the second kind ~n an algebraic variety. Ann. of Math. 62__, 56-91,

1955). Pour Y, hypersurface singuli~re quelconque, la condition a) eat le

th~or~me de comparaison de Grothendieck [4]. Dana [3], P. Delisne caractgrise par la condition a) une connexion r~guli~re ([3], proposition 6.8). Si on remarque que l'on a un isomorphe canonique dana D(Cx) ([19], proposition 6.1, chap. II), ~hom~__F[y](¢~)

, ~X)~-~

~hOm~X(¢~

, ~X~y)

o~ ~X~y = lira ~'X/ ~ yk eat le compl~t~ formel de X le long de Y qui est un ~X-mOdule ~ gauche, on constate que is condition b) eat due g P. Deligne (pour ~ =

~X )

(1969, non publiC) g Herrera-Libermann (X compact, Duality and the De Rham Cohomology of Infinitesimal Neighbourhoods.

Int. Math. 13, 97-126,

1971). Une autre d~mons-

tration eat due g Hartshorne (On the De Rham Cohomology of Algebraic Varieties. Publ. I.H.E.S. n°45, 6-99, 1975). Cependant, ces auteurs ne consid~raient que la cohomologie de Betti des vari~t~s alg~briques et ne consid~raient pas l'aspect syst~mes diffgrentiels. En dimension un, Malgrange caract~rise une ~quation diff~rentielle r~guligre par la condition b) : "toute solution formelle eat convergente" ([12], th~or~me 1.4). La condition c) eat en rapport ~ la thgorie des rgsidus et le complexe dualisant en g~om~trique analstique. Pour un diviseur ~ croisements normaux, elle a ~t~ gtablie par des calculs directs [14],[15]. Le cas ggn~ral a demand~ plusieurs armies d'efforts. Dana [16],[17], I° )

on a utilisg lea c i ~

ingredients suivants.

Le complexe~__~F[y](¢~) appartient ~ D(~)X) h s i ~ D appartient g D(~)X) h. C'est Ig un des r~sultats lea plus importants de la th~orie du polynSme de Bernstein-Sato ([I],[2],[9]) sous la forme que lui a donn~e Kashiwara [9] pour une hypersurface. Le cas de la codimension queleonque se ram~ne par une suite de Mayer-Victoria

101

alg~brique sous la forme de [17]. 2 =) ee complexe ~ h o m ~ ( ¢ ~ ,

~'X) est constructible si ¢ ~ appartient ~ D(~X) h

(Kashiwara [8]). 3 ° ) Le faiseeau constant CX est dualisant dans D(CX) c (Verdier [23]). 4 ° ) ee th~or~me de dualit~ locale Hans D(¢ X)

qui affirme que DR(~) e t ~ ( o ~ , ~ X ) C

sont duals l'un de l'autre dans D(¢X) c s i o ~ appartient g D ~ X ) h

([19], chap.

III, th~or~me 1.I). 5 =) ee th~or~me de biaualit~ locale Hans D(~X~ h qui affirme que " ~ X " est aualisant dans D(~)X~

([19], chap.Ill

th~or~me 2.1).

h Les ingredients 4 ° ) et 5 ° ) sont dQs ~ l'auteur ([19], th~or~mes |.let 2.1, chap. Ill). lls sont ~ l'origine du th~or~me de dualit~ globale [18]. Nous les avons con~nuniqu~s g Kashiwara en janvier 1978. Les cinq ingredients ont un int~r~t en euxm~mes. Une lois qu'on les a tous, il est facile de voir que les conditions a), b) et c) sont ~quivalentes ([19], proposition 3.1, chap.Ill). Mais un probl~me fondamental est de d~cider qu'~tant donn~ un systgme h o l o n o m e ~ ,

s'il satisfait ~ l'une des

conditions a), b) et c) le long d'un sous-espace convenable Y. A cause des singularity§ de Y, c'est i~ un probl~me difficile. Pour l'illustrer, il suffit de voir les efforts consid~rables qu'il a fallu pour d~montrer que le syst~me de De Rham ~ X est r~gulier le long de tout sous-espace analytique Y de X [17]. Ce r~sultat qui peut ~tre regard~ comme th~or~me de comparaison alg~brique-analytique locale a He nombreuses consequences [16]. Soit ~ u n

complexe He D(~X) h.

Supp(~) est un sous-ensemble analytique de X. Ii existe un sous-espa~e analytique ferm~ Sing(¢~0) He Supp(¢~) en mehors duquel la cohomologie de son complexe des solutions holomorphes est formic par des syst~mes locaux. On dira qu'un complexe de D(~X) h est r~gulier si ~ D e s t

r~gulier le long de Sing(~). Les complexes r~gu-

liers forment une sous-cat~gorie pleine de D(~X) h

que nous noterons D(~X)hr.

I0~

§.6 - DEMONSTRATIONS (INDICATIONS) Pour d~montrer l e th~or~me 4 . 1 ,

lement un complexe ~ d e

on d ~ v i s s e

l e eomplexe ~ p o u r

eonstruire

loca-

D(~X) h qui satisfait a la condition a) et done ~ la condi-

tion c) qui nous fournira le th~or~me. Pour cela, nous suivons la m~thode de i ~ X t C x ( ~ , G)

J.L. Verdier [23] qul d~montre que les faisceaux si ~ e t

sont constructibles

G sont constructibles sur un espace anal~ti~ue. Le eas des schemas est

traitg par Grothendieck dans (S.G.A.5, expos~ I). Les autres ingredients sont la r~____s61ution des singularit~s, le th~or~me de comparaison de ([17], th~or~me 1.2), l'extension canonique de P. Deli$ne [3], la r~$ularit~ des systgmes holonomes, le th~orgme d'images directes pour un morphisme projectif sous l'hypoth~se d'existence de bormes filtrations ~lobales. On invoque enfin le th~or~me de Grauert-Remmert comme Grothendieck [4] pour conserver la r~$ularit~ par un morphisme projectif.

On se

bornera ici g indlquer les trois ~tapes.

6.1 - l ~ r e ~ t a p e . R~duction de D(~X~ h ~ D ( ~ X ) h r

.

La categoric D(~X) c est triangul~e. On peut supposer que ~ e s t

un faisceau

constructible. Le probl~me est local sur X. On peut filtrer le faisceau ~

par des

sous-faisceaux tels que les quotients successifs soient localement constants sur un ouvert lisse d'un ferm~ analytique de X,compl~mentaire d'un fermg analytique. On peut donc supposer alors que ~

est localement constant sur un ouvert W lisse d'un

ferm~ analytique Y de X dont le complgmentaire Z dans Y est un ferm~ analytique de X et nul en dehors. Notons

j :W~Y

, ~ :Y~X

, B =~ o j

W~X

et y : V ~ X

les inclu-

sions canoniques si V d~signe le compl~mentaire de Z dans X. En utilisant le theorYme de comparaison ([17], th~or~me 1.2), on se ram~ne par des formules de dualit~ g : mhOmcX(~,

~)

: ~y-l(~x~x~r;

[Y](ex) @c B!L)

o~ L d~signe le faisceau localement constant constructible, surW, ~ho___m~w(J-l~-l~,$W).

103

Le point clef est la remarque suivante : si le complexe ¥-1(fRr[y](~X) (9 B!L) de ¢

D(~)w) h

se p r o l o n g e en un complexe "~.~" de D(,~X) h r f i g u l i e r l e iong de Z, a l o r s on

obtient le th~or~me 4.1. En effet, la condition c) nous donne : [RhOmEX(~, ~ X ) ~

fR~y-I(~ x O ~) '~x ~

~X

Mais IR l~m h O m e x ( ~ k k v e r t u de l ' i n g r f i d i e n t

,~)

k

IR l~m h O m ~ X ( ] Z '~ X k

appartient ~ D ( ~ ) h

puisque ~ a p p a r t i e n t g D(~)X) h en

I du p a r a g r a p h e 5. Par l a m~me o c c a s i o n , on aura r~pondu l o c a -

lement ~ la question (HR)2 puisque [R limk h°m~X( ~Zk ' o~) est r~gulier.

6.2 - R~solution d e s s i n g u l a r i t ~ s .

On peut trouver [7] un morphisme ~ projectif de vari~t~s analytiques complexes lisses :

XmYDZ tel que z soit un isomorphisme hors de Z et que ~=~-I(Y) est lisse, ~=n-l(Z) est un diviseur ~ croisemen~mormaux dans Y. C'est de cette faqon qu'on r~sout

locale-

ment les singularit~s d'un sous-espace Y de X qui j oue le r$1e d'espace a~iant lisse. Notons :

=~=n-l(w) ~

, ~ : ~Y ~ X~ ,

= ~

et y : V ~ X

les inclusions cano-

niques. La restriction ~ de ~-i B!L g W =Y\Z ~ ~ est un systgme local. ~ ri~t~ lisse Y e t

une connexion int~gr~le ~ ' ~

sur W

a donc une va-

compl~mentaire d'un diviseur

¢ g croisements no,aux. C'est la situation favorite de P. Delisne [3]. En vertu du thgor~me d'existence [3], cette connexion se prolonge en une connexion o ~ r~$uligre m~romorphe sur Y. Nous avons rappeler dans ([19], chap.V) la construction de cette e~tension canonique en l'adaptant au langage des ~ - m o d u l e s . ~est

~gale

En tant que ~ - m o d u l e

104

lim h o m a ~ ( ~ k ( ~ ) @ E -Z' ¢ og E e s t l a f i b r e de ~ en un p o i n t de W q u ' o n p e u t s u p p o s e r connexe. La c o n s i d ~ r a t i o n de l a monodromie l o c a l e p e r m e t de m u n i r c ~ d ' u n e s t r u c t u r e c e l l e de ~ tre part,

~ ~. ¢

de ~ - m o d u l e

qui prolonge

On p e u t o b s e r v e r que l e < ~ ) - m o d u l e c ~ e s t g l o b a l e m e n t f i l t r g .

il est rggulier

D'au-

en ce sens que l ' o n a l ' i s o m o r p h i s m e

DR@) __~ :DR__(3 ~ , ~- ~~ q u i c o r r e s p o n d ~ l a c o n d i t i o n a) du p a r a g r a p h e 5. Nous c h e r c h o n s un p r o l o n g e m e n t A laritfis

en t a n t que , ~ - ' m o d u l e

pour g v i t e r

les singu-

de Y. On pose

~

: ~÷~

Pour la dfifinition de ~ ÷ ~ , que A e s t

~)) 2[-codim

~]

.

voir [9]. On dgmontre par un lemme facile [19]

un prolongement r~$ulier de

~-I~-I(~F[y](~X ) O B!L) . ¢

6 . 3 - Images d i r e c t e s .

Le prolongement cherch~ sera

&

=

A.

ll faut v~rifier deux choses. D'une part, que ¢ ~ appartient a D(~X) h. Mais est $1obalement filtr~ et ~ est projectif. Ce sont les conditions suffisantes pour lui appliquer le th~or~me d'images directes de [9] et ¢ ~ appartient g D(~X) h. D'autre part, que ~ e s t

r~gulier le long de Zo Ceci r~sulte du th~or~me suivant qui a

un intgr~t en soi. Thgorgme 6.3. I.- L'homomorphisme canoni~ue de D(Cx) e st un isomorphisme :

m~r[z]

(~))

~

DR(~ z (~)).

Le th~or~me 6.3.1 ach~ve la construction de "l'extension o~" qui est r~guli~re

~05

puisqu'elle satisfait ~ la condition a). Pour d~montrer le th~or~me 6.3.1, on construit le diagramme N

R~. D__R(RF[~](~))

~(~

[z I (d%))

> R~. D__R(R.F~(¢~))

~

D~(R._.rz (O~b))

La formule de projection appliqu~e ~ ~ nous permet de montrer que la fl~che verticale de droite est un isomorphisme. C'est I~ un isomorphisme purement topologique. Maintenant, pour montrer que la fl~che verticale de gauche est un isomorphisme, le th~or~me de Grauert-Remmert (trivialit~ cohomologique des morphismes projectifs) nou~ r~duits ipso-facto ~ le v~rifier fibre par fibre (voir Grothendiec~ [4], note n°7, page 99). N

La fibre de ~ ~tant alg~brique, le syst~me ~ O r ~ g u l i e r est donc al~brisable.

On

est r~duit cormne pr~cgdemment ~ un raisonnement purement topologique pour la topologie de Zariski via la formule de projection. La fl~che verticale de gauche est un isomorphisme. Comme la premigre fl~che horizontale est un isomorphisme correspondant la r~$ularit~ de

le long de Z, on obtient que la deuxi~me fl~che horizontale

est un isomorphisme, d'oO la r~gularit~ de o ~ le long de Z et le th~or~me 6.3.1 donc le th~or~me 4.1 et en m~me temps une r~ponse locale du probl~me de Hilbert-Riemann pour les faisceaux constructibles. Pour t o u s l e s

dgtails, nous renvoyons g ([]9],

chap.V).

Remarque 6.3.2.- On peut voir que l'extension r~guligre " ~ "

comme objet de D ( ~ X ) h

ne d~pend pas de la r~solution ~ choisie g cause du fait qu'on peut coiffer deux r~solutions par une troisi~me. Les operations mises en jeu commutent. Remarque 6.3.3.- L'extension locale "¢~" peut ~tre globale dans certains cas. Par

106

exemple, si X est alg~brique et ~ e s t

al$~bri~uement constructible, alors il sera

globalement solution de "~' objet de D(~X) h r~gulier. Ceci r~sulte du fait q~e ~ peut se d~visser globalement et qu'on dispose pour les vari~tgs alggbriques de r~solutions globales des singularit~s. Remarque 6.2.4.- On espgre, en utilisant la mgthode de la descente cohomologique comme pour le th~or~me de dualitg ([19], chap. IV), "globaliser" l'extensiono~ dans D(~X)hr" R~sumons nos r~sultats par le diagra~e suivant :

¢

D(~X)hr [F(~

= rRhom X(~=o , ~'X)

~(~)

: ~hOm~X(~,

S(~)

= ~hom )X(~, ~ X )

~X )

O

~x

On a obtenu que ~ et ¢ sont inverses l'un de l'autre (= (HR)'I), que S, donc r,sont ioT

calement sur X surjectifs(= (HR)2 local). §.7

- APPLICATIONS

Nous allons voir quelques exemples du dictionnaire g~om~trique-analytique

ainsi

obtenu.

7.1

-

Dualit~

Soit 4

de P o i n c a r ~ - V e r d i e r .

un complexe de D(~X)c" II est donc solution d'un complexe de D(~)X~ h,~O°°,

En prenant les solutions globales, on trouve

107

[Hi(X ; ~ )

: fExti oo(X ;¢~O=° , ~X)

@x

De m~me, en prenant lea De Rham $1obaux, on trouve ([19], chap. IV) 2n-i, 2n-i ~ X t C x , c I X ; ~ ' CX ) : ~ X ~ x , c ( X

; ~X' o~

Le premier espace de gauche eat le dual alggbrique du deuxi~me espace de gauche. C'est la dualitg de Poincarg si ~

eat constant et de Verdier s i ~

eat quelconque.

Le premier espace de droite eat le dual alg~brique du deuxigme espace de droite. co

C'est la dualit~ pour lea ~x-m0dules [18]. Ainsi on obtient une description analytique de la dualit~ de Poincar~-Verdier qui a ~t~ g l'origine du probi~me de HilbertRiemann pour lea faisceaux constructibles ([18], remarque 5.1). Con~ne on le voit, il a fallu r~soudre le probl~me de Hilbert-Riemann pour traiter le cas d'un faisceau constructible ([19], chap. IV).

7.2

- Bidualit6

locale.

On sait que le faisceau "Cx" eat dualisant dana D(CX) c

[23] ; c'est-~-dire que

l'homomorphisme de D(CX) canonique eat un isomorphisme : > ~ h ° m c x ( ~ h ° m c x ( ~ ' CX ) ' CX) 1 C'est Ig un r~sultat sgomgtrxque. Son analogue analytique affirme que le syst~me " ~'X" eat dualisant dams D ( ~ X ~ h ; c'est-~-dire que l'homomorphisme canonique de D(~X~ eat un isomorphisme :

d~~

~hom~ (mhom ~ ( ~ ,

~X ) , ~ )

x

2~) X Sa d~monstration utilise en plus de l'analyse : Dualit~ de Serre,

toriels topologiques nucl~aires et d~finition c ohomolo$ique de

espaces vec-

X de M. Saeo ([19],

chap. Ill).

7.3 - R~gularit~

Soit ~ u n

et

irr6gularit~.

complexe de D(~X) h non n~cessairement r~gulier et ~ = m h o m ~JX ( ~ , ~ v~)

108 !

son eomplexe des s o l u t i o n s holomorphes. Par l a r~pomse l o c a l e de (HR)2, ~

solution d'un complete ae D ( ~ ) X ) h r ~ R r~gulier ~

fRh°m~)x( ~

e s t aussi

:

' ~X )

Mais, en vertu du th~or~me de bidualit~ precedent ("~'X" est dualisant dans D(~X)h)

, on a :

~o~ : ~X

0

~ ) --~ fRhOmcX(~ , ~ X ) ~

~D x

[R~.Cx(IRhom~)x(~ R , ~'X ) , ~XX)

oo

On o b t i e n t l o c a l e m e n t

~x ~ x

~x

C'est-~-dire que, par une transformation d'ordre infini, on peut ramener un syst~me irrggulier g u n

syst~me rggulier. C'est, semble-t-il, un r~sultat fondamental pour

Kawai-Kashiwara qui ont annonc~ le cas micro-local dans ce m~me colloque. C'est une simple consequence de (HR)2 et du thgor~me de bidualit~ locale. De m~me, une fois qu'on aura une r~ponse globale g (HR)2 , on d~duira l'analogue global. De toute fa§on, les r~sultats de ces auteurs n'englobent pas le th~or~me d'existence ni l'~quivalenee de categories entre D(¢X) c et D(~X~ h.

7.4 - Syst~me de Gauss-Matin. %

Soit ~ : X ~ X un morphisme d'espaee analytique o3 X est lisse mais o~ on ne fait pas d'hypoth~se sur le couple (X,~). Soit ~

un complexe de D(¢~)

et faisant l'hyc

poth~se que R ~ . ~

appartient g D(CX) c" C' e s t 1~ une hypothgse pas trop r e s t r i c t i v e

et qui englobe des situations non propres. Par la rfiponse ~ (HR)2 , il existe localement sur X un complexe de D

hr' soit

N

~,

dontle

complexe s o l u t i o n e s t ~ , ~

:

r~,g _..~ ah__Om~x(~, O"x) %

Nous a p p e l l e r o n s le complexe ~tb, syst~me de Gauss-Manin associfi au couple (X,~).

109

Sans les hypotheses de non singularit~ de X et pour ~ = ¢ ~

, c'est le syst~me de

Gauss-Manin de la singularit~ ~ (dimX= 1) . Pour le voir, on utilise le th~or~me de dualit~ relative pour les ~X-mOdules holonomes. Soit ~ o n s u p p o s e que X e s t

lisse,

tel

partient

g D(,~)X) h . C ' e s t

globales

[9], raais pas seulement.

que son intggrale

le cas si ~ est

projectif

un complexe de D ( ~ ) h ,

le long des fibres et ~

adraettant

o3

= eJ~ a p des filtrations

Th~or~me 7.4.1.- On a un isomorphisme canonique dans D(CX~c : ~n.~hom~(A,

Si ~ =

~'~)[dim X]--~-> ~hom~)x(¢~,

~,

alors ~ =

[ ~

~x)[dim X].

est le syst~me de Gauss-Manin dont le complexe so-

lution e s t N ~ , C ~ p a r l e lemme de P o i n c a r f i . Le thfior~me 7 . 4 . 1 p e u t s e d f i m o n t r e r ~ p a r tir

de s a v e r s i o n

absolue

[18].

Donnons un exemple de syst~me de Gauss-Manin o3 X n'est pas lisse. Soit (~,x) un germe d'espace analytique r~duit. Soit z : (~,x) ~ (¢,0) un germe de fonction analytique. On suppose que (X,x) est plong~e Hans (¢N,0) et ~ est la restriction de ~ :

(~N,0) ~ (¢,0) ~ (x,0) Fixons e, 0 < e ~ s 0

0 et 0 <~ ~e 0. Posons X = B e D~-I(D _ .) o3 D = D

n

est le disque de

0 rayon n centr~ ~ l'origine dans ¢, et B g est la boule ouverte de rayon e centr~e l'origine

dans N.

Notons encore ~ le morphisme induit

X ~ D. D.T. LE n o u s a c o m m u n i -

qu~ le r~sultat suivant [ll]:

Th~orgme 7.4.2.- Le complexe ~ . ~

appartient g D(¢D~ c.

L'int~r~t de ce r~sultat est qu'on ne fait aucune hypothgse de sinsularit~ sur !

ni de propri~t~ sur ~. Alors, par (HR)2 , il lui correspond un complexe de D(~D)hr,

110

soit ~ .

R~.¢~ Le complexe ~ d o i t

= ~hOm~D(~,

{~D)

rendre des services pour l'~tude de la singularit~

(~,x) ~ (¢,0)

tout comme la connexion de Gauss-Manin usuelle. BIBLIOGRAPHIE [I] BERNSTEIN I.N. - The Analytic Continuations of Generalized Functions with Respect to a Parameter. Functional Anal. Appl. 6, 26-40 (1972). [2] BJORK J.E. - Rings of Differential Operators~ North-Holland, Amsterdam (1979). [3] DELIGNE P. - Equations Diff~rentielles g Points Singuliers R~guliers. Lecture Notes in Math. n°163, Berlin, Heidelberg, New York (1970). [4] GROTHENDIECK A. - On the De Rham Cohomology of Algebraic Varieties. Publ. Math I.H.E.S. n°29, 93-103 (1966). [5] KATZ N. - Th-e Regularity Theorem in Algebraic Geometry. Actes Congr~s International des Math~maticiens de Nice, 1970, vol. I, 436-449. Gauthier-Villards, Paris [6] KATZ N. - An Overview of Deligne's Work in Hilbert's Twenty-first Problem. Proceedings of Symposia in Pure Mathematics A.M.S., voi.28, 537-557 (1976). [7] HIRONAKA I. - Resolution of Singularities on an Algebraic Variety over a Field of Characteristic Zero. I-II Ann. of Math n°1 and n°2 (1964). [8] K A S H I W A R A M. - On the Maximally Overdetermin~d Systems of Linear Differential Equations I. Fubl. R.I.M.S. Kyoto University lO, 563-579 (1975). [9] KASHIWARA M. - B-Functions and Holonomie Syst~s. Invent. Math. 38, 33-53 (1976). [10] KASHIWARA M., OSHIWA I. - System of Differential Equations with R'egular Singul~rities and their Boundary Value Problem. Ann. of Math. 106, 145-200 (1977). [II] LE D.T. - Some Remarks on Relative Monodromy. Nordic Summer School I.N.A.V.F. Symposium in Mathematics. Oslo, August 5-25, (1976). Sijthoff Noordhoof (1977). [12] MAKGRANGE B. - Sur les Points Singuliers R~guliers des Equations Diff~rentielles. Enseignement Math. 20, 147-176 (1974). [13] MALGRANGE B. - Le Po-TynSme de I.N. Bernstein d'une Singularit~ Isol~e. Lecture Notes in Math. n°459, 99-119. Berlin, Heidelberg, New York, Springer (1976). [14] MEBKHOUT Z. - La Valeur Prineipale des Fonctions ~ Singularit~s Essentielles. C.R. Acad. Sc. Paris 280, 205-207 (1975). [15] MEBKHOUT Z. - La valeur Principale et le R~sidu Simple des Formes ~ Singularit~s Essentielles. In Fonctions de Plusieurs Variables Complexes II. Lecture Notes in Math. n°482, 190-215, Berlin, Heidelberg, New York, Springer (1975). [16] MEBKHOUT Z. - La Cohomologie Locale d'une Hypersurface. In Fonetions de Plusieurs Variables Complexes III. Lecture Notes in Math. n°670, 89-119, Berlin, Heidelberg, New York, Springer (1977). [17] MEBKHOUT Z. - Local Cohomology of Analytic Spaces. Fubl. R.I.M.S. University 12, 247-256, suppl. (1977). [18] MEBKHOUT Z. - Th~or~mes de Dualit~ pour les ~x-Modules Coh~rents. C.R. Acad. Sc. Paris 285, 785-787 (1977). [19] MEBKHOUT Z. - Cohomologie Locale des Espaces Analytiques Complexes. Th~se de Doetorat d'Etat, University de Paris VII (1978), 126 pages. Soutenue le 15.2.1979. [20] MEBKHOUT Z. - Dualit~ de Poincar~. Sgminaire sur "Singularit~s" de Paris VII. 1977-1979. Publ. de l'Universit~ de Paris VII (~ paraitre). [21] MEBKHOUT Z. - The Poincar~-Serre-Verdier Duality. Proceeding of the Conf. of Algebraic Geometry, Copenhague (1978). Lecture Notes in Math. n°732, Berlin, Heidelberg, New York, Springer (1979). [22] SATO M., KAWAI T., KASHIWARA M. - Micro-Functions and Pseudo-Differential Equations. Lecture Notes in Math. n°287, 265-529. Berlin, Heidelberg, New York, Springer (1973). [23] VERDIER J.L. - Classe d'Homologie Associ~e ~ un cycle. S~minaire Douady-Verdier Ast~risque 36-37, 101,151 (1976).

Conditions

de positivit~ dans une vari~t~ (*)

symplectique

complexe

. Applications

~ l'~tude des microfonctions

Pierre SCHAPIRA D~partement

de Math~matiques

C.S.P. Universit~ Paris-Nord Avenue J.B. ClEment, 93430

Let boundary

~

VILLETANEUSE

be a strictly pseudo-convex

~

in a complex analytic manifold

mal bundle of

~

in

projects on

X

(T~X) +

outside of

and

A;

of

T~X

injective morphisms

associated to

A I.

A

( T ~ ° X) +

and

(T~IX)+

coordonates.

of sheaves of microfunctions

Ce texte est le m~me qu'une Note aux Comptes Rendus de l'Acad~mie des ~ paraltre en 1980.

A

is the

(*) Sciences,

is

We also give a condition for two

to be locally of the type

allows us to construct or

A

~ , and we prove the converse when

for two s.p.c, open sets, in a common choice of symplectic

Ao

the exterior conor-

in the sense of Melin and Sj~strand,

conormal bundle of a complex hypersurface. A°

X , (T~X) +

T~X . We prove that if a conic set

positive with respect to

manifolds

(s.p.c.) open set with real analytic

That

112

§I.

Soit

X

une vari~t~ analytique complexe , ~X

fonctions holomorphes

sur

X .

la l-forme canonique de

T~X '

op~rateurs micro-diff~rentiels de

T~X

sur

X

et nous

Soit

A

~X

le fibrg cotangent g

le faisceau sur

d'0rdre flni

d~signons par

nulle. Nous dirons qu'une partie si

T~X

A

de

D~finition |

:

d'une partie

complexe,

¢ -conique,

X ,

des

T~X

l'espace

T~X

est

~+-conique

R+

(resp.

T~X

~

la projection

priv~ de la section (resp.

¢×-conique)

¢~). On note

~x%

A.

Nous appellerons vari~t~

x

T~X

[6]. Nous notons

est invariante par l'action du groupe

l'enveloppeCX-conique

le faisceau des

lagrangienne dans

¢-Lagrangienne une varigt~ analytique T~X .

Nous appellerons vari~t~ ~-lagrangienne une vari~tg analytique r~elle, + R -conlque, lagrangienne dans la vari~t~ symplectique r~elle ((T~X)~,Re d~). Nous dirons d'une vari~t~ Im d~

~-lagrangienne

est symplectique sur

un complexifi~ de

Exemple ! vari~t~

:

A , ou

A

qu'elle est

l-symplectlque

ce qui est ~quivalent,

si

T~X

si

d~finit

A .

Soit

Y

une sous-vari~t~ analytique complexe de

C-lagrangienne

TSX

faisceau de microfonctions

(le fibr~ conormal ~ C ~yi X

dgfini dans ~6].

Y

dans

X . La

X)

est muni du

C'est un faisceau de

~X-mOdules.

Exemple 2 vari~t~

:

Soit

M

~-lagrangienne

microfonction de

Exemple 3

:

pour

et

l-symplectique

M. Sato [6~.

Soit

~

d~fini par l'~quatlon do # 0

une vari~t~ analytique r~elle de complexifi~ T~

est muni du faisceau

CM

des

C'est un faisceau de ~X-modules.

un ouvert strictement pseudo-convexe 0 < O , o3

P = 0 . Soit

X. La

(T;~X) +

P

(s.p.c.) de

X

est une fonction analytique r~elle, avec le flbr~ conormal ext~rieur g

~ ~

:

(T~X) + = {(z,~) E T~X ; f(z) = O, ~ = k d'f(z), k > O}

113

C'est une vari~t~

~R-lagrangienne et

du faisceau

, image r~ciproque par

C ÷~

des valeurs au bord sur de

~X-mOdules

Soit

~

T

$

et de

~

l-symplectique. ~

On munit

(T~X)

du faisceau

des fonctions holomorphes.

C'est un faisceau

~4 ~.

une transformation canonique complexe homog~ne qui ~change localement (T~X)

. On peut alors, loealement,

~X-modules

de

C M

sur

C +~

"quantifier"

(cf. [4],

~

[5]

en un isomorphisme

ou [i]

dans le

cadre des distributions).

§2. de

Soient

A°

T~X, et

A

et

:

deux vari~t~s

une vari~t~

voisinage d'un point

Th~or~me 1

AI

~-lagrangiennes

¢-lagrangienne,

et

l-symplectiques

ces vari~t~s ~tant d~finies au

p s T~X .

a) On peut trouver une transformation canonique complexe

homog~ne qui ~change

A°

en

(r~

X) +

et

A

en

rSx , oN

~o

est

o un ouvert s.p.c, de

X

et

Y

une hypersurface

b) On suppose la codimension de infgrieure

ou f i g a l e g u n .

On p e u t a l o r s

canonique complexe homog~ne qui ~change (T~IX)+

og

n°

et

~1

analytique complexe de

Tp A ° ~ Tp A 1 trouver

AO

sont deux ouverts

dans

Tp A o

une transformation

en

(T~

X) +

o de

s.p.c,

et

A1

en

X

§3.

Rappelons la d~finition de A. Melin et J. Sj~strand de la positivit~

[2],

E3].

de

Soit d'abord

V. On note

r~elle sur

V, ~

~

V

une vari~t~ analytique r~elle,

l'"application" z E W

associe

de

W

dans

TV

X.

W

un complexifi~

qui dans une carte locale,

(Rez, Imz) g TV . Si

~

est une l-forme sur V,

114

la fonction

z ÷ <~,~(z)>

est bien d~finie modulo

13(V) , l'espace

des fonetions nulles ~ l'ordre 3 sur V.

D~finition

2 ~3]

:

Soit

AO

une vari~t~

~-lagrangienne

et

+ tique de

T~X , A

positif en

une partie

p e AO

si l'on a sur

] - -~l < ~0 , ~) >

Si

A

o

= T • cn ~n

T~¢n

~ -conlque.

~

0

mod

(Ao,A)

On dit que

A , au voisinage

l-symplec-

de

est

p :

13(Ao).-

gtant muni des coordonn~es

(z,%), cette condition

s'exprime par : / -
pour

(z, ~ ) g A

Th~or&me

2

:

soit

A

Soit

~

§4.

N A) ~

>I-

de

C(II m z I +

IRe

~ I) 3

p , et pour une constante

un ouvert s.p.c, de bord analytique

une pattie

~(~

~ >

dans un voisinage

positif en un point que

z, I m

¢×-conique

de

T~X . On suppose

p. II existe alors un voisinage

c > O .

rgel et

((T~X)+,A) ~

de

p

tel

~ = ~ .

On peut construire

des vari~t~s

lagrangiennes

positives

de diff~rentes

manleres.

Th~or&me 3

:

Soit

hypersurface •

un ouvert

analytique

s.p.c,

de bord analytique

complexe ne rencontrant

pas

~

+

~

((T~,TyX) Soit

~

M

est positif en tous points de une vari~t~ analytique

fonction holomorphe

au v~isinage

est une fonction de type positif

(T~X)

~ . Alors .

rgelle de complexifi~ de

r~el, Y une

X ,

f

une

x ° g M , dont la restriction

E6, ch.

I, dgf. 3.1.4j.

115

Soit

Y

l'hypersurface

complexe des z~ros de

est positif en (x °, idf(x°))

Soit dans

~(z,8)

en

associe

p.

Soit

(w,@)

T~

applications

:

Th~or&me 4

:

(z, ~) g T~C n

par la relation

On suppose x

(T~n

A

T•

,

¢-lagrangienne).

Re ~ $ 0

¢~(z,e)

pour

~-lagrangienne

.

(z,0) g ~n x (i ~n). Alors

et

p .

l-symplectique

faisceau de microfonctions

(resp. sur

A

un

localement isomorphe, par transformation canonique

C M

qunatifi~e au faisceau

=

est positif en

Nous appellerons

~X-mOdules

~

Is r6sultat suivant, utile pour les

~n)))

une vari~t~

faisceau de

P ~ ~n x (i~ n)

e , et tangente ~ l'application

la transformation canonique qui a

On d6duit imm6diatement de ~3]

Soit

.

homog~ne de degr~ l en

= ¢~(z,e)

§5.

(TMX,TyX)

une fonction holomorphe au voisinage du point

~n x cn ,



f. Alors

(resp. C ~yIx

de l'exemple 2

de l'exemple

l)

On peut alors ~noncer : Th~or~me 5 A

:

Soit

une vari~t~

A°

une vari~t~

~-lagrangienne

a) : ~-lagrangienne

b) : ¢-lagrangienne.

On suppose

et

et

I-symplectique,

I-symplectique,

(Ao,¢XA)positif

en un point

et on suppose aussi, dams le cas de l'hypoth~se

a) ,

de codimension ~ l

dams

CA

microfonctions

Ao

sur

TpA O • Soit d A ° et sur

p ~ T~X ,

TpA o ~ TpA l

des faisceaux de

A . Alors on peut trouver dams un voisi

nage de

p , un morphisme injectif de

GAIAoN

AI

dams le faisceau

et

ou

~X-mOdules

£Ao N A I ( ~ A O) "

du faisceau

116

Ce th~or~me permet de d~duire des r~sultats d'hypo-ellipticitg analytique sur

A

pour des syst~mes micro-diff~rentiels de r~sultats de propagation

sum

Ao ~7 ] .

Nous sommes heureux de remercier J. SjSstrand avec qui nous avons eu plusieurs discussions fructueuses.

BIBLIOGRAPHIE

[1]

L.

BOUTET de MONVEL

:

Convergence dans le domaine complexe des

s~ries de fonctions propres. C.R. Acad. Sc. Paris, t. 287 (1978) 855-856.

[2]

A. MELIN, J. SJOSTRAND

:

Fourier integral operators with complex

valued phase functions. Lecture Notes in Math. 459 Springer (1975) 120-223.

A. MELIN, J. SJOSTRAND

:

Fourier integral operator with complex

phase functions and parametrix for an interior boundary value problem. Comm. in Partial Diff. Eq. 1 (1976), 313-400.

[4]

M. KASHIWARA

:

Syst~mes micro-diff~rentiels. (1976-77) et s~minaires U.P.N.

M. KASHIWARA, T. KAWAI

:

Publ. Universit~ PARIS-NORD (76-77) non publi~s.

Some applications of boundary value problems

for elliptic systems of linear differential equations. Ann. of Math. Studies, n ° 93, Princeton. Princeton Univ. Press. A paraitre.

[6]

M. SATO, T. KAWAI, M. KASHIWARA

:

Hyperfunctions and pseudodifferential

equations. Lecture Notes in Mat. 287, Springer (1973), 265-529 P. S CHAPIRA

:

Conditions de positivit~ dans une vari~t~ symplectique complexe. Applications ~ l'~tude de l'hypo-elliptioit~ analytique.

(R~sum~). Rencontre sur les E.D.P. lingaires

Saint-Cast, Mai 1979 et article ~ paraltre.

FUCHSIAN SYSTEMS OF LINEAR DIFFERENTIAL EQUATIONS ASSOCIATED TO NILSSON CLASS FUNCTIONS AND AN APPLICATION TO FEYNMAN INTEGRALS by Arno van den Essen (~)

§0.

INTRODUCTION

In 1963 Regge proposed to construct partial differential equations for Feynman integrals, that generalize the standard hypergeometric equation. Ten years later, in 1973, V.A. Golubeva remarked that most probably a Feynman integral will appear as solution of a system of linear partial differential equations, which forms a generalization to several variables of the theory of ordinary linear differential equations of Fuchsian type (see [ 4 ]) . In 1975 Malgrange pointed out the great interest of having a theory of partial linear differential equations having regular singularities in the framework of D-modules; the so-called Fuchsian modules

(see [

7

]).

In the meantime several authors have been working on this subject: Kashiwara, Kawai, 0shima, Mebkhout, Ramis. In my thesis, [ 2 ], I have introduced in a pure algebraic way a very general framework in which Fuchsian D-modules are defined. For example, we study simultaneously modules over the rings of differential operators over the convergent and formal power series, the Weylalgebra

An(k)

etc. etc.

In this note I will try to give an idea of some of the results obtained in my thesis and indicate that Nilsson class functions (and a f o r t i o r i Feynman integrals) are solutions of the Fuchsian D-modules introduced there. At the end of this note I will discuss the connection between the theory of regular singularities introduced by the authors mentioned before and my theory of Fuchsian D-modules. We now pick up again the questions of Regge and Golubeva on the construction of differential equations for Feynman integrals. Let

F

be a Feynman integral, or more general, a Nilsson class function (i.e.

is a multi-valued analytic function on

@n \V(q)

of so-called finite dimensional

determination and satisfying a temperated growth condition; zero polynomial with complex coefficients, i.e.

F

here

q6~[Zl,...,Zn] ,

q

denotes a non-

such that

q

ha!

no multiple factor, bee [ I] and [ 8 ] for more detailsj To such a Nilsson class function

F

we can associate the system of all linear

differential equations (with polynomial coefficients) which

(

F

satisfies

~)Supported in part by the Netherlands Organisation for the Advancement of Pure Research (Z.W.0.).

i.e.

118

L = {Q6An(C)IQF

: o}

(An(~) = $[z I ..... Zn][~ I ''' .,~n ] ; B i = Bz. , see [ I ], Chap. I.) i

So the

A -module n

A F n

is isomorphic to

A /L . n

Our main result is: THEORem4:

A F

is a Fuchsian

A -module, Fuchsian along the principal ideal

n

n

(q) = ~[z I ..... Zn]q there exist

m6~

i.e. for every

and

T6DerC~[z 1,...,z n]

ao,...,am_ 1 6 C [ z 1,...,z n]

such that

Tq6 (q)

such that

TmF + am_iTm-IF+...+a oF = o . In particular, there exist

p6~

and

aij6C[Zl,...,z n]

(qBI)PF + ap_1,1(q31)P-IF+...+ao1F

= o

(qBn)pF + ap-l,n(q3n)P-IF+'''+aonF

= o

To prove this result we also introduce on sheaf

~

such that

~n

a sheaf

~

of left ideals in the

of differential operators: Lz

{Q £

IQF

o}

A non-trivial result, proved by Kashiwara and Kawai, asserts that coherent sheaf of left Proposition: ~/!

~__/~ is a

~--modules and we prove:

is a Fuchsian

~_-module, i.e. each stalk

is a Fuchsian

~z/iz

D -module. Z

Our results on Fuchsian modules in

§I.

[ 2 ] imply that these systems are holonomic.

THE ALGEBRAIC THEORY OF FUCHSIAN MODULES

We now start introducing Fuchsian modules in a pure algebraic way. Let A denote either

C[z I ,...,z n] or 0 the ring of convergent power series in ^ n ' 0n, the ring of formal power series in z 1,...,z n over

z 1,...,z n

over

C.

we denote the ring of

By

D

C,

D = A[~I,...,Bn]. denoted by

An' Rn

or

If and

C-linear differential operators over

A = C[Zl,...,z n] Dn

or

respectively.

0n

or

0n '

Finally, if

R

then

D

A

i.e.

is usually

is a ring, then

denotes the set of all left R-modules which are finitely generated over

M(R)

R .

To avoid technical complications we shall restrict our attention to the case of D-modules without A-torsion.

119

I. I

D-modules and the ideal of singularitie_ss

Let

M

be a left D-module without A-torsion. Then we define its ideal of singulari-

ties, denoted I°

If

S(M),

M6M(D)

,

as follows

then S(M) = {g6AIMg6__M(Ag)}

Obviously

S(M)

2-q If

is arbitrary, we put

M

is a radical ideal in

S(M) =

1.2

Examples:

Observe that

A .

G S(Dm) m6M

If

M = A ,

If

M = Affe(~6@),

If

M = D,

S(M)

.

then

then

S(M) = A . then

S(M) = Af

o

S(M) = (o) .

is a principal ideal in these cases. Indeed by [ 2 ], Chap. II,

Th. 1.24 we get S(M) is a principal (radical) ideal,

Proposition 1.3: f6A)

i.e.

S(M) = Af

(for some

.

An important property of the ideal

S(M)

is, that it serves to describe holonomic

D-modules as follows (see [ 2 ], Chap. III, Th. 3.1). Proposition 1.4: M

Let

~£M(D).

is a holonomic D-module iff

1.5.

Then: S(M) # (o) .

Fuchsian D-modules

Now we introduce Fuchsian D-m0dules without A-torsion and give some properties. To make the definition more understandable we recall first Yu. Martin's definition of a Fuchsian D1-module. Definition 1.6:

Put

A left

(1.7)

0 = C[[t]], ~I = 0[d~ ] " ~1-module

z~ ^ d pm 0(t~-~) p=o

M

is called Fuchsian iff

6M(0) =

'

for all

m6M

In more concrete terms (1.7) amounts to say that for every integer

p, p ~ o

and elements

ao,...,ap_ 1 6 0

" m6 M

there exist an

such that

(t ~d) p m = aom + a1( td~) m +.. "+aP -I (t ~)d P-lm . Now back to the general situation.

Let

I

I) = { T 6 D e r ( A ) I T a 6 I ,

be an ideal in for all

A.

a61}

Define

120

For

wE R(i)

and

mE M

put

ET(m) = Z~= o ATPm . Definition 1.8: TE R(1).

M

M

is Fuchsian along

is Fuchsian iff

1.9 Examples:

M = A

Proposition 1.10:

and

M

M

I

iff

ET(m) E~(A),

is Fuchsian along

M = Aff~(~EC)

is Fuchsian along

for all

mgM

and all

S(M) .

are Fuchsian. I

iff

both

M

is Fuchsian and

I= S(M)

(see [ 2 ], Chap. II, Cor. 1.28 and Cor. 1.31) . Corollary 1.11:

M

Corollary 1.12:

If

Proof:

If

is Fuchsian iff M

So

is F u c h s i a n ~ S n g s o m e

is Fuchsian, then

S(M) = (o),

Der(A) = R(A).

M

M

ideal

I.

S(M) # (o) .

then E (m) EM(A) for all m E M and all T E T = is Fuchsian along A, implying A c S(M) = (o),

~((o)) = a contra-

diction. From Prop. 1.4 and Cor. 1.12 we deduce Proposition 1.14: 1.15

If

ME~(D)

and Fuchsian, then

M

is holonomic.

A criterion for Fuchsian modules

Let

ME~(D)

(without

A-torsion).

without multiple factor. Put say

d = dimkV . By

THEOREM 1.16: I) M

Suppose

V = k @A M,

S(M) # o

where

k

and let

f # o, f E S ( M )

is the quotient field of

[ 2 ], Chap. II, Th. 4.4 and Chap. I, Th. 1.3

A,

we get

There is equivalence between

is a Fuchsian D-module.

2) Ef~i(m)E~(A),

for all

mEM

and all

3) For every irreducible component 8p~Ap,

p

the differential operator

of

i. f

and every

T = p(Sp)-1~

on

DE Der(A) V

such that

has the following

property: If

e EV

and

is a cyclic vector of T in V (i.e. (e,Te, .... Td-le)

Tde = -aoe+...+a d iTd-le, -

where

§2.

then for all

i

a. E A l

p

is a k-basis of V)

(= {a/bE kiaE A , b ~ p)),

p = Ap .

ANALYTIC CONSEQUENCES

We shall now try to illuminate the above results by describing their analytic meaning.

121

Let

X

a coherent

be a complex analytic manifold of dimension sheaf of left

g--modules on

X

without

n

and let

0--torsion.

By

D/L

M = SM

be

we denote

the set S M = {zE XI(z,~)EVM, (here

VM

Let

denotes the characteristic

J(SM,X)

for some

variety of

be the sheaf of holomorphic

~ # o}

M, see [ I ].)

functions

on

X

vanishing

on

SM .

Then we get (see [ 2 ], Chap. IV, Prop. 2.1) Proposition Let

Y

2.1:

S(M~) = J(SM,X) x , for all

be an analytic

subvariety of

X

with dim(Y)

Th. 1.16 the following result can be proved THEOREM 2.2:

XE X . < n.

Suppose

SMCY.

By

([ 2 ], Chap. IV, Th. 2.3)

There is equivalence between:

I) Every component of

Y

of codimension one contains a point

y

such that

M Y

is a Fuchsian 2)

Mx

P -module. Y is a Fuchsian Px module for all

XE X .

2.3

Fuchsian modules and Nilsson class functions

Let

F

and

L

be as before. We already remarked that Z

sheaf of left and that zE@n

D-modules.

SMCV(q)

.

M =

D/L

--

--

It is not difficult to prove that

So to prove that

Mz

is a Fuchsian

q = z .

M

has no

Dz-mOdule

it suffices to prove this in the regular points of

Hence we can assume

is a coherent

--

V(q)

0-torsion

for all

by Th. 2.2.

But then, locally on some polydisc

A 'around

n

zEV(q)

we have

F = Z~0a,h z ~ ( l o g z n ) h where

Q0, h

along

0 z . z n

i s h o l o m o r p h i c on Finally,

A.

to prove that

It

(~E ~, h E ~, h _> o)

is easy to verify A F n

is a Fuchsian

local result for ~ -modules and apply Th. 1.16.

that

DzF

is Fuchsian

A -module, n

we u s e t h i s

For more details we refer to

Z

[3]

§3.

.

FINAL REMARKS ^

Remark 3,1. @[Zl,...,Zn],

In the^ theory above we introduced Fuchsian On, 0n-tOrsion respectively.

with torsion are introduced. rings

D

localization

without

In my thesis also Fuchsian modules

This is done by studying modules over quite general

of so-called universal differential

Fuchsian modules

An,Dn,Dn-mOdules

operators.

It is proved that

(of finite type) are holonomic and that they are stable under

i.e. iff fE A

Fuchsian D-module.

and

M

is a Fuchsian D-module,

then

Mf

is also a

122

Remark 3.2: M

Let M be a D -module of finite type without 0 -torsion. Suppose n n has regular singularities in the sense of [ 5 ] and [ 6 ]. The~ using Th. 2.2,

it can be proved that

M

sian ~ -module without n of modules of the form

is a Fuchsian ~ -module. Conversely, if M is a Fuchn 0n-tOrsion , then it can be shown that M is a finite sum ~n F,

where

F

is some (local) Nilsson class function.

Each such a module has regular singularities, so a finitely generated

0 -module without n M

is Fuchsian iff

M

also.

Summarizing.

If

M

is

0 -torsion then n M

has regular singularities.

For a discussion about Fuchsian

~ -modules with torsion and their relation with n V -modules having regular singularities we refer to [ I ], Chap. IV n REFERENCES [1]

J.-E. BjSrk, Rings of Differential Operators, North-Holland Mathematical Library Series (1979).

[2]

A.R.P. van den Essen, Fuchsian Modules (Thesis, 1979, Katholieke Universiteit Nymegen, The Netherlandsl.

[3]

A.R.P. van den Essen, C.R. Acad.Sc.Paris, t. 289, Ser. A, 103-1054 (1979].

[4]

V.A. Golubeva, Russian Math. Surveys, 31, no 2, 1976, 139-207.

[5]

M. Kashiwara and T. 0shima, Ann. of Math.'106,

[6]

M. Kashiwara and T. Kawai, On holonomic systems of micro-differential equa-

1977, 145-200.

tions III. Systems with regular singularities R.I.M.S-293, (19791. [7]

B. Malgrange, Springer Lecture Notes Vol. 459, 1976, 98-119.

[8]

N. Nilsson, Arkiv fSr Matematik, 5, 1965, 463-475.

Mathematisch Instituut der Katholieke Universiteit, Toernooiveld, 6525 ED Nymegen, The Netherlands Till I oktober 1980:

Matematiska Institutionen, Stockholms Universitet, Stockholm,

Sweden

PART I I

MISCELLANEOUS MATHEMATICAL DEVELOPMENTS

SINGULARITES

DES SOLUTIONS

DES EQUATIONS AUX DERIVEES PARTIELLES NON LINEAIRES Jean Michel BONY Math~matiques - Universit~ paris-Sud 91405 O r s a ~ France

1. Notations e t r 6 s u l t a t s

Nous u t i l i s e r o n s les espaces de Sobolev Hs dont nous ne rappelons pas la d ~ f i n i t i o n e t les espaces de HSlder Cp, d ~ f i n i s pour p E ~ \ propri~t~s suivantes : pour 0 < p < i , on a u E Cp ~

I qui j o u i s s e n t de

lu(y) - u ( x ) l < Ct~ l y - x l p ;

les op~rateurs p s e u d o - d i f f ~ r e n t i e l s d ' o r d r e k a p p l i q u e n t Cp dans Cp-k, pour p e t p - k ~ 7.

Nous nous int~ressons aux r ~ g u l a r i t ~ s " m i c r o l o c a l e s " des d i s t r i b u t i o n s f i n i e s comme s u i t . S o i t (x o, ~o) E ~ n x ( ~ n \ { o } ) ,

d6-

e t s o i t u une d i s t r i b u t i o n

d ~ f i n i e au voisinage de x o. On d i t que u a p p a r t i e n t microlocalement ~ cP[resp. Hs] en (x o, ~o), si pour t o u t op#rateur p s e u d o - d i f f ~ r e n t i e l K d ' o r d r e O, ~ symbole nul hors d'un p e t i t voisinage "conique en ~" de (x o, ~o), on a Ku £ Cp [resp. HS].

Soit ~[u]

= F(x,u(x) ......

~ u ( x ) . . . . ) = 0 une ~quation aux d~riv~es p a r t i e l

les non l i n ~ a i r e d ' o r d r e m, oQ F(x,u o . . . . . uB . . . . ) est une f o n c t i o n C~ des ses a r guments. On supposera ~ventuellement que F est q u a s i - l i n ~ a i r e , e t plus pr~cis~ment q u ' e l l e peut se mettre sous la forme : F(x,u ..... ) = >

>---- A ( x , u , . . . . ~ B u , . . ) ~ p ( k ) . ~ u

ko
+ Ako (x,u . . . . ~ u ) ~ k o

124

Bans c e c a s ,

lorsque les A

on n o t e d = Max(k o , k + p @ )

en f a i s a n t

]a convention

p(k)

= - ~

correspondants ne d~pendent que de x. On a d = m pour une ~quation

non q u a s i - l i n ~ a i r e ,

0 4 d 4 m - ½ p o u r une ~ q u a t i o n

quasi-lin~aire,

et

d = - ~ pour

une ~quation l i n ~ a i r e .

Si u e s t une f o n c t i o n r ~ e l l e de classe Cp avec p • d, les f o n c t i o n s non l i n~aires e t les p r o d u i t s a p p a r a i s s a n t dans l ' e x p r e s s i o n de F sont bien d ~ f i n i s , F(x,u . . . . . . . . .

et

) C Cp-d.

De m6me, si u C Hs, l ' e x p r e s s i o n ~ [u] e s t bien d ~ f i n i e si on a l a double condition

: s > n/2 + Max(ko,P(k)) e t s > n/4 + d.

Si u e s t une s o l u t i o n de ~ [u] = O, on d ~ f i n i t

Pm(X,~) =

s

l'~quation caract~ristique

~u-~F( x , u ( x ) , . . . , ~ 6 u ( x )

. . . . ) ( i ~ ) ~ = O.

Un p o i n t (Xo,~o) e s t d i t c a r a c t ~ r i s t i q u e si Pm(Xo,~o) = O. Les b i c a r a c t ~ r i s t i q u e s sont les courbes i n t ~ g r a l e s du champ h a m i l t o n i e n de Pm (ou de i Pm pour m i m p a i r ) , ~Pm ~Pm c ' e s t - ~ - d i r e les a p p l i c a t i o n ( x ( t ) , ~ ( t ) ) v ~ r i f i a n t ~ ( t ) = - ~ - , ~ ( t ) = - ~x

Th~orCme point sip

1 :

Soit u une solution

(Xo,~ o) non caract~ristique, + m - 2d ~ E. Si u E ~ ,

de classe C p de

~[u]

u est microlocalement

= O, p > d. Alors

en tout

de classe C 2p+ m - 2d

s > n/2 + Max(ko, P(k)) , s > n/4 + d, on a microloca-

lement u E H t pour t < 2s - n/2 + m - 2d.

Th~or~me

2 :

sus. Supposons varidt~

Soit u une solution

de classe H s, s vdrifiant

de plus que Pm(X,~)

soit de classe

caract~ristique,

tel que u appartienne

(Xo,~o).

~ ~

ci-des-

(Xo,~ o) appartenant

microlocalement

avec r < 2s - n/2 + m - 2d - 1. Alors u appartient tique issue de

C 2. Soit

les conditions

~ la

~ H r e__nn (Xo,~o) ,

l_~e long d_~e la bicaract~ris-

125 Sous l'hypoth~se u E Hs, on c h o i s i t p < s - n/2, v o i s i n de s - n/2. On a a l o r s u E Cp. Le gain de r 6 g u l a r i t 6 est p + m - 2d dans le th6or~me 1, q u a n t i t 6 p o s i t i v e d'apr~s les hypotheses. Le th6or~me 2 n ' e s t i n t 6 r e s s a n t que s i p

+ m - 2d > 1, ce

qui implique seulement Pm(X,~) E C1. On n'a plus a l o r s n6cessairement u n i c i t 6 des courbes i n t 6 g r a l e s du champ h a m i l t o n i e n . T o u t e f o i s , un r a f f i n e m e n t du th6or~me 2 assure qu'une s i n g u l a r i t 6 en (Xo,~o) se PrOlonge le long de l ' u n e au moins des demib i c a r a c t 6 r i s t i q u e s issues de (Xo,~o).

Remarques :

Les th6or~mes pr6c6dents ne s ' a p p l i q u e n t qu'~ des s o l u t i o n s ayant

d6ja une c e r t a i n e r 6 g u l a r i t 6 .

Pour des s o l u t i o n s moins r 6 g u l i ~ r e s , i l

peut se pro-

duire des ph6nom~nes du type "ondes de choc", oO le comportement des s i n g u l a r i t 6 s est tr6s d i f f 6 r e n t .

D'autre p a r t , les th6or~mes 1 et 2 ne contr61ent la r 6 g u l a r i t 6 m i c r o l o c a l e que jusqu'~ un c e r t a i n ordre de r 6 g u l a r i t 6 . Au delA, comme le montrent les exemples de B. Lascar [5] et J. Rauch [ 6 ] , des ph6nom~nes typiquement non l i n 6 a i r e s apparaissent.

Enfin, les c a r a c t 6 r i s t i q u e s d6pendent en g6n6ral de la s o l u t i o n u elle-m6me. Les th6or~mes 1 et 2 ne donnent une d e s c r i p t i o n g6om6trique exacte de la propagat i o n des s i n g u l a r i t 6 s que pour les 6quations q u a s i - l i n 6 a i r e s dont la p a r t i e p r i n c i p a l e est l i n 6 a i r e . T o u t e f o i s , dans le cas g6n6ral, des estimations p o r t a n t sur u f o u r n i s s e n t des estimations sur les domaines oO u est r 6 g u l i ~ r e .

Nous ne donnons i c i qu'un b r e f apergu des d6monstration des th6or~mes 1 et 2. Les d6monstrations d 6 t a i l l 6 e s f i g u r e n t dans [ 1 ] . Certains cas de ces th6or~mes ont 6galement 6t6 d6montr6s par B. Lascar [5] et J. Rauch [ 6 ] .

2. Un nouveau calcul symbolique S o i t a(x,~) une f o n c t i o n homog~ne de degr6 men C, de classe C~ en ~ pour ~,0,

126 support compact en x et de classe Cp e n x (p > 0 non e n t i e r ) . Posons :

TaU(X) = (2~) -n f e i X ' ~ x ( ~ ) ~ ( ~ - n , n ) Q ( n ) d n

oQ ~ est la transform~e de Fourier de a par rapport a la premiere v a r i a b l e , et oQ × est ~gale ~ 1 pour l~-nl -< ~ l l n l

et ~gale ~ 0 pour l~-nl > ~ 2 n, avec 0 < ~1 < ~2

< 1.

L'op~rateur Ta applique Ca dans C°-m , Hs dans Hs-m quels que s o i e n t o (non e n t i e r ) et s. Si ×1 et ×2 v ~ r i f i e n t

les conditions ci-dessus, la d i f f e r e n c e des

deux op6rateurs associ~s applique C°(H s) dans C°-m+p (Hs-m+P). On se ram~ne en f a i t , l ' a i d e de la d~composition en harmoniques sph~riques, plications"

~ l ' ~ t u d e des " p a r a m u l t i -

: op~rateur Ta associ~s a une f o n c t i o n a(x) de classe Cp. On ~tudie

ceux-ci notamment a l ' a i d e du d~coupage "en couronnes dyadiques" de R. Coifman et Y. Meyer [ 2 ] .

Dans un ouvert Q de R n , on note z m la classe des fonctions du type p am(X,~ ) + am_1 (x,~) + . . . + am_[p](x,~), d ~ f i n i e s dans ~ x ( ~ n \ { 0 } ) ,

o0 am_k est

homog~ne de degr~ m-k en g, C~ en ~, C° - k en x.

On d i t que A est un op~rateur p a r a d i f f ~ r e n t i e l

(proprement supportS) dans ~,

d'ordre m e t de classe Cp [A E Op(~pp)] si A est un op~rateur proprement supportS, et s ' i l

e x i s t e a(x,~) E zm(~) tel que, pour t o u t compact K de ~, quelles que s o i e n t

~1 et ~2 dans Co(a ) , ~gales ~ 1 au voisinage de K, l ' o p 6 r a t e u r A - ~2T i a, r e s t r e i n t aux d i s t r i b u t i o n s

~ support dans K, applique C° (resp. Hs) dans C°-m+p (resp. Hs-m+p)

L ' a p p l i c a t i o n qui ~ un op~rateur p a r a d i f f ~ r e n t i e l est bien d ~ f i n i e et s u r j e c t i v e .

A associe son s y m b o l e a = o ( A )

127

Proposition 4 :

m ml+m2 Soient A E 0p (Z 1)~et B E 0p (Z 2). Alors A.B E 0p (Zp ) et

on a :

o (AB) = ~(A) ~ ( B )

L'adjoint

A

= I~> l+~+k<[p ] ( 1 / ~ ! ) ~ ~ a

.D~b m ml-K 2-

appartientA Op(s 1), e t on a

=

( A ) = [o(A)]* = ~l+k~[p] ( i / ~ ! ) 8c~ D~ x aml_k

3. Fonctions non l i n ~ a i r e s

Proposition 5 :

Soient a E Cp e t u E Ca

a)

s i p > 0, s > O, on a au = Ta.U + Tu.a + r avec r E Cp+~.

b)

s i p > 0, o < 0, p + o > O, on a au = Ta.U + r avec r E Cp+°.

Th@or¢me 6 :

Soit u E C p, p > 0 et F de classe C~. Alors

F(u(x)) = TF,(u(x)).u(x) + r(x) avec r E C 2p

Plus g6n~ralement a on a :

F(u](x) ..... UN(X)) = ZTaF/Su..U i + r avec r E C 2p

si uj E C p. Si de plus u E H s, s > n/2, on a r E H 2s-n/2-~.

D'apr~s la proposition 5, on a u2 ~ 2Tu.U z T2u .u(mod C2P). Une application r~p~t~e de cette proposition et du f a i t que (Prop. 4), Ta . Tb z Tab (mod. p-r~gularisants) permet de d~montrer le th~or~me lorsque F est un polyn6me. Une estima-

128

tion soigneuse des restes, et les th~or6mes de Bernstein sur l'approximation des fonctions C~ par des polyn~mes permettent de conclure.

Corollaire 7 :

Soit u E C p, p > Max(ko, p(k)), une solution de ~ [ u ]

P u_nnop~rateur paradiff@rentiel

d'ordre m e t

= O. Soit

de classe C p+m-2d de symbole

(P) = ~ ~F/~u~ (x,u(x) .... )(is) ~ B>2d-p a)

Sip

>

d,

on a Pu

E

C 2p-2d

I

b)

Si u E H s, avec s > n/2 + p e t

s > n/4 + d,

on a Pu E H s+p-2d.

Dans le cas non quasi-lin~aire, on a imm~diatement, pour p > m

~'[u] z ZTBFIauB. ~Bu z TZBFI~uB (i~)~ . u (mod. c2p-2m).

Dans ]e cas q u a s i - | i n ~ a i r e , on applique d'abord la proposition 5 aux produits As (x,u . . . . ) ~ u , puis le th~or~me 6 aux termes A

(...).

~ndications sur les d~monstrations des th~or~mes 1 et 2

Soit donc, sous les hypotheses du th~or~me 1, (Xo,~o) un p o i n t non caract~ris. tique, et s o i t k(x,D) un op~rateur pseudo-diff~ren~iel classique, d'ordre O, dont le symbole s'annule au voisinage de la vari~t~ caract~ristique de P, et dont le symbole p r i n c i p a l est non nul en (Xo,~o).

Soit q E z-mp+m-2d tel que k = q ~=p, et Q E op ( p+m_2d) de symbole q. On a alors k (x,D) u = QPu + Ru 00 R e s t p + m - 2d r~gularisant. On a Q[Pu] E C2p+m-2d (ou H2s-n/2+m-2d-~) et Ru appartient au m6meespace, ce qui prouve le th~or~me.

En ce qui concerne le th6or~me 2, c ' e s t une consequence immediate du th~or~me

129

lin~aire suivant :

Th6or~me 8 :

Soit P E Op (E~), s > 1, dont le symbole principal Pm est r~el et

de classe C 2. Soit u E H s tel ~_2_Pu E H t a_~uvoisinage d'un arc de bicaract~ristique. Pour r ~ Min (t + m - I, s + ~ - 1), si u est de classe ~ l'arc de bicaract@ristique,

u est de classe ~

en un point de

au voisina~e de l'arc entier.

La d~monstration de ce dernler th6or~me est assez technique, mais voisine dans son principe de la d~monstration de H~rmander [4] dans le cas pseudo-diff~rentiel, fond~e sur une in~galit~ "d'~nergie". Un point-clef de cette d~monstration est le th~or~me suivant (in~galit6 de Garding pr~cis~e pour les op~rateurs paradiff~rentiels), que nous d6montrons par une m~thode inspir~e de [3].

Th@orCme 9 :

Soit S E Op (~),

~ > O. O__nnsuppose l~e symbole principal

sm (x,~) ~ O. I1 existe alors p > 0 tel que

Re (Su, u) ~

-

ctelulm/2_V/2

.

BIBLIO~RAPHIE [1] [2] [3] [4] [5] [6]

J. M. Bony : Calcul symbolique et propagation des singularit~s pour les ~quations aux d~riv6es partielles non lin~aires (Pr~publications Universit~ Paris-Sud). R. Coifman, Y. Meyer : Au-dela des op~rateurs pseudo-diff~rentiels, Ast~risque 57 (1978). A. Cordoba, C. Fefferman : Wave packets and Fourier integral operators. Comm. P.D.E. (1978). L. Hormander : On the existence and the regularity of solutions of linear pseudo-differential equations. L'enseignement Math. 17 (1971) 99-163. B. Lascar : Singularit~s des solutions d'~quations aux d~riv~es partielles non lin~aires. C.R. Acad. Sc. Paris 287 A (1978) 527=529. J. Rauch : Singularities of solutions to semilinear wave equations. J. Math. Pures et Appli. (1979).

COMPORTEMENT

SEMI

CLASSIQUE

HAMILTONIEN

DU

SPECTRE

D'UN

QUANTIQUE.

J. CHAZARAIN.

I - Introduction. On consid~re d a n s ~ n l'op@rateur elliptique (hamiltonien quantique en physique) h2 Q = - TA + V(x) d@pendant du param~tre h ~ ] 0,1] (la constante de Planck en en physique).

On suppose que la fonction potentielle V e s t

2 0 et v@rifie pour Ixl --~ + ~ les conditions suivantes

o(Ixl2-1 ~V(x)

X

=

l

)

pou

0(1)

dans C ~ ( ~ n )

A valeurs

:

o

pour I~l ~ 2

et V(x) 2 clxl 2

avec c > 0.

h2 L'exemple le plus simple est l'oscillateur harmonique - ~ - A + En utilisant les r@sultats de Beals

[ 3 ] et Robert

lxl 2.

[19 ] , on v@rifie que l'on

peut associer A Q un op@rateur auto-adjoint positif dans L 2 ( ~ n )

; son spectre

est constitu@ de valeurs propres (Xj)j 2 1 telles que : 0 < XI ~ X 2 ~

-~. + ~. J Un probl~me constant en m@eanique quantique est d'essayer d'exprimer le comportement asymptotique,

...

avec

Xj

quand h -* 0, des grandeurs li@es A Q en fonction de grandeurs

de la m@canique c]assique, c'est-A-dire d@finies A partir du champ hamiltonien

Hq(X,~) =

~q(x,~)~ x - ~xq(X,~)~

[4 ] et Voros

(ici q(x,~) = - - 7 - +

[20 ] ). Lorsque le champ H

q

V(x))

(cf. Berry et Mount

est compl~tement int@grable, on a de

nombreux r@sultats sur l'expression asymptotique de familles de valeurs propres (Maslov

[18 ] , Duistermaat

[12 ] , Colin de Verdi~re

[9 ] , Leray

[17 ] .... )

131

2. R~sultats Soit

@t le flot du champ hamiltonien Hq, on n o t e ~ l ' e n s e m b l e

des p@riodes des

solutions p@riodiques des ~quations de Hamilton Jacobi (x'(t), ~'(t)) = Hq(x(t),~(t)) La mesure spectrale de Q est d~finie par

~(~) =

z jh]

~(~- ~j),

on lui associe, par transformation de Fourier, la distribution S(t) =

Z

exp(-ih-lt

lj) E6P'(~).

jAI On ale Th@or~me

I

Soit

:

w° CIR et

O E Co(B)

tel que supp p r ] ~ =

@ . Alors la

fonction de h -i T Ih =

< Sh(t),

D(t) e

h-lt o

>

est 0(h~), c'est-~-dire que pour tout N CI~ il existe C N tel que

IIhl

< cN hN, h E lo,ll

.

On peut donner une formulation plus parlante de ce th~or~me en utilisant la notion d'ensemble de fr@quence d'une distribution asymptotique au sens de Guillemin et Sternberg

[ 15 ] . Ici, Sh(t) est une distribution de 6g'(B)

qui d@pend du para-

m~tre h ; on dit que le point (to,To) de T~]R n'est pas dans l'ensemble de fr@co

quence de S (ensemble not@ F[S]

) s'il existe

p E Co(lq) avec

p(t o) # 0 tel

que < Sh(t),p(t)e-iTh-1>

= 0(h ~) pour T voisin de

To .

Alors le th~or~me I permet de montrer l'inclusion ~(F[Sl) Ce t y p e

d'inclusion

C~

ressemble

, oh ~ : T ~ au r@sultat

--~.

concernant

la

f o r m u l e de P o i s s o n

un op@rateur elliptique Q sur une vari@t@ coml~acte (cf. Chazarain Duistermaat

et Guillemin

pour

[6 | ,

[ 13 ] ). Mais l'analogie est purement formelle, car ici

132

la g@om@trie d@pend essentiellement de V alors que dans le cas de la formule de 2 Poisson, c'est la partie principale de Q, ~ savoir ~L~2 , qui intervenait. En fait, ce th@or~me est plutSt ~ rapprocher des travaux de Balian et Bloch et Berry et Tabor

[2 ]

[5 1

Dans le cas o~ le flot hamiltonien est compl~tement p@riodique, on a un ph@nomSne de concentration des valeurs propres au voisinage d'une progression arithm@tique, c'est ~ rapprocher des r@sultats de Duistermaat et Guillemin [21 ], Colin de Verdi~re

[13 ] , Weinstein

[ 11 | avec la diff@rence qu'il s'agit ici d'un compor-

tement quand h --+ 0. Th@or~me 2 : On suppose

ct p@riodique et soit T la plus petite p@riode positive.

Alors, il existe M ~ 0 et spectre

avec

~k(h) = < Y +

~ E ~ tels que

Q C

U kEN

[~k(h ) - Mh 2, ~k(h) + Mh 2]

2~ 1 ~--k + ~ ~T)h et o~ y = ~

moyenne du Lagrangien L(x,x' ) =

pT I L<x(s), x'<s))ds d@signe la Jo

Ix'l 2 2 - V(x) sur une trajectoire de p@riode T.

En plus de l'oscillateur harmonique, il y a de nombreux cas o~ les hypotheses du th@or~me 2 sont satisfaites. Voici une fa@on d'en construire quand n = I. Soit 0(X) une fonction C ~ ( ~ ) ,

paire, born@e, ~ d@riv@e born@e par une constante

k < I. Soit X(x) la fonction r@eiproque de la fonction strictement croissante x = X + @(X). On pose V(x) = ~(X(x)) 2, alors toutes les solutions de x"(t) + V'(x(t)) = 0 sont p@riodiques de p@riode 2~. Si on suppose de plus que pour tout entier j ~ 2, on a

@[J)(x) = 0 (~), alors V(x) satisfait aux condi-

tions de croissance ~ l'infini. 3. Esquisse de la d@monstration du th@or~me I. Iine

peut ~tre question de donner en une conf@rence la d@monstration de ces deux

th@orSmes, aussi on va seulement expliquer celle du premier th~or~me. Soit U(t) = exp(-ih-]tQ) le groupe unitaire solution de l'@quation de SchrSdinger (*)

ih~tU - Q.U = 0,

U(0) = I

133

Alors U est ii6 ~ la distribution S par la relation "S(t) = tr U(t) qui signifie que pour tout

0 E 6 ~ ~) l'op@rateur U 0 < S(t),

O(t)

= tr

=JU(t)

0(t) dt est ~ trace et v@rifie

(Uo).

Pour d@montrer le th@or~me, il suffit de prouver que pour tout N E ~ on a O

N

Ih = 0(h o) ; aussi on est conduit ~ construire une solution approch@e E(t) de (~) modulo h N avec N assez grand. On commence par construire E(t) pour

It I ~ T avec T assez petit, ce qui permettra

de d@montrer le th@or~me pour supp p C I-T,T [. On construit E(t), au moins formellement, sous la forme (E(t)u)(x) = (2Z) -n

//

e+ih-1(S(t'x'n)-Y'q)a(t,~n;h)u(y)dydn ~n x ~n

o~ la phase S(t,x,~) est solution del'@quation caract@ristique

~t s + q(X,SxS)

= 0

sit= o = x.n

N

et o~ l'amplitude a(t,x,~;h) = h -n

Z aj(t,x,q)h j, avec a. solution de l'@quation j=o J

de transport

8t aj + 9x Sgx aj

+ 1

1

~ AS.aj = - 2 A aj_ I

ajlt= ° = 6o, j

~ > 0 ..... N

et a_1 = O.

Une construction analogue a @t@ faite @galement par Fu~iwara

[14 ] et Kitada

[16 ]. Pour d@passer le stade formel, il faut pr@ciser le comportement en t, x, n des fonctions S e t

a.. Posons J consiste ~ d@montrer la

X(x,~) = (I + Ixl 2 +I~12) I/2, la pattie technique

Proposition : Ii existe T > 0 et c > 0 tels que S(t,x,~) est d@fini et v@rifie l~tS(t,x,~) I _> c 2 ~2(x,~), ~

~,qS(t,x,~) = 0(~ p) pour Itl ~ T, (x,n)E ~2n,

P

2, ~multi-indice. De plus, on a

et

~.

~

~x,n aj(t,x,~) = 0(~ p) pour tout p_> 0

Cette proposition permet de montrer que, si on pose Jh = trace ( [ E(t) p(t) e -iTh-lt dt), J

134

on a l'expression Jh = ( 2 ~ - n

flf

eih-1(S(t'x'q)-x'q)

p(t)a(t,x,~;h)dt

dx d~

et de plus, on peut appliquer le th6or~me de la phase non stationnaire,

pour prou-

vet clue Jh = O(h~)" N Ensuite, il fau~£~,v 6 r i f i e r

que I h a l e

m~me comportement asymptotique (modulo h o)

clue Jh' c'est-a-dlre que l'erreur F(t) = U(t) - E(t) n'apporte qu'une contribuN tion O(h o) ~ la trace. Pour cela, on utilise un th6or~me de Asada et Fujiwara [ I ] qui permet de montrer que pour llF(t)ll

Cette majoratio~, telle que

Itl < T on a

~(L2,L 2 )

= o(hN).

combin6e avec le fait que l'op~rateur Q-n poss~de une trace

Itrace (Q-n) I = O(h-2n), entraine l'estimation

trace (

;

F ( t ) p(t) e - i T h - l d t ) = O(h o)

pour N assez grand vis ~ vis de N . o Enfin, quand p n'est pas ~ support dans

I-T,T [ on peut toujours,supposer

son support est dans un intervalle du type

que

]kT, (k+1)T [ et on utilise dans cet

intervalle l'op6rateur E(t - kT).(E(T)) k comme approximation de U(t). Ces r6sultats ont 6t@ annonc6s dans des notes aux C.R. Acad. Soc. REFERENCES [I ]

[7 ] et

[8 I •

: K. ASADA

et D. FUJIWARA

Jap. J. of Math. Vol. 4, n ° 2, 1978, p. 299-361.

[2 ]

R. BALIAN et C. BLOCH

Ann. of Physics, vol. 85, 2, 1974, p. 514-545.

[3 I

R. BEALS

Duke Math. J. 42, I, 1975, p. 1-42.

[4 ]

M.V. BERRY et K.E. MOUNT

Rep. Prog. Phys., 35, 1972, p. 315-397.

[5 I

M.V. BERRY et M. TABOR

J. Phys. A, vol. 10, n°3, 1977, p- 371-374.

[6 I

J. CHAZARAIN

Inv. Math. 24, 1974, p. 75-82.

[7 ]

J. CHAZARAIN

C.R. Acad. Sc. 288, 1979, p. 725-728.

135

i81

J. CHAZARAIN

C.R. Acad. Sc. 288, 1979, p. 895-897.

[91

Y. COLIN DE VERDIERE

Inv. Math. 43, 1977, p. 15-52.

[101

Y. COLIN DE VERDIERE

Duke Math. J. 46, 1979, p. 169-182.

[111

Y. COLIN DE VERDIERE

C.R. Aead. Sc. 286, 1978, p. 1195-1197.

[121

J.J. DUISTERMAAT

Comm. Pure Appl. Math. 27, 1974, p. 207-281

[131

J.J. DUISTERMAAT et

Inv. Math., 29, 1975, p. 39-79.

V. GUILLEMIN

[14]

D. FUJIWARA

Proc. Jap. Acad. 54, 1978, p. 62-66.

[151

V. GUILLEMIN et

Geometric Asymptotics A.M.S.,

1977.

S. STERNBERG

[161

H. KITADA

On the fundamental solution for schr~dinger equations (~ para~tre).

[171

J. LERAY

Analyse lagrangienne et m@canique quantique, Coll~ge de France 1976-77.

[181

V.P. MASLOV

Th@orie des perturbations et m@thodes asymptotiques, Dunod, 1972.

[191

D. ROBERT

Comm. Partial Diff. Equat. 3, 1978, p. 755-826.

[201

A. VOROS

D@veloppements semi-classiques, th~se, Orsay, 1977.

[21 ]

A. WEINSTEIN

Duke Math. J. 44, 1977, p. 883-892.

RATIONAL AND PADE APPROXIMATIONSTO SOLUTIONS OF LINEAR DIFFERENTIAL EQUATIONSAND THE MONODROMYTHEORY by G.V. Chudnovsky (CNRS

-

Paris and Columbia U n i v e r s i t y , New York)

Introduction. The purpose of t h i s paper is to investigate rational and Pad~ approximations to solutions of l i n e a r d i f f e r e n t i a l equations from the perspective of the monodromy theory. The classical Pad6 approximation problem - the construction of a l i n e a r combination of a given function having zeroes of high order - is reduced to the analysis of local exponents of Fuchsian l i n e a r d i f f e r e n t i a l equations. The recurrent relations between Pad~ approximants are reduced to Riemann's contiguous relations [ i ] , [2]. This reduction forces us to concentrate on monodromy properties of Fuchsian l i n ear d i f f e r e n t i a l equations [ 2 ] , [12]. We continue isomonodromy studies [ 2 ] , [ 1 2 ] , [ 1 3 ] , and contiguous relations in this respect are nothing but special cases of the B~cklund transformation [ 1 2 ] , [ 1 7 ] . The present paper is important f o r applications; one possible a p p l i c a t i o n is the a n a l y t i c continuation of solutions of d i f f e r e n t i a l equations represented by the asymptotic series ( i n the s p i r i t of S t i l t i e s ) .

Another application is to

transcendental numbers, where the tool of Pad~ approximants [ 2 ] , [ 1 5 ] , [ 1 6 ] ful in proofs of i r r a t i o n a l i t y

is most use-

and transcendence of values of a n a l y t i c functions

( e . g . , e, 7, log 2 or ~(3)[17]).

In §I we introduce our main object: the Riemannian

module of multi-valued functions on C~I which s a t i s f y the monodromy conditions. In §2 we repeat known facts on Fuchsian equations in the dimension one. The r e l a t i o n between monodromy and exponential matrices and c o e f f i c i e n t s of l i n e a r d i f f e r e n t i a l equations is investigated in §§3,4 (in

§3, equations with regular [ I 0 ] , in §4, with

i r r e g u l a r s i n g u l a r i t i e s ) . Then §§5-7 are devoted to Fuchsian and Riemannian modules; we study the relations between scalar and matrix l i n e a r d i f f e r e n t i a l equations. We i n v e s t i g a t e local exponents of solutions of l i n e a r d i f f e r e n t i a l equations and apply t h i s knowledge to rational approximations. In

§8 proper d e f i n i t i o n s of p-point Pad~

approximations are introduced, and we prove the "perfectness" of the system {xm~ . . . . . . , x ~n} at a f i n i t e set in ~i. Then in §9 a simple formula f o r rational approximations of vector solutions of matrix l i n e a r d i f f e r e n t i a l equations is proposed in terms of monodromy data. In §I0 the recurrences between one-point Pad~ approximants are w r i t ten down e x p l i c i t l y

in terms of the c o e f f i c i e n t s of l i n e a r d i f f e r e n t i a l equations. At

l a s t , in §11 the conditions for perfectness and "almost perfectness" of the Pad~ approximations to solutions of l i n e a r d i f f e r e n t i a l equations are formulated.

137 §1. The Riemann modules of systems of functions s a t i s ~ i n ~ monodro~ conditions. l.l.We repeat now the basic assumptions of Riemann's c l a s s i c studies [ 1 ] , f o l l o w i n g our paper

[2].

We d e c i d e d

to follow our function - t h e o r e t i c approach instead of

the modern algebraic, methods of Fuchsian modules.

Moreover, we prefer to c a l l the

Corresponding modules not Fuchsian, but rather Riemannian; repeating the Lappo-Danilevski [3] notations of the "corps de Riemann". 1.2.We s t a r t from m f i n i t e

s i n g u l a r i t i e s a I . . . . . am and we consider the Riemann sur-

face F(a I . . . . am) which we get from the x-complex plane {p1 by cutting along m curves

a~

a2

a3

1.3.We pick up some fixed representation of the fundamental group 1 ( c p l ~ a l . . . . . am}) in G L ( n ; ~ ) . In other words we can take m non-singular matrices (1.3.1)

V1 . . . . . Vm .

We assume of course that Vi is connected with the point a i ; we associate with the matr i x Vi the l i n e a r transformation: (1,3.2)

z~Viz ,

and so Vi is called an integral s u b s t i t u t i o n associated with a i : i=1 . . . . . m.

Our main

object becomes the f o l l o w i n g ordered object:

(1.3.3)

h .....,a W \al,

.

1.4.Now we introduce the main a n a l y t i c

object connected with ( 1 . 3 . 3 ) . This is the

Fuchsian module of n-plets of the functions a n a l y t i c in F(a I . . . . . am) defined over C[x]. D e f i n i t i o n 1.4.1. Let F(a I . . . . . am) be the space of n-plets ( f l ( x ) . . . . . fn(X)) of anal~ t i c functions, each of them being uniform and holomorphic in F(a I . . . . . am) with the exception ° f the curves ( a l ' ~ ) . . . . . (am' ~)" I 1 D e f i n i t i o n 1.4.2. For any given sequence VI . . . . . Vm we define the following module I'

(l

.4.3)

over ¢ [ x ] .

,a~

!/VI

. . . . . Vml S~aI . . . . amJ The module

. Vml consists of n-plets # ( x ) = ( f l ( x ) . . . . . fn (x) ) from S[ ~ i . . ..,am]

F(a I . . . . . am) such that f o r a ~ j = l . . . . . m and any point x 0 on the curve ( a j , ~ ) , i t values 4_

f (Xo)=(fz(Xo) . . . . . fn(Xo)) and 4÷

+

+

f ( X o ) = ( f l ( x o) . . . . . fn(Xo)),

the lim-

138 when x tends to XO¢(aj,~) from F(a I . . . . . am) in negative (-) and p o s i t i v e (+) directions, are l i n e a r l y related via the transformation ( 1 . 3 . 2 ) : (1.4.4)

~(Xo)t=vj'~+(Xo)tat

XO~(aj,~ ): j = l . . . . . m..

Instead of the functions ~ ( x ) = ( f ( x ) , , f (x)) defined on F(a . . . . am), we can conI ' , ~ "" sider the multi-valued functions ~(x) on Cpl with s i n g u l a r i t i e s ( i . e . , branch points and logarithmic s i n g u l a r i t i e s )

at {a I . . . . . a_} and, possibly, at co.

we speak about the representation of I

Instead of V1 . . . . . Vm

(C~i~{al . . . . . am,w} ) in GL(n;C), and f o r a

fixed basis Y1 . . . . . ym,y~ of ~l(~IPl\{a I . . . . . am,~}); y l +.,.+-~m+,Y=O; we take the matrices Vyi corresponding to Yi ~ i=1 . . . . . m,~.

Then Vy1 .. .Vym-V¥~=I and we replace (1.4.4)

with (1.4.5) ~(x) ~y Vy~ (x) when x makes a c i r c u i t y around the closed path on ¢~l\{a I . . . . . am,~}, and V ~GL(n;@). Y There for Y=nlYl+...+nmYm+nY ~ we have V = [ ~ m . . v n i . v n ~ N a t u r a l l y V stands f o r Vi in (1.4.4): i=1 . . . . . m. IV1 . . . . ,Vml 1.5. Riemann's [ i ] main r e s u l t (theorem 1.6.3 below) shows that the moduleS[a I . . . . am} is at most n-dimensional over ¢ [ x ] . Moreover the construction of J. Plemelj [4] shows /Vl1,. . . . . ,aml Vml t h a t the module S ~a

is indeed n-dimensional over $ [ x ] .

imply a certain Fuchsian l i n e a r d i f f e r e n t i a l

Conditions (1.4.4)

equation of the order n s a t i s f i e d by ~(x).

One should remember though, t h a t there is no single l i n e a r d i f f e r e n t i a l equation of ( V 1, l'" the order n, s a t i s f i e d by a l l elements ~(x) from S ~a ,'Van~ . A c t u a l l y the sequences HI

~(x) from ( 1 . 4 . 3 ) , which have the same monodromy properties (1.4.4) may have local ex/V 1I .. .. .. .. . Vm~ ponents at x=a k that d i f f e r by integers. We c a l l n-plats ~ ( x ) , from S \a amJ contiguous to each other. Then the system of l i n e a r d i f f e r e n t i a l equations of order n s a t i s f i e d by contiguous systems of functions is parametrized b y ~ n ( m + l ) - l . /v II ...... 1.6. Let's take some basis of S \a . . . am) : (1.6.1)

@k(X)=(@k,l(X))l=l . . . . . n : k=l . . . . . n.

Indeed i t ' s enough to assume that ~l(X) . . . . . ~n(X) are l i n e a r l y independent over ¢ [ x ] . /v II ...... Let's denote by ~(x), the fundamental matrix of S \a . . . am) : (1.6.2)

÷t ÷t t ~(x)=(~Z . . . . . d#n)=(q~k,l (x)) k,l =1 . . . . . n.

Now we can present the simple proof of the Riemann theorem about the dimension of /v I ..... Vml S \ a I . . . . amJ : . . . . .

Theorem 1.6.3

For any e l e m e n t : ~ ( x ) = ( f l ( x ) . . . . . fn(X)) of S ,. ,am,] we have the representation of ~(x) as a l i n e a r combination of ~l(X) . . . . . ÷~n(X) with c o e f f i c i e n t s from

[11.

C(x):

(1.6.4)

n

~(x):Ej=lqj(x)~j(x),

139 where uniform functions qj(x) in x ( i . e . r a t i o n a l ) : j = l . . . . . n are defined in the f o l lowing way: det ($1 . . . . . S j - l ' # ' $ j + l . . . . . Sn ) (1.6.5) qj(x)= > : j = l . . . . . n. det ($1 . . . . . Sj . . . . . Sn ) Proof.

I t is t r i v i a l

that functions qj(x) defined by (1.6.5) indeed s a t i s f y the

equation (1.6.4). Now qj(x) is analytic in ?(a I . . . . . am). For x 0 from ( a i , ~ ) : i=l . . . . . m we consider x 0 to be a l i m i t point of x from F(a I . . . . . am) in the negative (-) and posi t i v e (+) directions. Then we have:

(1.6.6)

qj(Xo-)=

det (Vi)-det ($1(x0+) . . . . . ~(Xo+) . . . . . Sn(Xo+)) + + det (Vi).det ($1(Xo) . . . . . Sj(Xo+) . . . . . Sn(Xo ))

+ = qj(x 0 ).

In other words, qj(x) are rational functions: j=l . . . . . n. 1.7. We can consider now any fundamental matrix V . . . . Vm (1.7.1) @\a I . . . . am x) = ($1(x) t . . . . . Sn(X)t) ÷ {V 1 . . . . . V ) for linear independent elements @l(X) . . . . . Sn(X) from i ~al . . . . a ( i . e . l i n e a r l y independent over $(x)). Following Lappo-Danilevski [3] we call @~Vl . . . . . Vm \al,...,am xj a basis of (1.4.3). As a consequence of Riemann's theorem 1.6.3 we get: Theorem 1.7.2. For any two bases ( a1,VI''" " i ~ I x)

~,.,,,

Vml x) @ ( Vell ' . . . . ..,am

of (1.4.2) we have:

~(Va~,::Vaml, x)= o~,~a,, ......amVmI x).~,x,

f o r a r a t i o n a l matrix-function Proo.f. Indeed for

~,,°~

and

P(x) (from GL(n;C(x))).

~x~ = o~v~', l,a,, .'Vaimn x)'. ,~V,,a,...... .a~Vmx)

we have for x÷xO- and XOe(ai,-) j i=1 . . . . . m: P(Xo-) = @(Xo-)-1@(xo-) = @(Xo+)Vi-lvi@(Xo +)

: P(Xo+). Thus (1.7.4) is a rational function in x. The formula (1.7.3) is proved. y_ 1.8. We remark that the d e f i n i t i o n of the basis of S (V:l . . . . . V=m) is the following major property of the nonsingular matrix @fV1 . . . . Vm ~ ' l ' ' w i ~ m e l e m e n t s being functions ~a i . . . . am analytic in F (a I . . . . . am):

~,~

°(~ ..... avmI xo)vo~Vi ~ ~a~ ..... v°~mI x°+)

for x0 on (ai,~): i=1 . . . . . m where Xo+ means the l i m i t x÷x0 from inside of F(a I . . . . . am) in a positive (+) or negative (-) direction. 1.9. The d i f f e r e n t i a l equations s a t i s f i e d by an element of S (VI . . . . . Vm'~ are again \a I . . . . am}

140

consequences of Riemann's theorem 1.6.3. Indeed i f ~ j ( x ) is an element of S fV1 . . . . . Vm'~ thend~ Sj(x) is also an element of \az, 'am} ' Vm~ since (1.4.3) can be d i f f e r e n t i a t e d . S fVl If ~l(X),. ,$n(X) is the basis \al,

of

.,am]

'

. .

s f\val t ". . . . .,a . m... , then according to Theorem 1.6.3:

(1.9.1)

d ÷ ~-~$j(x) = ~z(x)Pzj(X)+...+~n(X)Pnj(X),

j=Z . . . . . n; where Pij(x) are rational

functions in x: i , j = l . . . . . n. In other words, for a basis @fV1 . . . . . Vml x~ of S f V l . . . . . Vm~we have the following ~a I . . . . am l ~a I . . . . am} linear d i f f e r e n t i a l equation with rational coefficients:

(1.9.2)

d-~-@ ~Vl..... V x~ = ~a I . . . . a~

@~I:

....'am Vm X~R(x)

for R(x) from GL(n;$(x)). 1.10. The rational matrix-function R(x) in (1.9.2) has poles in CP1 of the maximal order k~l (the case of k=O is completely t r i v i a l ) . Then k-1 can be called the order of i r r e g u l a r i t y of (1.9.2). For k=1, the system (1.9.2) is called a regular one, and we write the canonical regular system of the equations (1.9.2) in the form [3] m Ui (1.10.1) d~¢(x) = ¢ ( x ) - ~ i=1 x-a i for nxn matrices Uj called d i f f e r e n t i a l substitutions. The generality of the equation (1.10.1) is explained by results of Plemelj [4] and Lappo-Danilevski [3]: Theorem 1.10.2. [3],

[4]. For any ~a am} m there is a basis Y \al"(Vl'" " a' m'Vm x) of _#Vz.I .......... V,,,~

the module (1.4.3) satisfying the canonical regular system of the equations. (1.10.3)

~_~yfa I

d v

,V m . . . .

,

,a m

x) =

{V I . . . . Vm \a I ,

-,a m

Ix).Ei=lxm

Ui

ai

As a consequence of 1.7.2 we obtain Corollary 1.10.4. Two canonical regular systems of equations d~ d-{ : ~'~-~l:l

Ui x-a i

Ui* and d~* = ~ * - > , m ~ i=1 x-a i

belong to the same module

S IV1 . . . . . Vm~ i f and only i f there exists a rational matrix-function G(x) such that \ a 1, ,am/ dG(x) = ~ m mui*-miG . dx i=1 x-a i 1.11. We want to remark that i f the module S { V l . . . . . Vm~ is considered over $(x), \ a I . . . . am} then 7(x), in addition to branch points at x=a i , may have poles in c ~ l \ { a l . . . . . am,~}. S t i l l we speak about F(a I . . . . . am); this ambiguity w i l l cause serious troubles l a t e r . §2. The Classical theory of scalar Fuchsian linear d i f f e r e n t i a l equations. 2.1. We repeat some known facts about Fuchsian scalar linear d i f f e r e n t i a l equations. Though the best way to consider these equations is to use matrix linear d i f f e r e n t i a l equations of the f i r s t order (cf. E. H i l l e [5]), we formulate the results of L. Fuchs'

141 theory independently.

For the proofs see E. H i l l e

[5], A. Forsyth [6],

E. l~ice [7],

B. Golubev [8]. 2.2. Let us consider a l i n e a r d i f f e r e n t i a l

equation of the order n w i t h r a t i o n a l

func-

tion coefficients (2.2.1)

w (n) (z)+Pz(Z)w(n-l)(z)+...+Pn(Z)W(Z)

f o r w =~ i )" ("z )

di/dz i'w(z)

and P j ( z ) ~ ( z ) :

(2.2.1) w i t h r e g u l a r s i n g u l a r i t i e s are s i n g u l a r i t i e s

of P j ( z ) :

only.

j = l . . . . . n.

(2.2.1) is c a l l e d apparent [6] i f i t

= 0

j = l . . . . . n.

We are interested in equations

F i r s t of a l l ,

all singularities

We remind t h a t the s i n g u l a r i t y

is a s i n g u l a r i t y

of s o l u t i o n s

of the system

of some of P j ( z ) :

j = l . . . . . n but

not of the s o l u t i o n s of ( 2 . 2 . 1 ) .

The notion of a r e g u l a r s i n g u l a r i t y

defined in many e q u i v a l e n t ways.

The simples way is described by the roots of an i n -

dicial

f o r (2.2.1)

is

equation [ 6 ] , [ 7 ] :

D e f i n i t i o n 2.2.2.

Let a be a s i n g u l a r i t y

of ( 2 . 2 . 1 ) ,

i.e.,

a singularity

We look f o r formal s o l u t i o n s w(z) of ( 2 . 2 . 1 )

of one of

the P j ( z ) :

j = l . . . . . n.

(2.2.3) invertible

w(z) = (z-a)r°wo(Z) = ( z - a ) r - ( a o + a z ( z - a ) + . . . ) , ao#O, where Wo(Z) is an element of C [ [ z - a ] ] . S u b s t i t u t i n g w(z) i n t o (2.2.1) we obtain some algeb-

r a i c equation Pa(r)=O c h a r a c t e r i z i n g possible values of r. an i n d i c i a l

equation of (2.2.1) at a

and has degree < n.

a c t l y n, then z=a is c a l l e d a regular s i n g u l a r i t y I f z=a is a r e g u l a r s i n g u l a r i t y of the form (2.2.3) e x i s t s . Pa(r) d i f f e r

by r a t i o n a l

of w(z) (see [5],

[6],

of ( 2 . 2 . 1 ) ,

of the form

This equation is c a l l e d I f the degree P a ( r ) i s

ex-

of ( 2 . 2 . 1 ) .

then a fundamental system of s o l u t i o n s

However one must be aware of the case, when the roots of

integers and l o g a r i t h m i c terms may appear in the [7]).

expansion

Using the method of G. Frobenius and L. Fuchs [9], we

obtain the fundamental set of s o l u t i o n s at z=a of ( 2 . 2 . 1 ) as products of (z-a)r.logk(z-a) Theorem 2.2.4. dicial

by a convergent power series at z=a: Let z=a be a s i n g u l a r i t y

equation Pa(r)=O of (2.2.1) at z=a.

of (2.2.1) and r I . . . . . r n be roots of an i n Let { r l , I . . . . . r l k } . . . . . { r l , 1 . . . . r l , k l ~

be 1 blocks of elements of { r I . . . . . r n } , n = k l + . . . + k I such t h a t ~n each block { r i , 1. . . . . r i , k . }

elements are e q u i v a l e n t (mod E), and elements of d i f f e r e n t

nonequivalent Imod ~ ) ,

i.e.

ri,p-rj,qp/7

i f and only i f

blocks are

i = j f o r p=l . . . . . k i ; q=l . . . . . kj.

Let the order of r I . . . . . r n be chosen in such a way t h a t r i , l ~ . . . . . ~ r i , k i .

i=l ..... I.

Then there e x i s t s a fundamental system of s o l u t i o n s of (2.2.1) having the form (2.2~5)

Wi,p(Z) = ( z - a ) r i , P u ~ , p ( Z ) + ( z - a ) r i , p + Z . l o g ( z - a ) - U ~ , p + l ( Z ) + . . . • .. +(z-a) r i ' k i

where uP l,q

log k i - p (z-a)-u p (z) i,k i (z) are a n a l y t i c functions at z=a: p~q~ki; p=l . . . .

The o r i g i n of the p a r t i t i o n form of the monodromy m a t r i x

ki; i=l ..... I.

of { r I . . . . . r n} i n t o blocks is connected w i t h the normal of the system of (2.2.1) at z=a, [ I 0 ] ,

[Ii].

Exactly,

l e t ~(z) = (Wl(Z) . . . . . Wn(Z)) be the fundamental system of s o l u t i o n s of ( 2 . 2 . 1 ) .

We

consider now any simple closed contour y s u r r o u n d i n g a, and not c o n t a i n i n g any singularity

of (2.2.1) other than a.

Let the f u n c t i o n W(z) = (Wl(Z) . . . . . Wn(Z)) be

142 what ~(z) becomes a f t e r z has described the c i r c u i t y.

Since the coefficients

Pj(z): j = l . . . . . n are unaltered by the description of the c i r c u i t y, the equation as a whole is unchanged. Thus W(Z) is expressed in terms of ~(z): (2.2.6)

W(Z) t = Vy.~(z) t

)

Let the canonical Jordan form of Vy be the matrix SyVySy- I : ( I X I "

.. i~ l ' where JX"i

is a Jordan block corresponding to the number ~ i ' and of the sizes ki: i=1 . . . . . 1 and kl+...+k I = n. Now as is well-known [2], [7], [9], the roots r j of the indicial equation are connected with eigenvalues ~j of Vy by a famous formula ~j = e2~Vq-irJ: j = l . . . . . n. Now { r I . . . . . r n} is divided into 1 blocks corresponding to the 1 Jordan blocks J~i of SyVySy-I according to a formula e2~VrZiri,p = ~i: p=l . . . . . ki; i=l . . . . . I. 2.3. There is anotiler characterization of a regular s i n g u l a r i t y due to L. Fuchs [9]: Lemma 2.3.1. The equation (2.2.1) has a regular s i n g u l a r i t y at its singularity z=a i f and only i f Pi(z) = ( z - a ) - i - p i ( z ) for pi(z) is analytic I t is easy to get the most general form of the equations s i n g u l a r i t i e s in the whole z-plane ~ i (including z=~) [6], are called Fuchsian, and t h e i r form is as follows: Theorem 2.3.2. Fuchsian linear d i f f e r e n t i a l equations of ing form: (2.3.3)

at z=a. (2.2.1) haviDg only regular [8], [9]. These equations the order n have the follow-

QI (z) w(n-Z)(z) Q2(z) w(n-1)(z) Qn(Z) w(z) = 0 w(n)(z) + p ~ + P(z) 2 + P(z) n

for P(z) = ~ m j = l ( z - a j ) , of the degree m and polynomials Qi(z) of degree < i(m-1): i=1 . . . . . n. Now we make a few remarks about the local monodromy of Fuchsian linear d i f f e r e n t i a l equations. Due to the form (2.2.5) of the fundamental system of solutions of (2.2.1), we call the roots {r I . . . . . r n} of the indicial equation of (2.2.1) at z=a, local exponents of (2.2.1) corresponding at z=a. I t ' s clear from the form (2.2.5), that the local exponents of (2.2.1) corresponding to z=a are independent of the choice of the fundamental solution of (2.2.1). Let's consider the local exponents of the equation (2.3.3). We have the set of all s i n g u l a r i t i e s of (2.3.3) as {he set {a I . . . . . am} , possibly including ~. If

Qi(z) _ \ P(z) ~

/

then the indicial

.m s=l

Pm,s (Z-as) i

I

+

Qi (z) p(z) i-1 : i=l . . . . . n,

equation corresponding to z=a s has the form

(r)n + ~ V = I Pis ( r ) n - i

= O,

where (r) k = r . . . ( r - k + l ) . Let's denote by ~i(as): i=1 . . . . . n, local exponents of (2.3.3) at Z=as: s=l . . . . . m, and by ~i(~): i=1 . . . . . n, the local exponents of (2.3.3) at z =~ (with the local parameter i / z for a large z). Then we have

143

Pls

n(n-l) 2

n (as) = - E i=1 ~i : s=l . . . . . m

and Em n(n-l) s=1Pls 2

n =Ei=l

Xi(~)

Corollary 2.3.4. The total sum of all local exponents of (2.3.3) corresponding to : all s i n g u l a r i t i e s at ~ 1 is (m-1)-n(~ - I ) E~=IE

ni=l x i ( a s ) + E n i = l

X i ( ~ ) = n(n-l)2 (m-i).

Another way to understand local m u l t i p l i c i t i e s is to reduce the scalar Fuchsian equation (2.3.3) to a matrix linear d i f f e r e n t i a l equation of the f i r s t order with poles of the f i r s t order [5]. ~3. The Global and local monodromy theory of matrix linear d i f f e r e n t i a l equations with regular s i n g u l a r i t i e s . 3.0. In this chapter we describe the monodromy theory of matrix linear d i f f e r e n t i a l equations with regular s i n g u l a r i t i e s following the algorithmic approach of Lappo-Danilevski [ i 0 ] . We formulate Poincare-Lappo-Danilevski solutions of the d i r e c t and inverse monodromy problems in terms of expansions in polylogarithmic functions• In our presentation we follow papers [12] - [14], where some of the applications of the algorithmic approach to Feynman integrals are outlined. 3.1. We Consider a canonical regular system (1.10.1) with d i f f e r e n t i a l substitutions Uj at regular s i n g u l a r i t i e s aj: j = l . . . . . m: (3•1 i) •

dY(x) = ~ n dx ~ ,j=l

YUj x-aj

The system (3•1.1) with regular s i n g u l a r i t i e s at x=a~ representation IVI . . . . . Vml of ~1(¢~1\{a . . . . . a }), o~ ~aI . . . . . am} ± " m Y(x) as a matrix function onP(a I . . . . . am) s a t i s f i e s the (cf. (1.4.4)): (3.1.2)

gives rise to a certain fixed 1.3 and 1•4. This means that --following monodromy property

Y(Xo-) = Vj.Y(Xo +)

for x 0 on the l e f t (-) and r i g h t (+) sides of the cut (aj,~): j = l . . . . . m. In other words, l e t us consider Y(x) to be a multi-valued function on ¢~I with branch points at a I . . . . . am,~, and l e t ¥i . . . . . Tm,T~ be a fixed basis of ~l(£~l\{a I . . . . . am,~}), TI+...+Y m+T~ = O. Then a f t e r enclosing a c i r c u i t along ¥ i ' the i n i t i a l value Y(x) changes via a linear transformation (1.3•2): (3.1.3)

Y(x)~-TVjY(x) : j=Z . . . . . m

and (3.1.4)

Y(x)~VJ(x)

with

(3.1.5) Vi...Vm.V ~ = I. 3.2. The method of Lappo-Danilevsky enables us to find monodromy matrices Vj (socalled integral substitutions) through the coefficients Uj and vice versa. The natural

144 tools for this are polylogarithmic functions [10], which were studied beginning with Euler and Abel. For a fixed b different from a I . . . . . am,~ we define as in [I0] time system of polylogarithmic functions (3.2.1)

Lb(ajz . . . . . a. Ix) Jv

(Jl . . . . . j =1 . . . . . m; ~=1,2,3 . . . . )

by induction

f x dx1

x-ajl

Lb(ajllX)=jb

x~

1 - log b_aj 1

Lb(ajz,...,aj

Ix)=Fb x kb(ajz'"

(3.2.2)

..,a. IXl ) dx 1 J~-I X =a.

Theorem 3.2.3. The fundamental solution Yb(X) of (3.1.1) satisfying (3.2.4) Yb(b) = I is represented as a series (3.2.5)

m) U j I ' " U J ~ " Lb(ajl . . . . . a j J x ) Yb(X)=l+ v ~=i F " " J(Il ' . . . . . 'Jv

,

that is entire in Uj and is uniformly convergent with respect to x in any f i n i t e domain D i n P ( a I . . . . . am). Now we w i l l express the monodromy matrices s i m i l a r l y . Let ¥i be closed contours from x 0 traveling around a i and returning back to x 0 : i=l . . . . . m. Then one defines [t0]:

fy dx = I~V~-1 : j=jl ; Pj(ajllb) = j x-ajI JPJi

(3.2.6)

f Lb(ajl..... ajv_11x)dx Pj(ajl ..... ajvlb) = Jyj x-ajv Theorem3.2,7 [10]. The monodromymatrices Vj in (3.1.3) which correspond to yj and to the fundamental solution Yb(X) of (3.1.1) satisfying (3.2.4) are defined by (3.2.8)

Vj

I

~'~:i

~ . ( 1Jl .. .. .. .. m) J~ U j l . . . U j

.Pj(ajl,

,aj~Ib) : j=l . . . . . m.

3.3. For applications i t is necessary to consider together with monodromy matrices Uj, the so-called exponents Wj at x=aj: j=l . . . . . m. Definition 3.3.1. Let Y(x) be a (fundamental) solution of (3.1.1). We call a matrix Wj the exponent of Y(x) at x=aj i f (3.3.2) Y(x) = (x-aj)WjHj(x) where Hj(x) and Hj(x)- 1 are holomorphic at x=aj: j = l . . . . . m. At the i n f i n i t e point the exponent W is defined by (3.3.3)

Y(x) = (x-1)W~H (x),

where H (x) and H (x) -1 are holomorphic at x=~. In the case when the exponents Wj of Y(x) e x i s t , the relation with the corresponding

145 monodromy matrices V. of Y(x) in ( 3 . 1 . 2 ) , 3 (3.3.4) e 2 ~ W J = Vj : j = l . . . . . m,~. Theorem 3.3.5 [ I 0 ] . Let d i f f e r e n t i a l of a zero matrix.

is very simple;

s u b s t i t u t i o n Uj in (3.1.1) be in the neighborhood

For a fixed b we consider the monodromy matrices Vj of the solution

Yb(X) of (3.1.1) defined in (3.2.8). Then Vj are in the neighborhood of a u n i t matrix and we can define the following matrices: (3.3.6)

Wj

I

- 2~Wz-i-

f o r j = l . . . . . m,~.

log Vj

I

- 2~i

~

(-1) ~-1

~'~=1 ~

(Vj -I)~

Then Wj = Wj(b), defined by ( 3 . 3 . 6 ) , i s the unique exponent of the

solution Yb(X) of ( 3 . 1 . 1 ) , s a t i s f y i n g ( 3 . 2 . 4 ) , at s i n g u l a r i t y x=aj: j = l . . . . . m,~. Moreover, we can find exponents Wj of Y(x) at x=aj: j = l . . . . . m in the general case, when Uj are no longer close to a zero matrix.

According to Theorem VII~ Ch. I I

the exponents Wj are meromorphic functions of the d i f f e r e n t i a l

[I0]

s u b s t i t u t i o n s UI . . . . . Um-

The only s i n g u l a r i t i e s of Wj as functions of UI . . . . . Um, are those values of matrices Uj f o r which there are two eigenvalues of Uj which are d i f f e r e n t by a non-zero r a t i o n a l integer.

In addition to a fundamental solution Yb(X) of (3.1.1) normed at x=b: (3.2.4),

we can consider the so-called metacanonical solutions e j ( x ) of (3.1.1) normed at x=aj [3].

These solutions have the form

(3.3.7)

Oj(x) = ( x - a j ) K j - o j ( x )

f o r Oj(aj) = I . Lemma 3.3.8. I f U. don't have eigenvalues that d i f f e r 3 there e x i s t s a unique metacanonical matrix (3.3.9)

by non-zero integers, then

U. Oj(x) = ( x - a j ) J - O j ( x ) , Oj(aj) = I : j = l . . . . . m.

The same statement is true f o r x=~ i f we put (3.3.10)

U = _~m

j=l

Uj

'

and so (3.3.11)

0 (x) = ( x - l ) U ~ . o j ( x ) ,

Oj(~) = I .

3.4. We devote t h i s part of the paper to e s t a b l i s h i n g some simple r e l a t i o n s between differential,

integral and exponential matrices of the system ( 3 . 1 . 1 ) .

we define a d i f f e r e n t i a l

F i r s t of a l l ,

s u b s t i t u t i o n U at x = a =~ by (3.3.10), so

(3.4.1)

~ m , ~ Uj = 0 z.j=l Theorem 3.4.2 [10]. For any solution Y(x) of (3.1.1) and monodromy problem (3.1.2): the eigenvalues of the monodromy matrix Vj are equal to the eigenvalues of e2~V~-IUJ; and eigenvalues of the exponent Wj are equal to eigenvalues of Uj: j = l . . . . . m,~. Other r e l a t i o n s between Uj, Vj and Wj are contained in ( 3 . 1 . 5 ) ,

(3.3.4) and ( 3 . 4 . 1 ) .

To s i m p l i f y computation we r e f e r the reader to [ I 0 ] , where the theory of functions of the matrix argument is presented.

Here are some of the examples.

I f W=S diag (h I . . . . .

146

. . . . . ~n).S -1, then xW=S,diag(x ~1 . . . . . x~n).s -1. Wj satisfying equations

We can determine all possible exponents

e2~C"2IWj=vj. Let Vj=S.[Jpi(~i) ..... Jpl(ml)] .S-1 where Jpi(mi)

are Jordan blocks of size Si corresponding to ~i: i = l , . . . , l ; pi+...~=n. Then for a 2 r~W~ fixed solution Wj of the equation (3.3.4), all other solutions W~j of e ~¢-i j =Vj can be represented as (3.4.3)

W~ = W. + S - [ I p i - r I . . . . . I p l - r l ] . S - 1 J J

where Ipi is a unit matrix of the size Pi and r i is a rational

integer: i=l . . . . . I.

Also matrices Wj and Wt.j commute for all solutions Wj, W{j of (3.3.4). 3.5. Let us make a few remarks about the solution of the inverse monodromy problem (namely the Riemann-Hilbert problem) of the reconstruction of the system of functions satisfying the given monodromy properties (1.4.4) or (3.1.3). The solution of this problem was proposed by Lappo-Danilevski [I0] as the result of the inversion of the series (3.2.5), (3.2.8) and (3.3.6). First of a l l , for any object (1.3.3): (~I . . . . . Vm~ ,al,. .,aj and V satisfying (3.1.5), there exists a system of d i f f e r e n t i a l substitutions Uj: j = l . . . . . m such that the solution Y(x) of the equation (3.1.1): U, (3.5.1) Yx : y ~ m j j = l x-aj satisfies the monodromy properties (3.1.2). Moreover, these d i f f e r e n t i a l substitutions Uj: j = l . . . . . n can be chosen in such a way that the fundamental solution Y(x) of (3.5.1) has exponents Wj at x=aj: j = l . . . . . m,~ and e2~TL-IWj=vj: j = l . . . . . m. Also in [i0] the problem of the reconstruction of (3.1.1) having exponents Wj at x=aj is considered. F i r s t of a l l , the exponential matrices Wj at x=aj: j = l . . . . . m,~ uniquely determine the system (3.1.1). For given matrices WI . . . . . Wm we can always find a system (3.1.1) having exponents Wj at x=aj, having monodromy matrices Vj.=e2~V'/--1Wj at x=a.: J j = l . ... . m, and having some exponent W at x=~ . The exponent W is not arbitrary since e2nV~-IWl..,e2~V%-IWm-e 2nivrZ-IW~ = 1. However, due to (3.4.3), the choice of W depends on an n-plet of rational integers. The variation of W with the fixed WI . . . . . Wm is exactly equivalent to the construction of the one-point Pade approximation at x=~. §4. The exponents of matrix linear d i f f e r e n t i a l equations in the irregular case. 4.0. We consider here the formal expansion of the solution of an arbitrary linear d i f f e r e n t i a l equation, with rational function c o e f f i c i e n t s , at the neighborhood of the irregular singularity [10]. 4.1. Let's s t a r t with the most general matrix linear d i f f e r e n t i a l rational function coefficients: y.u.(r) (4.1.1)

dY

d~ = / ' Z ~ j = I

r=Z (x_aj)r

'

equation with the

147

then matrices Uj (1) . . . . . Uj (s) are called d i f f e r e n t i a l s u b s t i t u t i o n s of rank s-1 at x=aj. Let Y(x) be a fundamental solution of (4.1.1) normed by (3.2.4) at x=b. Then making a c i r c u i t Yi around the s i n g u l a r i t y a i , we transform the fundamental solution Y(x) l i n e a r l y as in (3.1.2) - (3.1.5): (4.1.2)

Yb(X)~iViYb (x)

for the basis Yi of ~l(c~l\{al . . . . . am,~}), yl+...+ym+y~ = O. Then formally the solution Yb(X) can be represented in the neighborhood of x=aj as

(4.1.3)

Yb(X) = Gb(J)(x).~b(J)(x)

where Gb(J)(x) satisfies the system of equations dGb(J)(x) (4.1.4)

=~,s

Gb(J)(x)-w.(r)j

dx

r=l

(x-aj) r

and

w(1) (4.I.5) where

G~J)(x) = (x-aj) J

(4.1.6)

G~J)(x) = I + ~ = I

.Gb(J)(x); Cj (p) •

i (x-aj) p

-i

is an entire function of(x-a~) . I n (4.1.3) both Yb(J)(x) and Yb(J)(x) -1 are holomorphic r J at x=a.. The matrices W. ( ): r=l . . . . . s are called exponents (of the rank s - l ) at x=a.: J J (1) J j=1 . . . . . m,~. 0nly the exponents Wj are important for the monodromy matrices Vj: (4.1.7)

e

2~vc-_iW.(1) j = Vj : j=1 . . . . . m,~.

4.2__z. Again, in the case when U~(r) ( j = l . . . . . m; r=l . . . . s) is in the neighborhood of the zero matrix, a l l the exponents w~(r): j = l . . . . . m; r = l l . . . . s e x i s t , and representaJ tions (4.1.3) - (4.1.6) take place. However in this case the exponents are not unique and we have: Theorem 4.2.1. Let's assume that Yb (x) s a t i s f i e s ' ( 4 . 1 . 3 ) - (4.1.6) at x=aj and Yb(X) admits also representation (4.2.2)

Yb(X) = G'(x) Y ' ( x ) = (x-aj) Wj(I~ - G ' ( x ) Y ' ( x ) ,

(4.2.3)

~ CIp (x-aj)-P G'(x) = I +~.p=Z

and G1(x) is an entire function in (x-aj) -1 and Y'(x) and Y ' ( x ) -1 are holomorphic at x=a.. Then G'(x) s a t i s f i e s a n equation J G' (x)Wj (r)' dGt (x) =~--~s (4.2.4) dx Lr=l (x-aj) r with e2~V~-iWJ(I) = e2~V~-iWj (1)I = Vj and the equation

148 #

(4.2.5)

dX =~.S'r=l ~

X'w'(r)J - Wj(r)X (x-aj)r

admits a solution X(x) such that X and X- I are polynomial in x. admit a solution X such that X and X- I are polynomials in x.

Oppositely, l e t (4.2.5)

Then Yb(X), together with

representation (4.1.3) - (4.1.6) admits another representation of the form (4.2.2) (4.2.4). 4.3. The investigation of the system (4.2.5) is closely connected with studies of contiguous relations and recurrent formulae for Pade approximations.

In both cases we

must consider the following matrix equation on an unknown matrix X[IO]: (4.3.1)

[X,U] = ~X.

I f U = S diag(o I . . . . . On)-S - I , then a solution is possible only for I = oi-~ k. this case l e t ( C ) i j = 0 i f i t j and { i , j } ~ { k , l } . I f C = SCS- I , then (4.3.2)

In

X = (~I-~k)-[C,U] + [C, [C,U]]

satisfies (4.3.3)

[X,U] = (~l-~k)X.

§5. Scalar Fuchsian l i n e a r d i f f e r e n t i a l equations and t h e i r r e l a t i o n with matrix regular l i n e a r d i f f e r e n t i a l systems. 5.0. One of the most crucial problems in the study of n-point Pade approximations is the analysis of the local monodromies of solutions of Fuchsian equations.

We w i l l con-

sider systems of the scalar Fuchsian l i n e a r d i f f e r e n t i a l equations s a t i s f i e d by systems of contiguous functions (see §2).

We are in the same situation as in §I when the sys-

tem of n scalar Fuchsian equations is equivalent to the matrix l i n e a r d i f f e r e n t i a l equation with regular s i n g u l a r i t i e s .

However passing from the scalar to the matrix

case and vice versa, we are losing local exponents and get apparent s i n g u l a r i t i e s . E.g., the Schlesinger equations [12] in the matrix case correspond to variations of the matrix coefficients with respect to real s i n g u l a r i t i e s ; while in the scalar case Painlev~V[12] correspond to the variation of the positions of the apparent singulari t i e s as functions of real s i n g u l a r i t i e s . 5.1. F i r s t of a l l we want to imbed the scalar Fuchsian equation (2.3.3) into the canonical regular matrix l i n e a r d i f f e r e n t i a l equation (1.10.1).

Let's consider the

fundamental system ~(x) = (Yl(X) . . . . . Yn(X)) of the equation (2.3.3) of the order n: (5.1.1)

y(n)(x)

+~n

Qi (x)

y(n-i

)(x) = O,

i=1P(x) i where P(x)=~mj=l (x-aj) and d(Qi ) <_ i ( m - l ) : i = 1 , . . . n.. Let . Rk(X): . . . k=l

,n be poly-

nomials r e l a t i v e l y prime with P(x), whose degrees s a t i s f y i n e q u a l i t i e s (5.1.2) (5.1.3) and Rn~l.

d(Rn_i+l) > d(Qi) - d(P): i = l . . . . . n; d(Rk+z) + d(P) > d(Rk): k=Z . . . . . n-1 The system (5.1.2) - (5.1.3) is t r i v i a l l y

solvable e.g., with d(Rn_i+ 1) =

149 = (i-l)(m-1):

i=1 . . . . . n.

(5.1.4)

We define

@i,k(X) = y i ( k - 1 ) ( x ) - P ( x ) k - l . R k ( X ) :

k=Z , . . . , n ;

i=l,...,n.

Then we have (5.1.5)

¢i,k,x

=

Rk + ¢i,k+l P-Rk+1

¢i,k

Px 1 : k=l . . . . . n-1 Imk~x JRk + k~-

and (5.1.6)

P

~i,n,x = ¢i,n "n ~

7 n (-1)

-,

,j=l

QJ

P'Rn-j+l"¢i'n-j+l

The system (5.1.5) - (5.1.6) for the functions (5.1.4) can be represented in the canonical regular form ( i . i 0 . i ) . Indeed, l e t {c I . . . . . c d} be the set of (simple) zeroes of the polynomials Rk(X)=O: k=l . . . . . n-I and (5.i.7)

@(x) = (¢i,k(X))~,k=l.

Then the system (5.1.5) - (5.1.7) is equivalent to the canonical regular system for

¢(x) (5.1.8)

d-d-~¢(x) = @(x)

j=l t:l where the expressions of Uj, At follow from (5.1.5) - (5.1.6) and x=c t are, of course, apparent s i n g u l a r i t i e s . 5.2. Let us consider the situation when all Ri are r e l a t i v e l y prime polynomials with simple zeroes, so that d = ~ n (n-k)(m-1) = (m-l)~(~- I ) We can easily compute the k=l eigenvalues of exponential matrices of the system (5.1.8) for (5.1.7) at x=a : ~=i . . . . . m,~and x=ct:t=l . . . . . d. x=a : ~=i . . . . . m,~ (with a =~).

Let X l ( a ~ ) , . . . , X n (a~) be local exponents of (5.1.1) at Then the form of (5.1.4), (5.1.7) shows that the ex-

,m. ponential matrix WV of (5.1.7) at x = a has eigenvalues ~1 (a~) ,~n(a~): ~=I,_ The exponential matrix W of (5.1.7) at x=~ has eigenvalues ~l(~)-(n-l)(m-1} . . . . . . . . Xn(~)-(n-l)(m-l), and the exponential matrix W^ of (5.1.8) at x=c~ has eigenvalm,~ + d ues: 0,0 . . . . . 0,1 for t=1 . . . . . (m-l) . By (3.4.~) we have ~ j : l Uj ~ t : l At: 0 and according to (3.4.2), the eigenvalues of d i f f e r e n t i a l substitutions are the same

as those of exponential matrices. Thus the sum of the eigenvalues of all exponential matrices of (5.1.8) is zero. This gives us ~ m ,v : l~C n

xi(av) = n(~-l____~)(m-i) i=1 and so we obtain another proof of (2.3.5). In this way we transform a Fuchsian linear d i f f e r e n t i a l equation (5.1.1) of the order n with m f i n i t e regular s i n g u l a r i t i e s into a canonical regular matrix system (1.10.1) of the order n having m regular s i n g u l a r i t i e s and ( m - l ) ~ a d d i t i o n a l gularities.

apparent sin-

5.3. I t is quite natural to transform any canonical regular system ( l . l O . 1 ) i n t o n scalar Fuchsian linear d i f f e r e n t i a l equations of the order n, satisfied by contiguous systems of functions.

I f we do t h i s , we again add systems of apparent s i n g u l a r i t i e s

150 to regular s i n g u l a r i t i e s . mann theorem 1.6.3.

Such reduction is carried on in §6 (or §7) based on Rie-

This reduction can be done straightforwardly taking up to n-1

derivatives of (1.10.1). §6. Matrices of the local exponents of l i n e a r d i f f e r e n t i a l equations. 6.0. Now we want to define eigenvalues of the general matrix system of l i n e a r d i f f e r e n t i a l equations with regular s i n g u l a r i t i e s of the form (1.9.2).

We s t a r t f i r s t

of a l l with the most general matrix l i n e a r d i f f e r e n t i a l equation (6.0.1)

dV(x) _ Y(X)'A(X) dx

for the n x n matrix A(X) with rational function coefficients.

We assume that

{a I . . . . . am} are the only singularities of the coefficients of A(X).

The concept of

the monodromy group of the equation (6.0.1) is applied to an arbitrary equation with

regular (§3) or i r r e g u l a r s i n g u l a r i t i e s (§4).

I f {Y1 . . . . . . T m , y j is the basis of

~ l ( { p l \ { a I . . . . . am,W}), Tl+...+ym+y~ = O, then a fixed fundamental solution Y(x), by analytic continuation along a closed path Tv undergoes a l i n e a r transformation (1.3.2): (6.0.2)

Y(x)yF-~ V Y(x): ~=i . . . . . m,~.

The formula (6.0.2) defines a representation of l ( { p l \ { a

I . . . . . am,~}) in GL(n;C1);

and the corresponding subgroup GL(n;{ 1) generated by {V 1 . . . . . Vm,V~} is called the monodromy group of (6.0.1). (6.0.3)

The only general r e l a t i o n is

Vl...VmV ~ = I.

The existence of the monodromy matrices at x=a~ makes i t possible to define exponent i a l matrices W~ at x = a as in 3.3.1 or as in (4.1.5) - , ( 4 . 1 . 7 ) . Our main i n t e r e s t i s concentrated on the eigenvalues of W only.

The exponential

matrix Wv doesn't necessarily e x i s t nor is i t unique, but the eigenvalues of Wv do e x i s t . E.g. in the case of the canonical regular system (1.10.1) the eigenvalues of W are simply eigenvalues of U by the Theorem 3.4.2 (v=l . . . . . m,~). Again by the Theorem 4.2.1, the eigenvalues of W ( I ) are determined by the system (6.0.1) up to the n-plet of rational integers (cf. (3.4.3)). 6.1. Let us proceed with the most general system (6.0.1) in the case of regular sing u l a r i t i e s only, and with the singular set {a I . . . . . am,~}.

For a given fundamental

solution Y(x) of (6.0.1) we have the monodromy matrices V1 . . . . . Vm,V~ at x=a I . . . . . am,~. If (6.1.1)

÷t

Y(x) : ( ~ l t , . . "'Yn ) '

then each Yi(X)=(Yi,l(X),. . . , Y i , n ( X ) ) is an element of S (Vl . . . . . Vm~ corresponding to ~.a I '

Vl . . . . . Yam): i = i . . . . . n. al' ' m

, am/

Now Yi(X) i s , according to §1, 1.6.3 or [2],

the system of

solutions of the Fuchsian linear d i f f e r e n t i a l equations ci{Y}=0 of order n: (6.1.2)

Li { Y i , j ( x ) } = 0 : j=1 . . . . . n; i=1 . . . . . n.

We expand {a I . . . . . am,~} to include the set of a l l apparent singularities of a l l the equations (6.1.2).

The monodromy substitutions corresponding to apparent singularities

151 are t r i v i a l ; tiplicities

however the addition of apparent s i n g u l a r i t i e s by additive integers.

For each system Yi(X),

V1 . . . . . Vm,V~ at x=a I . . . . . am,~ are the same.

may change the local multhe monodromy matrices

Let V =S - l [ J k l ( P l v) . . . . . J k i ( P i V ) ] - S

and

pl (a~) . . . . . ~n (a~) be eigenvalues of V arranged in 1 blocks of sizes k I . . . . . k I. The local exponents (roots of the i n d i c i a ] equation) Xi,~ ( a v ) , . . . , X i , m (a~) of the equation (6.1.2) at x=a~ are related to the eigenvalues of V~ by the famous formula (6.1.3)

e2~d~-1-Xi,j(a~ ) = pj(av): i,j=Z . . . . . n.

of the local exponents of (6.1.2) at x=a by: We define the vector + Xi {av) ,

(6.1.4)

~i (av) = (Xi,Z (a v) . . . . ,X i ,n (av)),

and according to (6.1.3), (6.1.5)

one gets

~i(av) ~ ~i,(av)(mod Z n) : i , i ' =

1 . . . . . n; ~=1 . . . . . m,~ .

According to Corollary 2.3.4 we have (6.1.6)

~v=Z,...,m,

~(~i(av))

_ n(n-Z)2 (m-l)

for the trace ~(.) of $n-vector: o(~(av) ) : ~ n

X (a~) j=Z i , j The r e l a t i o n between ÷Xi (av) and the eigenvalues tained in (6.1.3): "

(6.1.7)

"

exp{2~v~-l-diag ~i (a~)} = diag ~(av):

~(av) = (pi(av) , . . . ,~n(aV)) is coni= I . . . ".

n; v=l . . . . " m, ~ "

The analytic structure of Y(x) at x = a is characterized by the normal form of the monodromy matrices V . We have (6.1.8)

S Y(x) = Z (x)

together with following structure of Z (x) based on the description of 2.2.4 and (2.2.5) of the system of solutions ÷Y i S ~ of (6.1.2). Following our agreement about the order of indexes in the local monodromy, we can w r i t e f o r Z (x) in (6.1.8): (6.1.9)

Z (x)ij

= d i j ( ~ ) ( x ) ( x _ a ) X j , i (a~). ~ v , i j ( x ) :

i,j=l ..... n

for @v i . ( x ) being holomorphic and non-zero at x=a The following is the exact s t r u c , J, ~" fa ~ ture of d~ .(x). Let i=I . . . . . n be fixed, and l e t ' s consider X . . ~ ~J that correspond to the s-th block Jks(Ps ), 1.e. kl+...+ks_ 1 1 ~ j ~ kl+...+k s and e

'J

=Ps "

Now i f we arrange Xi , j (av) in an order such that X.~,j (a~) > X.~ , j + l (a~): J=kl+. " ' + k s - i + +I . . . . . kl+...+ks-1

(this order may be d i f f e r e n t

for a d i f f e r e n t

the form ( h s = k l + . . . + k s _ l + l ) : di, h (x)= ai,hs~; and for u - 2 ~1

log(z-a ),

i),

then d i ~ ( x ) ~J

has

152

.

(6.1 10)

dVi,hs+l(X) = avi,hs+l .

.

(x) + a~.

1,hs

x(a~) ~(a~) .u.(x-a ) i , h s - i,hs+l '

"

.

,(av) ~(a~) d~i,hs+ks_l(X) = av

i,hs+ks_l(X)+"-

+a~

~u ~ i,h~,ks-lJ'(x-a

)

^i,hs-"i,hs+ks-i

where a V i , j ( x ) are regular at x=a . With this form of d~ . we have the complete desl,J cription of the Y(x) at x=a~. We define the matrix of the local m u l t i p l i c i t i e s of Y(x) at x = a by /x(aM)

1,1 "" .

~(a~) \ "'"~,1

Y(X)~ I x i a v ) \

.iav)]

at x=av

1,n . . . . An,n /

or by (6.1.11)

Y(x) ~ (~+l(av)t,.:.,~n (a~)t

at x=a .

In other words, we determine e x p l i c i t l y the analytic structure of Y(x) at x=a , knowing only the matrix of the local m u l t i p l i c i t i e s (also with the knowledge of the normal form of V , i f necessary). 6.2. We define now the matrix of the local exponents of the system (6.0.1) in the case, when (6.0.1) is of canonical form, e.g., of canonical regular form (1.10.1). In this case (6.0.1) may have an exponential matrix W at x = a as in 3.3.1. The existence of the exponential matrix implies p a r t i c u l a r l y simple structure of the matrix of the local exponents of Y(x) at x=a . In p a r t i c u l a r , r i g h t side in (6.1.5) changes to O; this is explained by the presence of apparent s i n g u l a r i t i e s in (6.1.2) in addition to s i n g u l a r i t i e s of (6.0.1) (as we have seen in §5). Let's consider a fundamental =Y(x)A(x) having at a regular s i n g u l a r i t y x=a the solution Y(x) of (6.0.1): dY(x) dx monodromy matrix Vv. (6.2.1)

Let

V = S - l - [ J k l ( P l ~) . . . . . Jkl(Pl~)]-Sv

for blocks Jki(Pi v) of sizes ki: i=1 . . . . . I; kl+...+kl=n. For any given pi ~ we have k i eigenvalues ~ i , l . . . . . Ui,ki of the exponential matrix Wv at x = a such that e2~¢~--l~i,J = piv: i=l . . . . . I. Here we assume that as in 3.3.1 the exponential matrix W determine the structure of Y(x) at x=a : W

(6.2.2)

Y(x) = ( x - a ) VH (x)

for H (x) and H (x) -I being holomorphic at x=a . according to ~1,1 . . . . . Ul,k I . . . . ; P I , I . . . . . Ul,k I W we can write (§3 and [ I 0 ] ) :

Let eigenvalues ~i (av) of W be orThen looking at the normal form of

153

(6.2.3)

S J ( x ) : Z(x)

with the matrix Z(x) having the canonical form (cf. 6.1.): x.(a~) (6.2.4) Z ( x ) i j = d!~)(x)-(x-a ) 1 lj @~,ij (x) for @~,ij(x)=

c(~),j+O(x-a

) -

being an analytic function at x = a and the matrix D

,(v) n (~,I~ =(a..1,j(x))i,j=l has the block s t r u c t u r e D ~ i )where Du,sareks x n matrices: s=l . . . . ~,I . . . . I. In these notations we define: (D~, s ) l j = aij + (~)a2j+ " " + ( K7s - l ) a k s J ' (6.2.5)

(Dv,s ) 2j = a2j + (~)a~.+ I Jj . ..+(~s_2)aksJ, . . . (D~,s)ks j = aks j

where u= 2 ~I

log(x-a ).

In fact the terms d(V)(x).(x-a ) Xi really determine ij

the

main contribution to C Y(x): (6.2.6)

det(~,ij)i

n

,j=llx=a

= det(cl~))ni , j = l # O

Thus (6.2.6) shows that for ~(av)= (Xl(a V). . . . . Xn(aV) ) the matrix of the local exponents of Y(x) at x = a can be represented as (6.2.7)

Y(x)

~

(~ (a~)t

,.

..,~ (a~)t )

at

x=a.

If we assume that Y(x) has exponential matrices W for any x=a : v=l . . . . . m,= (see 3.3.5), then following 3.4.1 - 3.4.2 one gets that the sum of the eigenvalues of all W : ~=l,...,m,~ is zero: (6.2.8)

~'~:I ..... m,~,~:l

xi(av) = 0

This "disagreement" with (6.1.6) is explained by the presence in (6.1,2) apparent singularities ( i . e . , in (6.1.2) the set of a is larger than in (6.0.1)) and by the fact that in 6.1 the equation didn't have a canonical form or exponential matrices. In the case when Y(x) doesn't have any exponential matrix (or has several, like for the irregular singularity §4) the structure of Y(x) is s l i g h t l y more complicated and was discussed in 6.___1_i. More precisely, instead of the canonical form (4.2.6) we had in (6.1.11), (6.1.4): Y(x) ~ (~ (a~)t,.,.,~(a~)t)(mod Mn(Z)) at x=a . 6.3, Bases of Riemannian modules. ~V I . . . . . Vm~ NOWwe can choose the basis of the Riemannian module S~a I, ,am) in many ways. All of these bases of @(x) are characterized by the following data. For the module

154

S a I,

' m

we can choose according to the discussion in §1 and 6.2 above, a single yV 1 . . . . . V~ vector ~(av)=~a~) . . . . . ~ a ~ ) ) c c n corresponding to the local exponents~of S~a I . . . . a~ ~=I . . . . . m,~. Here for the eigenvalues p~ . . . . p~ of V~ we have e 2 ~ a ~ = p i ~ : i=1 . . . . . n; ~=I . . . . . m,~ and (6.3.1)

~'~=l,...,m,~

ni=l ~ ( a ~ ) : O.

Now for any basis @(x) of S ( V l . . . . . V am~We have the matrix of local m u l t i p l i c i t i e s at x=a of the form \ a l . . . . mY (a) (a). (~I,i

(6.3.2)

""~n,1 1

@(X)~~ia~) ia~)] at x=a . \~l,n

~n,n /

For any ~=1 . . . . . m,~ we have "(a~)t (E)) at x=a . , .... ~ )(mod Mn (a~) ÷(a~ ) t ,~(a~ )t I t is obvious that for any matrices A (aV) such that A z(~ .... )

(6.3.3)

@(x) ~

(~ (a~)t

fv I

(mod Mn(Z)), there exists a basis @(x) of S~a 1, ,aVm) m- having A (a~) as its matrix of local m u l t i p l i c i t i e s at x=a : v=l . . . . . m. However, according to 1.7.2, all of these bases are transformed into one another by r i g h t multiplication on a non-zero matrix from GL(n~(x)). These transformations: a)may change A (a~) to A(a~)+K~ with K EMn(~); b) may add new s i n g u l a r i t i e s

to {a I . . . . . am,~} (either apparent or poles).

§7. Riemannian and Fuchsian modules. 7.0. We return now to the study of Riemannian modules started in §i. For this we use also the concept of the Fuchsian module of ~9] (see lectures at this volume) when instead of the family of all Fuchsian equations with a given monodromy group, we consider a system of all solutions of a single Fuchsian equation, which is a module i t self. +

I .....

7.1. Let f(x) = ! f l ( x ) . . . . . fn(X)) be any element of the space S \ a 1, ,am} introduced in §1. The f(x) is a system of multivalued functions (or is uniform a t P ( a I . . . . . . . . am)) with branch points at x = a and monodromy matrices V for v=l . . . . . m,~ and VI...VmV =I.

Now we can consider a module c ( ; ) generated by { f l ( x ) . . . . . fn(X)} over {.

7.2. We already know that ~(x) naturally generates a scalar Fuchsian linear different i a l equation (cf. §I): IV 1 . . . . . Vm~ Lemma 7.2.1. I f ~(x) belongs to S \a I . . . . am/ , then all elements of L(~) satisfy a scalar Fuchsian linear d i f f e r e n t i a l equation (with rational function c o e f f i c i e n t s ) : (7.2.2)

L {y} = 0 for yE~(f).

155 Here the order of L{.} is equal to dim£c(f),-÷ and c ( f ) is the set of a l l solutions of (7.2.2). Proof. ketdim~(~)=n; then we consider ~ ( i ) ( x ) = ( f ~ i ) ( x ) . . . . . fn(i)(x)): i=0,1 . . . . . n. Naturally, according to (1.4.4), a l l ~ ( i ) ( x ) also belong to S (XI ~" "" 'Vm~ ,a~] • The theorem 1.6.3

implies that the system ~ ( i ) ( x ) : i=O,l . . . . . n is l i n e a r l y dependent over ÷i { [ x ] : ~ ni=O Pi(x)f()(x)=O for the polynomials P i ( x ) ~ { [ x ] which is equivalent to (7.2.2). I f dim{L(f)=r
One of the best tests is based on the knowledge

Here is the simplest c r i t e r i o n showing when any non-zero n-plet of function #(x) (V1 . . . . . Vm~ on I-7(a1 . . . . . am) from S \a I . . . . amj is a fundamental system of solutions of the equation (7.2.2), i . e . , when dim~L(~)=n. We have: Theorem 7.3.1. Let M be a monodromy group corresponding to a fixed representation ~ l ( c ~ l \ { a l . . . . . am,~})~GL(n,~ 1) and V1 . . . . . Vm are generators of M, corresponding to a fixed basis y l , . . . , y m of ~ l ( { [ l \ { a I . . . . . am,~}). Let's assume that among the elements of M there exists a matrix M with the following property: for any root ~0 of the charact e r i s t i c polynomial XM(~) of M there is only one elementary d i v i s o r (~-~0) ~ of XM(~) corresponding to M.

Then for any system ~(x) from S

+

~]~',am~

elements of f ( x )

are C - l i n e a r l y independent, which means dim{L(f)=n. The proof follows immediately from the description of the structure of the subgroups of the system of f u n c t i o n s ~ f ) in the v i c i n i t y of s i n g u l a r i t i e s following Jurgens, Hamburger, and Fuchs studies as presented in [~]. 7.4. By the d e f i n i t i o n of 7.1, ~(x) is a system of generators of c(~), and we have I --~1 ., . . .i , V Lemma 7.4.1. For and ÷f ( x ) -from S ( 1 amm~ and any matrix C from G~n,£), the system f.Ct is a system of generators of L ( f ) and belongs to the system S (CVIC-I . . . . . CVmC-I~. \ aI , ,a m / Tb~ischangeof the variables doesn't a f f e c t the eigenvalues of the CV~C- I and gives one the d e f i n i t i o n of local exponents of f ( x ) or of

L(f)

at any point x=b.

First

of a l l , at x=a~ the local exponents ~1 ( a ~ ) , . . . , ~ n (a~) of c(~) are defined e x p l i c i t l y in §2, Theorem 2.2.4 as roots of the i n d i c i a l equation of the d i f f e r e n t i a l equation (7.2.2) s a t i s f i e d by a l l elements of L(~). exponents corresponding to an a r b i t r a r y point x=b. dim~L(~)=n, when we have n exponents.

Moreover we can now describe local We consider only the case when

In the case when ~(x) is a basis of L ( f ) and

x=b is d i f f e r e n t from singular points of the Fuchsian equation (7.2.2) s a t i s f i e d by

156

a l l elements of r,(~), the existence and uniqueness of the s o l u t i o n of any i n i t i a l ue problem at x=b is guaranteed.

val-

This means t h a t we can f i n d a transformation of L(~):

~ - * f - C b reducing ~ to the form ~.Cb = (@ib(x), (x-b)@2b (x) . . . . . (x-b)n-l~pnb(x)) w i t h ~pib (x) regular and non-zero at x=b: i=1 . . . . . n. Thus we define local exponents of 7

L ( f ) at x=b as (0,1,2 . . . . . n - i ) . not the s i n g u l a r i t y

I f x=b is a s i n g u l a r i t y

of the equation ( 7 . 2 . 2 ) ,

but

of any of the functions from L ( ~ ) , then the local exponents of ~(~)

are again r a t i o n a l integers ( r I . . . . . rn). Lemma 7.5.1. I f x=b is an apparent s i n g u l a r i t y

of the system ~.(~) ( i . e . ,

a singular-

i t y of (7.2.2) but not of r.(~)), then local exponents ( r I . . . . . r n) of T.(~) at x=b are n distinct

nonnegative r a t i o n a l

i n t e g e r s , and at l e a s t one of them is greater than or

equal to n. In other words, an apparent s i n g u l a r i t y

x=b is characterized by the existence of an

element ~(x) from L(~) such t h a t qb(x) has a zero of an order > n at x--b, but qb~0. C o r o l l a r y 7.5.2. Let S be any f i n i t e

subset of (~IPI c o n t a i n i n g a l l the s i n g u l a r i t i e s

(real and apparent) of the equation (7.2.2) s a t i s f i e d

by a l l the e l e m e n t s o f L ( ~ ) .

dimL(~)=n, and the local exponents of L(~) at x=c are (h~c)" , . . . , - - ~ c ) ) ,

(7.5.3)

n

EccSEi=l.

~(c)_ n ( n - l )

-

2

If

then

(Isl-z).

The formula (7.5.3) f o l l o w s from (2.3.5) since f o r a r e g u l a r p o i n t x = b , ~ , n hi(b) = i=1 = n(n-1) Moreover we can e a s i l y generalize the C o r o l l a r y 7.5.2 in the case of the 2 Fuchsian l i n e a r d i f f e r e n t i a l equations w i t h algebraic c o e f f i c i e n t s ; or Fuchsian equat i o n s defined on a Riemann surface T of the genus g_>0 [191.

We have n a t u r a l l y by

7.5.2 and arguments of [19], and [6]: C o r o l l a r y 7.5.4. Let T be a Riemann surface of the genus g>0, and l e t L(~) be the n-dimensional space of s o l u t i o n s of the system of Fuchsian l i n e a r d i f f e r e n t i a l t i o n s of the order n w i t h algebraic c o e f f i c i e n t s corresponding to T. finite

set in T c o n t a i n i n g a l l the s i n g u l a r i t i e s

of the equation s a t i s f i e d

of ~ ( f ) then f o r the local exponents (h c) , . . . , h (7.5.5)

n ~cES~i=Z

§8 The d e f i n i t i o n

~i (c)

=

n(n-l) 2

equa-

I f S is any by elements

of L(~) at x=c at c~T we have

(ISl+2g-2)

of Pad~ approximations and perfectness p r o p e r t i e s .

8.0. We s t a r t w i t h the problem of a general p - p o i n t Pad~ approximation [ 2 ] , [ 1 6 ] , [20] to a system of functions f ( x ) = ( f l ( x )

. . . . . f n ( X ) ) defined in some domain of ¢ [21],

[20]: D e f i n i t i o n 8.0.1. Let f ( x ) = ( f l ( x )

. . . . . f n ( X ) ) be a system of n f u n c t i o n s regular at

complex points x=~ I . . . . . X=~p , and l e t mI . . . . . mn be f i x e d non-negative i n t e g e r s . the polynomials Pl(X) . . . . . Pn(X) of

Let

degrees mI . . . . . mn be chosen such t h a t the remain-

157

der function (8.0.2)

R(x) =

~,ni=z P i ( x ) f i ( x )

s a t i s f i e s the following conditions on the orders ord~kR(X) of zeroes of R(x) at X=~k: (8.0.3)

~,~=I °rd~kR(X) > {~ -

ni = l

(mi+l)} - I

In order to concentrate on n o n t r i v i a l situations we can assume ~ n

i=l

(mi+l)>np+l and

change (8.0.3) to (8.0.4)

~ Pk=l ord~kR(X)~n ord~kR(X ) _> { ~

ni= 1 (mi+l)} - 1.

Then the system of polynomials P(x)=(Pl(X) . . . . . Pn(x)) is called an (mI . . . . . mn)-a pproximation to ~(x) at p points ~ l , . . : , ¢ p , ÷ a n d R(x) is called a remainder function. We can write (8.0.2) shortly as R ( x ) = ~ ( x ) - f t ( x ) . Lemma 8.0.5. For any n-plat (mI , .. . ,mn)~Z~ s a t i s f y i n g ~ n i=l(mi+l)~np+l there exists a non-zero system of polynomials ~(x)=(Pl(X) . . . . . Pn(x))~O that is a system of (mI . . . . . . . . . mn)-approximations to ~(x) at ¢ l , ' ~ ' , ~ p ~

Moreover for any hl>n_. . . . . hR>n_ and any

(mI . . . . . mn)~Z~, there exists a system P(x)~O of polynomials of degrees (mI ..... ,mn) such that for R(x)=~(x).~(x) t we have (8.0.6)

ord~iR(x) ~ hk : k=l . . . . . p,

provided only that (8.0.7)

~ ni:l (mi+l) -> { ~ = I

hk} + i .

Remark 8.0.8. I t is reasonable to weaken the condition of a n a l y t i c i t y of ~(x) at ~VI . . . . . Vm~ x=~ k. In the case of the system of functions ~(x) from S ~a I . . . . am~ we can look at the remainder R(x)=~ ni= I P i ( x ) f i ( x ) in the neighborhood of the branch points x=av, and we can formally define orda R(x) as the exponent with the smallest real part among a l l exponents in the development of R(x) at x=ag. one t r a d i t i o n a l l y considered.

This concept is d i f f e r e n t from the

On the other hand, the ordCkR(X) is defined in the nor-

mal sense i f R(x) is analytic at x=~ k.

I t ' s possible to show that in the general

situation and for ~(x) from S [ VI . . . . . Vm~ the function R(x)=~(x).~(x) is a n a l y t i c ÷ ~ a l . . . . amy' at x=av only in the case of f ( x ) uniform at x=a~, i . e . , when V~=I. 8.1. In this paper we follow the classic studies of Pad6 approximations [20], [25] where only diagonal or subdiagonal cases were considered.

In other words, we consider

the cases when either ml=...=mn=N or a l l differences [N-mil: i = l . . . . . n are bounded by a constant depending only on the system of functions f ( x ) = ( f l ( x ) . . . . . fn(X)). 8.1.1. The most important problem for applications, in p a r t i c u l a r for Number Theory, is the problem of the normality and perfectness of the Pade approximation [20] - [22]. The system of Pad~ approximations to ~(x) corresponding to (mI . . . . . mn) at x=¢1 . . . . . ~p is called perfect i f for any (mI . . . . . mn)-approximation system ~(x)=(Pl(X) . . . . . Pn(X))

158

to ~(x) at x=~ 1 , . . . , ~ p , we have equality in (8.0.3): ~

~=1 °rd~kR(X)={~:_1 ( m i + l ) } - l "

II

The system of Pad6 approximations is called almost perfect'[20] approximation ~(x) to #(x) at x=~ 1 . . . . . ~p, we have (8.1.2)

i f for an (mI . . . . . mn)-

I~, ~=I °rd~iR(x) - ~,hi= I (mi+1) + 11 < C(~)

for a constant C(~) depending only on ~(x) and, possibly, on ¢1. . . . . ~p. more refined notions in the case of the one-point Pad6 approximation.

There are Following [22]

we call m=(mI . . . . . mn) a normal point of the one-point Pad~ approximation to ~(x) at x=¢I i f the system of (mI . . . . . mn)-Pad~ approximation ~(x)=(Pl(X) . . . . . Pn(x)) to ~(x) at x=¢, is unique up to a scalar factor. There are only a few examples of normal or perfect systems of Pad~ approximations (see [20] - [23] for their descriptions); these well-known examples follow from the Hermite construction of the normal and perfect system of the Pad6 approximation to ~tx ~nx the exponential functions (e ..... e ) at x=0 [23]. 8.2. Results on "almost perfectness" of systems of functions satisfying linear d i f ferential equations are proved and reproved foralongtimesince C. Siegel (1929) work

on E-functions.

These results were summarized in [23] where the most general

result was proved: Proposition 8.2.1. Let ~(x)=(Wl(X) . . . . . Wn(X)) be a solution of a system of l i n e a r l y differential

(8.2.2)

equations:

d ~(x) = ~(X)'A(X) dx

for A(X) from Mn(~(x)). Let Pl(X) . . . . . Pn(X) be polynomials of degrees d(p 1) . . . . . d(p n) and l e t ~(x) be analytic at x=c and Wl(X) . . . . . Wn(X) be l i n e a r l y independent over C[x]. Then for v(x)=Pl(X)Wz(X)+...+Pn(X)Wn(X) we have ordcr(X) < n.max {d(Pl) . . . . . d(Pn)} + min {ordcWl(X) . . . . . ordcWn(X)} + (8.2.3)

+ n-1 + n ( @ D . Here D is the degree of A(X), i . e . , for g(x) being the common denominator of the

elements of A(x), D is the max{d(g(x)), d ( b i , j ( x ) ) } with n ( x ) g ( x ) = ( b i , j ( x ) ) ~ M n ( £ [ x ] ) . One of the main results of this paper is the establishment of the generalization of (8.2.3) for an a r b i t r a r y set of points in ~ I . Also the sense of D becomes a l i t t l e more clear, namely, in general, D is "the number of poles of A(X) in {~I counted with a m u l t i p l i c i t y minus one" (this is a s l i g h t improvement of the d e f i n i t i o n of D in (8.2.3)). The statement 8.2.1 is very important for the applications, especially in Transcendental Number Theory. E.g., l e t the conditions of 8.1.1 be satisfied and r ( i - l ) ( x ) = P l , i ( x ) Wl(X)+...+Pn,i(X)Wn(X): i=l . . . . . n. I f ordcr(x) ~ (n-1).max{d(Pl),.. . . . . d(Pn)} + C' for C' depending on the equation (8.2.2), then linear forms r ( i ) ( x ) are l i n e a r l y independent: i = 0 , 1 , . . . , n - I and d e t ( P i , j ( x ) ) i , j = i n ~ 0. New developments in investigation of Pad6 approximations to solutions of linear d i f ferential equations belong to Ch. 0sgood, W. Schmidt, who studied in detail rational

159

approximations to algebraic functions and considered rational approximations to solutions of algebraic d i f f e r e n t i a l equations. Very importnat recent progress in the study of "almost perfectness" of algebraic approximants f o r the solutions of l i n e a r d i f f e r e n t i a l equations (or special nonlinear d i f f e r e n t i a l equations s a t i s f i e d by e l l i p t i c functions) belongs to D. Brownawell

and D. Masser [26].

well-Masser generalized early results of Nesterenko.

Results of Browna-

The estimates proved in the

present paper refined considerably previous results on upper bounds of orders of rational approximations of system of functions s a t i s f y i n g l i n e a r d i f f e r e n t i a l equations. 8.3. As an example of the usefulness of our approach based on the Riemann and Fuchsian modules, we show that the system of functions {x ~I . . . . . x~n} is "perfect" with respect to any f i n i t e set in { \ { 0 } .

The results of [21] correspond only to the case

x=l. Theorem 8.3.1. Let ml . . . . . mn be any system of n d i s t i n c t (mod ~) complex numbers. Let {~I . . . . . ~D} be any set of non-zero complex numbers. Then the system of function {x ml . . . . . x n} is "perfect" with respect t o ~ l . . . . . ~p}" This means that for any non-

negative integers mI . . . . . mn and polynomials Pl(X) . . . . . Pn(x) of degrees mI . . . . . mn we have an upper bound of the order of zero of the remainder function (8.3.2)

R(x) = ~ ni = 1 P i ( x )

xwi

of the form (8.3.3)

~' ~=i °rd~kR(X)

~, ni - i mi + p(n-l) Proof. I t is obvious that R(x) s a t i s f i e s a scalar Fuchsian l i n e a r d i f f e r e n t i a l equa-

tion of the order n; we denote the corresponding Fuchsian module of the solutions by (f~. Let y be the simple closed contour in a f i n i t e part of the x-plane enclosing zero. Then continuing R(x) along any closed curve homotopic in C~1\{0,~} to y . k , ke~, we have

(8.3.4)

R(x)

e2

Tk 'x Pi(x)"

Let ~i=e2~VZ-l-~: i = l . . . . . n and (8.3.5)

Rk(X) = ~

n ~ikx~iPi(x) : kE~ i=1 Then by (8.3.4) a l l Rk(X) belong to a module(/~.

Let ~ (x)=~, n-1 k=O qi kxk be Lagrange

interpolation polynomial of the degree n-i such that (8.3.6)

Q i ( ~ ) = ~i,j

: i , j = l . . . . . n.

Then we have by (8.3.5) - (8.3.6) (x)x~i = ~ jn= l ~ ( x ) x ~ e i ( ~ ) :S Thus P i ( x ) - x ~

= ~ n

E n-1 k=O qik k

(x)

=

n-I k=O qik " ~ (x). belongs to a module(~as w e l l .

The module(~has real s i n g u l a r i t i e s

at x=O,~ and a certain set Sing of apparent s i n g u l a r i t i e s . d(~ )=~ >0, dimc(~=n.

Since ~ (x) x ~ e ~ a n d

160

Let ~ D_Sing; 0,=~ ~ and S=~u{O,=}; for bE{ we denote by hi (b) . . . . . ~n(b) the local exponents of~/~ at x=b. (8.3.8)

X~O) _>coi

According to (8.3.7) we have Xi(=)>_ -mi- ~ ,

;

for i = l . . . . . n.

Also for ~c~ we have by 7.5.1 the following:

(8.3.9)

, Xi(~) (n-2~(n-Z) + ~ ni=l XI(~) > O+l+...+(n-2) + maxi= I . . . . . n -> +ordER(x)

Now taking into account 7.5.2 we have ~S

~, ni=1

Xi(~) = n(n-l) 2 (I S I - 2).

I s l - 2 = l ~ l we get by (8.3.8) - (8.3.9) the following: n(n-Z) IQ I > _ ~, n mi + i ~ i . (n-2)(n-Z) + 2 i-i 2 Thus

~ ~EQord~R(x)<_ ~ n i = l mi+l~l(n-1) or

~,~E~

ordeR(x)

"

~ ~c~ {ord~R(x)-n+l}<_ ~'~=i mi.

The inequality (8.3.3) can be improved by a more careful analysis.

E.g. when

ord{kR(X)~n for a l l k=l . . . . . p; we can change p(n-1) in r . h . s , of (8.3.3) to ( n - l ) . §9 The construction of the Pads approximation to the solutions of l i n e a r d i f f e r e n t i a l equations. 9.0. In this chapter we describe the e x p l i c i t construction of the PadG approximation to the solutions of l i n e a r d i f f e r e n t i a l equations formulated in terms of monodromy data. In our presentation we follow the papers [2], [17], [18] in which the reduction of the Pads approximation problem to studies of contiguous system of functions was performed. 9.1. F i r s t of a l l we begin with the systems of l i n e a r d i f f e r e n t i a l equations in --yV I . . . . . Vm~ canonical regular form ( i . 1 0 . 1 ) . Let's consider a basis in @(x) in S ~a I . . . . . amJ s a t i s f y i n g a canonical regular system of equations (3.1.1): (9.1.1)

d@(x) m dx =~v=l

O(x)Uv x-av

We assume as in 3.3.1 and 3.3.5 that for @(x) there e x i s t exponential matrices Wv at any point x=av: v=l . . . . . m including the exponent W= at i n f i n i t y x=a~:=.

The con-

ditions for the existence of the exponential matrices are given in 3.3.5; however these can be removed because the only thing we actually need is to know the eigenvalues of Wv defined by (3.3.6) as well as the normal form of Vv: v=l . . . . . m,~.

We have

by (3.3.2) - (3.3.3): (9.1.2)

O(x) = (x-a~)Wv-H~(x)

for Hv(x) and Hv(x) -1 being analytic at x=av. Let ~i (av) . . . . . ~n (av) be eigenvalues of Wv and l e t ~(a~)=(~l(av) . . . . . ~n (a~)) with the order of eigenvalues corresponding to the normal form of Vv.

Then from 6.2 i t follows that the matrix of the local exponents

161

of @(x) has the form: (9.1.3)

@(x) ~(~(a~)t . . . . . ~(a~)t) at x=a~ for ~=I . . . . . m,~.

We consider now a fixed set S in { for which the Pade approximation problem w i l l be solved. We assume that SC{aI . . . . . am} since can always consider any regular point b of (9.1.1) to be a virtual s i n g u l a r i t y of (9.1.1) by putting Ub=O and Vb=l. In order to construct Pade approximation to @(x) we consider isomonodromy deformation of @(x) changing eigenvalues of exponential matrices Wa at a~S by rational integers. 9.2. Starting with @(x) and the exponential matrices Wa of @(x) at x=aES we can construct d i f f e r e n t contiguous bases of S ~VI al .. .. .. .. . Vm~ amy , connected with @(x) by polynomial relations. For this we consider the most general case when Wa is changed to an arbitrary solution Wa' of the equation (9.2.1) 2~L-~,W a = Log Va: a~S. Then for Va=Sa-l,[Jki(Pl a) . . . . . Jkl
Wa'= Wa + S~ 1 [~Z. rl~a) ,.

,

ZkI. r [a) I a ]'Sa

for arbitrary 1.~rati°nal integers rl(a). . . . . ~ ) . (9.2.3)

r~

We assume however

0 : acS.

At i n f i n i t y we choose an exponential matrix W~ with (9.2.4)

W~ = W~ + S~lo[Ikl.m~ ~) ,. " , I k l Z

m!~(~) ]'S~"

Let all eigenvalues of Wa' be (~l(a)+ rl(a) . . . . . ~ J a ) + r J a ) ) values of W~ be (~l(~)+m~ ~) . . . . . ~J~)+mJ~)). a(m(~))=~i (9.2.5)

for a~S and l e t eiqen-

I f as in 6.1 o ( r ( a ) ) = ~ i

~a),

mi(~ ' then the formula e2~V=-l-Wl'...e 2~{~-IW~' =I implies: ~acS

a(~(a)) + a(~(~)) = O.

Now we choose a nonsingular matrix solution ~(x) of the canonical regular systems having exponential matrices Wa' at x=a~Su{~} and Wa at f i n i t e x=a~S. This ~(x) is a (V I . . . . . Vn~ basis of S \ a 1, .,any and its existence follows from [4] or [ i 0 ] . From (1.7.2) we have (9.2.6)

@(x).P(x) = ~(x)

for n x n matrix P(x) with rational function c o e f f i c i e n t s . to (9.1.2): W' (9.2.7) ~(x) = (x-a) a . ~ , (x)

Here we have in addition

for Ha'(x), Ha'(X) -I beino, regular at x=a and we define Wa'= Waf°raeC\S" Theorem 9.2.8. Let us suppose that ~(x) s a t i s f i e s (9.1.1) - (9.1.3) and ~(x) satisfies (9.2.6) - (9.2.7) with Wa' for aeSu{~} defined in (9.2.1) - (9.2.5). Then P(x)

162 is a matrix with polynomial coefficients. (9.2.9)

Moreover in the notations of (9.2.4), l e t

N = -min {mi(~) }.

Then the polynomial P(x) is of

degree N.

Proof. According to (9.1.2), (9.2.6) - (9.2.7) we have P(x)=@(x)-l~(x) = =|la(x)-l(x-a)-Wa.(x-a)Wa'.Ha'(X).

Since Wa and Wa' commute (see 3.4 above) we have

P(x)=Ha(x)-l.(x-a)Wa'-Wa.Ha'(X).

Because Wa'-Wa=Sa-l.diag(~(a),...,rn(a))-S a we get

(9.2.10)

P(x) = (Sa'Ha(X))-Z'diag ((x-a) rz(a) ,. . . , ( x - a ) rn(a)~1"Sa'H a , (x)

For any f i n i t e point a the condition (9.2.3) gives ri(a)>o (where r i ( b ) z o when bE$\S) for i = l . . . . . n

Thus P(x) is regular in C and is a polynomial.

On i n f i n i t y

we have (9.2.11)

P(x) = (S~H~(x))-l.diag (x -mz(~)

,x-mn(~)).S~.ll~'(x)

Since H~(x) - I , H~'(x) are holomorphic at x=% P(x) is polynomial of the degree max{-ml (~) . . . . . -mn(~)}=N. The theorem 9.2.8 is the basis of our construction of the Pad~ approximation to vec. tor solutions of (9.1.1). 9.3. I f we consider the exponential matrices Wb' for b being a regular point of (9.1.1), then Vb=l and the set of a l l possible solutions W of 2~vs-1~,J=Log I has the form W = S.diag (r I . . . . . rn).S-1 for an a r b i t r a r y matrix S and an a r b i t r a r y sequence (r I . . . . . r n) of rational integers. 9.4. We use Theorem 9.2.8 to construct rational approximation to a solution ~(x) of (9.1.3) U

(9.4.1)

d__ dx ~(x) = ~(x)-E~:I x-~v

We consider as in 9.2 Pade approximations to ~(x) at a fixed set SK{aI . . . . . am}. In order to apply the d e f i n i t i o n of a Pad6 approximation we must assume that ~(x) is analytic at any x=a~S. We leave for the future the discussion of the case when the same solution ~(x) has zero exponents at several real s i n g u l a r i t i e s x=av, because t h i s implies the existence of the new relations among V~.

Thus we assume that S con-

tains only one, say a 1, real s i n g u l a r i t y of (9.1.1) at which monodromy is n o n t r i v i a l (Vv#l); however S may have many regular points.

We may assume, without Ioss of gen-

e r a l i t y that ~(x) corresponds to the exponent ~1(al)=0 at x=al: (9.4.2)

~(x) is regular at x=a I and at any x=a~S

According to §6 this means that i is an eigenvalue of V1 and there is a simple normal factor (X-I) of V1. =

Let SI be the matrix which reduces VI to the normal form: VI=

S~IEI ( a l ) , Jk2(p11),...,Jkl(p11)] SI with a unit Jordan block for ~1 (al)

If

163

Sl@(X)=(~l(al)t . . . . . y n ( a l ) t ) t, then we can i d e n t i f y y(x) with ~! ( a l ) .

We use 9.2.8

to construct a rational approximation to 3, which is not exactly Pad6 approximation since order of zero of the remainder function is less than in 8.0.1 - 8.0.3. 9.5. Let's consider the system of exponential matrices Wb' for bES satisfying the following conditions of the type (9.2.2) - (9.2.3) (9.5.1)

Wb' = ~ - l . d i a g

for rational

(kb,0,0 . . . . . 0),$ I : bES

integers kb~0: b~S.

satisfy (9.2.4),

Then for x =~ we define an exponential matrix W~' to

(9,2.5):

(9.5.2)

W~' = W~ + s~-l.diag (-m . . . . . -m) S~ and

(9.5.3)

~ bcS kb =nm

Theorem 9.5.4. Let @(x) have exponents Wa (for a~CPI) and l e t exponential matrices Wa' : aESu{~}be defined as in (9.5.1) - (9.5.3) for a given sequence of integers kaY0 (a~S). Let ~(x) satisfy (9.2.6) - (9.2.7) for given matrices Wa' : a~Su~}and

~(x) =@(x).P(x). We denote (9.5.5)

81@(x) = (yl(al)(x) t ..... yn(al)(x)t)t;

(9.5.6)

P(x) = (Pij(x)) ni,j=l,

(9.5.7)

Sl~(X) = ( R i , j ( x ) ~ , j = I

Then Pll(X) . . . . . Pnl(X) are polynomials of degrees at most m giving rational approximation to ~(x)=~i(al)fx), at x=a for any aES. The function R11(x)=~,ni=1 P i 1 ( x ) ~ l (x) has order ordaRll(X)~k a for any aES and (9.5.3) is s a t i s f i e d . For the proof i t ' s enough to notice that by the choice of ~(x) and SI~(X)=SI@(X)P(x), Sl@(X).P(x ) = diag{xka,1 . . . . . 1}-Ha'(X) for a~S.

The bound for the degree of Pil(X) follows from 9.2.8.

§i0. E x p l i c i t expressions and recurrent formulas for one-point Pad~ approximation. i0.0. We analyze below the nature of the one-point Pade approximation to solutions of linear d i f f e r e n t i a l equations. The e x p l i c i t expressions for Pade approximants are known now only for classical orthogonal polynomials [24]. We reduce the recurrent formula for Pad6 approximants to Riemann's contiguous relations of the form 1.6.3 or 1.7.2. Using this reduction we propose the e x p l i c i t contiguous relations for onepoint Pade approximations for the solutions of matrix linear d i f f e r e n t i a l equations. Following Lappo-Danilevski [I0] we put the point x=~ of the Pad~ appoximation at infinity. I 0 . i . In view of §9, Pade approximations to the solutions (1.10.1) at x=~ mean that we consider contiguous systems of functions having the same exponential matrices Wv

164

at the f i n i t e changes.

points a v : v=l . . . . . m.

The exponential matrices W is the only one t h a t

We consider system (3.1.1) together w i t h a l l

the notations of §3, e.g. we

demand the fundamental s o l u t i o n to be normed at x=b by Yb(b)=l. (10 1.1) "

dY dx

m - ~,j=l

Uj Y'x-aj

and we assume t h a t the normed s o l u t i o n Yb(X) Of ( I 0 . I . I ) at x=av : V=l . . . . . m, ~.

Thus

has exponential matrices Wv

The main r e s u l t in the r e g u l a r case is the f o l l o w i n g .

Theorem 10.1.2. ( i ) Let us consider another system of the form (1.10.1): (10.1.3)

dY'

m

Y'o

dx =~ j=l

Uj

x---a~-

w i t h the s o l u t i o n Yb'(X) normed at x=b.

The necessary

and s u f f i c i e n t

Yb ' ( x ) to have exponential matrices Wv at x=av f o r v = l . . . . r i x W~( at x --~, is the existence of the s o l u t i o n P(x) of (10 1.4) •

dP m d~=~j=l

condition for

m and some exponential mat-

PU'.-U.P J J x-aj

such t h a t P(x) and P(x) - I are polynomial matrices and P ( b ) = l . Theorem 10.1.5. ( i ) Let us suppose t h a t a s o l u t i o n P(x) o f (10.1.4) which corresponds to matrices Uj I : j : l . . . . . m in (10.1.3) e x i s t s . (m I . . . . . mn) of r a t i o n a l (10.1.6)

Then there e x i s t s a sequence

integers such t h a t

ml+...+m n = 0,

and i f U ~ = - ~ m j = l Uj has eigenvalues (T 1 . . . . . Tn), then the eigenvalues of U~ '= = - E mj=l Uj' are (Tl+m I . . . . . %n+mn).

Moreover, i f V~ is a monodromy matrix at x =~

(common f o r both (10.1.1) and (10.1.3)) and S reduces V~ to a normal form: W = = S ' [ . . . ] ' S -1, then the exponential matrix W~' has the form: (10.1.7) (ii)

W~' = W~ + S'diag (mI . . . . . m n ) ' S - l .

The matrix polynomial P(x) has degree max{-m I . . . . . -mn}

and the polynomial

P(x) -1 has degree max {m I . . . . . mn}. We can get d i r e c t proofs of Theorems 10.1.4 - 10.1.5 using Theorems 1.7.2 and 9.2.8. The matrix P(x) is defined by P(x) = Yb(x)-Z. Yb'(X);

P(x) - I = Y b ' ( x ) - l - Y b ( X ) .

Then by the theorem 1.7.2, P(x) is a matrix w i t h r a t i o n a l by 9.2 8~ both P(x) and P(x) - I are polynomial matrices.

function coefficients,

and

Since the monodromy matrices

V v : v = l . . . . . m are the same f o r (10.1.1) and ( 1 0 . 1 . 3 ) , V~=(Vl...Vm )-1 is the same f o r (10.1.1) and ( 1 0 . 1 . 3 ) . by the r u l e ( 3 . 4 . 3 ) . (10,1.8) then

Thus the exponential m a t r i x W~' f o r (10.1.3)

is determined

I f S is any matrix which reduces V~ to the normal form:

V~ = S ' [ J r 1 ( ~ 1) . . . . . J r s ( ~ S ) ] . s - l ,

165

(10.1.9)

W~' = W~ + S [ I r l - n I . . . . . I r s . n s ] . S - l .

Here in agreement with (10.1.7) we have [ I r l . n I . . . . . Irs.ns ]= diag (mI . . . . . mn). Applying Theorem 9.2.8 to ~(x)=Yb(X ) we get deg(P(x))=max{-m I . . . . . -mn}, and applying 9.2.8 to ~(x)=Yb'(X) we get deg(P(x)-l)=max{ml . . . . . mn}. 10.2. I f eigenvalues of Uj doesn't d i f f e r by rational integers, then exponents W. e x[l-s-t- and a system of integers (mI . . . . . mn) s a t l s f y l n g 10.1.5 exists. I f U ~ = - ~ j =mI j Uj also has eigenvalues d i s t i n c t (mod I ) , then for a given (mI . . . . . mn) with (10.1.6), the solution is unique. However, as in [i0] the system (10.1.3) doesn't e x i s t necess a r i l y for any (mi . . . . . mn). I f U~ is a r b i t r a r y , then the system (10.1.3) depends not only on (mI . . . . . mn) but on S in (10.1.9) as well (cf. 9.3 for V~=I). 10.3. We present now expressions for P(x) and Uj in the case max(Imll . . . . . Imnl)=l [ i 0 ] . The general case follows from this by iterations only. We assume mi=l, mj=-l, mk=O for k ~ i , j . Then P=l+A(x-b), P-l=l-A(x-b) and A2=0. Substituting expressions for P(x), P(x) -1 into (10.1.4) we get Lemma 10.3.1. All solutions of (10.1.4) with deg(P(x))=deg(P(x)-l)=l have the form .

.

.

.

I

(10.3.2)

U.'3 = [ l - ( a j - b ) A ] - U j ' [ l + ( a j - b ) A ]

with A satisfying the system of equations (10.3.3)

A2 = 0; A ' ~ mj=l Uj - ~j=im Uj-A = A - A ' ~ = I

ajUj-A

There U.'j is similar to Uj: j = l , . . . , m . The matrix A has zero eigenvalues and in i t s normal form Jordan blocks are of sizes at most two. Let A correspond to one Jordan block ( ~ ) at the place ( i , j ) and other zero eigenvalues are simple. Then (10.3.3) is eauivalent to: A2=0 and [U~,A]=~A with ~=tr(SA)-i and S=~ m ajUj Using the result of 4.3 we obtain: Lemma 10.3.4. Let U~=-~, v=l m Uv has a normal form U~=S~-I diag(~l(~) . . . . ,~n(~)) .S. j = l

and ~ mV=l av(S~UvS -1)i j ~ 0. (10.3.5)

"

Then the system (10.3.3) has a solution

A(i,j)

= (~.(~)-~.(~)+Z)(~,v=lm av(S~ U v S ~ ) i j ) - I - s ~ - I ' E i j ' S ~ 1 3 for ( E i j ) i , j , = 6 i i , ' ~ j j , . This solution corresponds to a case mi=l, mj=-I and U~' has eigenvalues (~1 (~) . . . . . ~i(~)+1 . . . . . ~ j ( ~ ) - I . . . . . ~n(~')). 10.4. For the most general equation (4.1.1) the formulae for one-point Pade approximations are almost the same. We must change only (10.1.4) for (10.4 I) "

dP dx

m S ~j=l~r=l

P u j ( r ) ' - U j (r)P (x_aj)r

and (10.3.3) for A2 : O,

[ A , ~ , ~ : I Uj ( I ) ] : A - A ' ~ : I

(ajUj(1)+Uj(2)).A.

Similarly in (10.3.5) we change avU~ for avU~(1)+Uv (2).

166 §11. Perfectness and "almost perfectness" of the p-point Pad6 approximation to solutions of l i n e a r d i f f e r e n t i a l

equations.

11.0. We study perfectness and "almost perfectness (in the sense of §8) of Pade approximations to solutions of l i n e a r d i f f e r e n t i a l ities.

equations with regular s i n g u l a r -

For t h i s w e use the r e s u l t s of §§5-7 on local exponents of Fuchsian

modules.

11.1. The most natural system ~(x)=(Yl(X ) . . . . . Yn(X)) to be Pad~ approximated is the system of functions s a t i s f y i n g the system of l i n e a r d i f f e r e n t i a l equations of the f i r s t order: d÷ ÷ d~ y(x) = y(x)A(x)

(11.1.1)

f o r A(x)cMn(¢(x)). Y(x) of (11.1.1):

Let's consider ~(x) to be a row in the fundamental matrix solution

(11 1.2)

dY(x) = Y(x)A(x) and dx

(11 " 1.3)

Y(x) = ( ~ l t ( x ) .

"

÷. t (x) ."'Yn . ).t =. (fi÷. t .(x)

÷fn t ( x ) ) .

e can conslder the Pade approxlmation to Yl(X) (or the simultaneous system of Pade

W

"

•

,

~

÷

"

÷

÷

approxlmatlons to Yl(X) . . . . . Yn(X)). Let P(x)=(P1(x) . . . . . Pn(X)) be a system of polynomials of degrees (mI . . . . . mn), and we introduce n remainder functions (11.1.4)

Ri(x) = P(x) "Yi ÷ t (x) = E jn= l P i ( x ) Y i , j (x) : i = l . . . . . n.

11.2. The Fuchsian module associated with the system of polynomials P(x)=(Pl(X) . . . . . . . . Pn(X)) and the system (11.1.1) is introduced: (11.2.1)

P = {~(x).~t(x):

f o r a l l solutions ~(x) of ( 1 1 . 1 . 1 ) } .

Lemma 11.2.2. In the notations of 7.1, the ~-module p is the Fuchsian module L(R), where ~(x)=(Rl(X) . . . . . Rn(X) ) . Let Sing be the set of a l l s i n g u l a r i t i e s of (11.1.2) ( i . e . of A(x) including ~) and we add to Sing a l l the set of a l l apparent s i n g u l a r i t i e s of each of the Fuchsian modules m ( f l ), . . . , L ( f. n) f o. r n columns . fl'

' n# of

Y(x) in (11.1.3). Let Sing={a I . . . . . am,~} and Vvbe a monodromy matrix of Y(x) at x=av: v=l . . . . . m,~ as in (6.0.2) corresponding to a fixed basis Y1 . . . . . ym,y~ of 1 ( ~ I \ \ { a I . . . . . am,=}),¥1+...+ym+Y~=0. The discussion of 6.0 - 6.1 gives us: Lemma 11.2.3. Let the singular set Sing of (11.1.2) be {a I . . . . . am,~} and V~ be the monodromy of Y(x) at x=av: v=l . . . . . m,~. yV I . . . . . Vm~ S~a I . . . . amy •

Then R(x)=(Rl(X) . . . . . Rn(X)) is an element of

Proof. By (11.1.3) - (11.1.4) we have ~=~.yt

or

~t=y.~t.

Thus by ( 6 . 0 . 2 ) , a f t e r

e n c i r c l i n g a c i r c u i t yv we get : ÷R(x) t ~-~Vv.Y.P÷ t = VvR(x) ÷ t : v=l . . . . . m,~. Since P=L(R) is a Fuchsian module, according to 7.2.1. the set p is a set of a l l solutions of a Fuchsian l i n e a r d i f f e r e n t i a l

equation of the order dimsP~n.

We can,

167 following 7.4, denote by OrdcP the sum of all local exponents of elements of P at x=c for c in ¢~I. 11.3. Let Vv=sv-l.[Jkl(Pi ~) . . . . . Jkl(PlV)]-S~ as in 6.__! so that the fundamental matr i x Y(x) changes into Vv.Y(x)=Zv(x) with Z~(x) having the canonical form (6.1.9) (6.1.10). Following this structure of Z~(x) in 6.1we defined the matrix of the local exponents of Y(x) at x=a~ in the form (6.1.11) (cf. (6.1.4)):

(11.3.1)

Y(x) . .(~1 . (a~)t . . .

÷~n (a~)t)

(11.3.2)

÷li(av) z ~(av)(mod ~n) :

at x=a~;

i=1 . . .. n.

Lemma 11.3.3. For any f i n i t e s i n g u l a r i t y a in Sing we have (11.3.4)

OrdaP >_ ~ni=z min { l j l ( a )

: j=Z . . . . n},

where, as in (6.1.4), ~j(a)= (~J,1 (a) . . ,~j,n . (a)) . . : j=l, ÷t Proof. We have ~t=y.p , so by (6.1.8)

..n.

÷t = Sv-Y.P ÷t = Zv(x)-P(x) ÷ t. (11.3.5) SvR ÷t =(rl(x) . . . . . rn(X)) t, Since SV is nonsingular, SvRt is a generator of P and for SvR (Ii.3.6)

ri(x)=~jn = l Pj(x)Zv(x)ij = ~ nj=1 Pj(x).dij(a)(x ) x (x-a) l j , i (a)"@a,ij'ix~•

with the sturcture of dij(a)(x) described in (6.1.10). Then from (11.3.6) i t follows that ri(x) belongs to exponents l j , i (a)=- l i(a)(mod E) and the ordari(x) is properly defined. Then (11.3.6) gives us for a f init e a: (11.3.7)

ordari(x) ~ min{~j,i(a) : j=l . . . . . n}:

i=i . . . . . n; which immediately implies (11.3.4). Lemma 11.3.8. At i n f i n i t y we have (11.3.9)

Ord~P >_ ~, ni= I min { ~ j i (~) - d(Pj) : j=l . . . . . n}.

Proof. We apply (11.3.6) for a=~ and V~={VI...Vm}-I. Then expressions (6.1.10) for dij(-J)(x), and the fact that for a polynomial P(x), ord~P=-d(P), give us ord~ri(x) ~ min{~j, i (~) - d(Pj) : j = l , . . . . . n} Applying Corollary 7.5.2 we obtain for any f i n i t e set fl in £~Icontaining Singu{~}:

~ae~

OrdaP ~ n(~-l_____))(]~] _ 2)

(since dimcp
...,n}

-

~a~Sing~=l

min{Xji(a): j=Z . . . . n}.

168

One

should

take

(11.3.12)

into

account

agSingu~}

that,

~

according

n ~(a) i=l ji

to

7.5.2,

n(n-l)ISing 2 (

I -i)

for any j=l .... ,n. Then choosing an a p p r o p r i a t e Y(x) with different

~(a~) ~(a~) 1 ''''' n we

get canonical bases of contiguous functions which

give normal and "perfect" Pad~ a p p r o x i m a t i o n s to y(x). Results (11.3.11)-(11.3.12) 11.4.

are a c o n s i d e r a b l e

improvement over

In particular we can prove normality

(8.2.3).

in the case of ~y(x),...

.... y ( n - l ) ( x ) } f o r y(x) satisfying a F u e h s i a ~ l i n e a r differential equation

(5.1.1) of order n. For this we use the reduction of a scalar

equation to a m a t r i x form (5.1.4)-(5.1.8).

Taking into account the

computation of local exponents in 5.2, we come to the following conclusion: Theorem

11.4,1.

equation

L{y~

larities

of

= L{y~

Pl(X),...,Pn(X) i=l,

. . . ,n. R(x)=P

we

Let 0 of = of

Then

y(x)

satisfy

order

0 and

n,

and

P(x)=

degrees

a Fuchsian

d(Pl)

let m ~i:l(x-a

linear

al,...,a i

..... d(P n

. We

m

be

differential all

consider

finite

and d(Pi)=N+(n-i)(m-l):

for

l(x)y(x)+p

2(x)P(x)y'

have

(x)+...+Pn

x ) P ( x ) n - l y (n-l) (x)~0

n

°rdaR+ -

singu-

polynomials

i=1 d( i )'

169 ReFerences

1°

B. Riemann, Oeuvres Mathematiques, A Blanchard, Paris, 1968.

2. G. V. Chudnovsky, Pad~ approximation and the Riemann problem, Lectures, Cargese summer school on mathematical physics, June 1979, Reidel Publishing Company, Dordrecht, Holland, 1979, 62 pp. 3.

J. A. Lappo-Danilevski; Mat. Sbornik, 34 (1927), 113-148.

4.

J. Plemelj, Problems in the sense of Riemann and Klein, John Wiley, 1964.

5.

E. H i l l e , Ordinary d i f f e r e n t i a l equations in the complex domain, John Wiley, 1976.

6.

A. R. Forsyth, Theory of d i f f e r e n t i a l equations, VI volumes, Dover, 1959.

7.

I. L. Ince, Ordinary d i f f e r e n t i a l equations, Dover, 1959.

8. V. V. Golubev, Lectures on the a n a l y t i c theory of d i f f e r e n t i a l Moscow, 1950. 9.

L. Fuchs, Gesammelte Mathematische werke, Bd. 1-3, B e r l i n ,

equations, GTIL,

1900-1906.

I0. J. A. Lappo-Danilevski, M6moires sur la th6orie des syst~mes des 6quations d i f f e r e n t i e l l e s l i n e a i r e s , Chelsea, NY, 1953. 11. L. Markus, Group theory and d i f f e r e n t i a l equations, Lecture notes, U n i v e r s i t y of Minnesota, 1959-1960. 12. D. V. Cbudnovsky, Riemann monodromy problem, isomonodromy deformation equations and completely integrable systems, Lectures, Cargese summer school on mathematical physics, June 1979, Reidel Publishing Company, Dordrecht, Holland, 1979, 61 pp. 13. D. V. Chudnovsky, Proceedings of the I n t e r n a t i o n a l Meeting "Nonlinear evolution equations and dynamical systems," Lecce, I t a l y , June 1979, Lect. Not. Phys., Springer, 1979 (to appear). 14, D. V. Chudnovsky and G. V. Chudnovsky, CEN-Saclay Preprint DPh-t-79/115. Seminar on the Riemann problem. 15. G. V. Chudnovsky a) C.R. Acad. Sci. Paris, 288A (1979), pp. A-607 - A609; b) C.R. Acad. S c i . , 286A (1970), pp. A-965 - A-967. 16. G. V. Chudnovsky, a) C.R. Acad. Sci. Paris, 288A (1979), pp. A-IO01 - A-1004; b) J. Math. Pure Appl. v. 56 (1979), 32 pp. 17. G. V. Chudnovsky, a) Proceedings of the I n t e r n a t i o n a l Meeting "Nonlinear evolution equations and dynamical systems," Lecce, I t a l y , June 1979, Lect. Not. Phys., Springer, 1979 (to appear), 60 pp.; b) Lect. Not. in Math., Springer, 1979, v. 751, pp. 45 69. 18. G. V. Chudnovsky, C.R. Acad. Sci. 289A (1980), pp. A-15 - A-17. 19. Equations d i f f 6 r e n t i e l l e s et syst~mes de P f a f f dans le champ complexe ~Lect. Not. Math., v. 712, Springer, 1979. 20. K. Mahler, Composito Math., 19 (1968), 95-166. 21. H. Jager, Proc. Kon. Nederl. Wet. 67 (1964), 192-249. 22. J. H. Loxton, A. J. van der Poorten, Rocky Moun. J. Math., v. 9, 393.

3 (1979), 385-

23. K. Mahler, Lectures on Transcendental Numbers, Lect. Not. Math., v. 546, Springer, 1976. 24. Bateman, A. Erdelyi. Higher Transcendental Functions, v. 1-3, London, 1953. 25. D. Brownawell, D. Masser (to appear).

METHODE

DE

LA

PHASE

STATIONNAIRE

ET SOMMATION

DE

BOREL

Bernard M A L G R A N G E Laboratoire de Math~matiques Pures - Institut Fourier d~pendant de l'Universit~ Scientifique et M~dicale de Grenoble associ~ au C.N.R.S. B.P. I16, 38402 ST Martin D'Heres, France

I.

-

Les quelques

remarques

de Balian-Parisi-Voros d'exarniner,

dans

un cas particulier simple,

tiellement petites marion,

("instantons"),

permettent

tiques obtenus Soit

r~els, v~rifiant

n

si

suivantes

de resomasyrnpto-

:

(P2)

les points critiques r4els

x I ,....x k

isol~s",

au voisinage

c'est-~-dire

convenable

(x I .....x n) = x , ~ coefficients

P(x)-- +oo

Soit alors

x ~+~

corrections e x p o n e n -

des d~veloppements

(P ])

que

des

stationnaire.

variables

les deux hypotheses

~quations

d4rons

~

par le travail

; elles ont pour but

jointes ~ un proc4d~

de la phase

un po l y n O m e

[2]

comment

de d~duire une formule exacte

par la m~thode

P

qui suivent m'ont 6t~ sugg~r~es

sur l'oscillateur quartique

;

que,

~--~P(x) = 0 5xj

n'admettent

de

P de

sont

"alg~briquernent

x £ (~ -~ 1 .....k) , les

pas d'autre solution complexe

x = x~ £0

une fonction

C OO

~ support compact

sur

]Rn , et consi-

i'int~grale I (~) = ~ e iTp(x)~0(x)dx ~0

II est classique loppement

que,

asymptotique

(A0) : I 0 ( T ) ~

pour

T

, (dx = dXl...dx n) . r~el tendant vers

de la forme

suivante

eiTdef~(T)

,

f~(T) =

~

+~

a

, I (T) ~p

admet

un d@ve-

Tk(log T) ~

avec i)

~d" = P(x ~)

(les

de

sont les "valeurs

critiques" de

P

correspon-

171

dant aux points critiques r~els) ; ii) les

k

sont rationnels < 0 , et forment

(autrement dit, il existe un e n s e m b l e soit de la forme iii) 0 ~ ~ < n-i

kq+r

, q = i, ....p

i. - En fair, ce r~sultat n'exige

d~montr~,

par exemple,

critique, c o m m e

facilement

dans

consequence

du th~or~me

fini

"fini modulo

k I .....kp

tel que tout

2Z" k

, r entier ~ 0) ;

.

Remarque

quence

un e n s e m b l e

[5]

dans

hypoth&se

le cas o4

P

que

P

d'Hironaka.

P

. Ii est

seule valeur

lui-m~me

conse-

Le cas g~n~ral s'en d~duit

n'a qu'un nombre

(cette derni~re assertion est simplement

sur

n'a qu'une

d'un r~sultat de ~eanquartier,

de d~singularisation

si l'on remarque

aucune

fini de valeurs

la version alg~brique

critiques

du th~or&me

de

Sard). Maintenant, pour x£

e

de l'hypoth&se

fix~ ne d~pendent

que des valeurs de

(pour simplifier l'expos~,

ple l'introduction de

[5]

(P2), on d4duit facilement

je suppose

. Si donc

~

qu e les

ak,p, 8

~0 et de sen d~riv~es

au point

d I .....d k

est une fonction

le d~veloppement

porte quelle fonction points

C~

asymptotique

de

~ support compact

C c°

; voir par e x e m quelconque

~ eeiTP(X) iTP(X) ~(x)dx

R n , on pourra d~finir "l'int~grale asymptotique" par d~finition,

distincts)

~

comme

~0(x)dx ,

~gale ~

~

sur ~tant,

~0 n'im-

au voisinage

des

x I .....x k

Prenons

en particulier pour

LEMME 2.

- L'int~qrale

demi-plan

Im T > 0

~

un p o l y n O m e

Q

JQ(-~) = ~ ei~P(X) Q ( x ) d x

converge dans

et d~finit une f o n c t i o n h o l o m o r p h e d a n s

le

le d e m i -

plan. En effet, l'hypoth&se pour

x

r~el assez

raisonnement, Noun

(P I) entraine qu'il existe un P(x) > IIxllr , avec

voir [4] , Appendix).

Le l e m m e

verrons un peu plus loin que

fonction analytique encore

grand

en tout point

meme

x~

r> 0

tel qu'on ait,

(pour ce type de

s'en d~duit imm~diatement.

la fonction

-~ £ R-{0}

IQ ; en fair, elle se prolonge

llxll2 = ~

IQ

se prolonge

en une

, fonction qu'il est naturel de noter en une fonction multiforme

sur

172

(E-{0]

. Le probl@me

consiste d o n c

T >> 0 , et le d @ v e l o p p e m e n t

Pour ~tudier x ~ R n ; pour

IQ

l'ensemble

P(x) < t

tout

. Un calcul

t £ R

(3)

~Q(~) = S

asymptotique

, notons

t ~ d 1 ..... d k

,

classique

+~ eiTtdt

la surface de niveau

P(x) = t

est lisse ; orientons-la c o m m e

y(t)

(P 1)

IQ(~),

(AQ) .

y(t) (t6R)

. L'hypoth&se

implique

montre alors

que

qu'on

y(t)

,

le bord de

est c o m p a c t

pour

a,

dx

~

--

Q d~ ' y(t)

l'int~grale figurant au s e c o n d

If. -

~ @tudier la relation qui existe entre

membre

Pour ~tudier l'int&grale

~tant absolument

convergente

pour Im-~ > 0

(3) , un point essentiel consiste ~ s~parer les

contributions des cliff,rents points critiques. O n

op~re pour cela de la mani~re -i : il existe un e n s e m b l e fini F c (E tel que, en dehors de P (F) , ~n l'application P : ~ (E soit une fibration C localement triviale, les fisuivante

bres ~tant non singuli~res

(volt par e x e m p l e

[3] , th. 6.13).

Les points de

F

peuvent etre de trois sortes : i)

les points

d I .....d k ;

ii) les valeurs critiques de

P , correspondant a u x points critiques

non r~els ; iii) les points

f 6 ~

tels que,

au voisinage de

f ,

P

ne d@finisse

pas ~ l'infini une fibration localement triviale. U n e x e m p l e derni~re situation est, par e x e m p l e et

donn~

f = 0 (je laisse le lecteur examiner

Naturellement,

un point

par

de cette

P(x,y) = (xy-l)2+x 2

les d~tails).

f @ (U peut aussi appartenir &

F

pour plu-

sieurs des raisons ci-dessus.

Si l'on a un c h e m i n

(C)

dans

la fibration se trivialise a u - d e s s u s

un

isomorphisme

d6pend donc

pas on

classe long

de

a une

Hn_I(P

d'homologie

6(t 2) 6 Hn.I(P-](t2),G) de

(C) "

, d'origine

de ce c h e m i n

choisie,

mais

qui

peut

"d@duite

de

et d'extr~mit~

on

d@pendre

6(t I) 6 Hn_l(P-l(tl),(~) , dite

tI

t2,

; on en d~duit, e n particulier,

-l(tl) '(E) ~ H n _ 1 (p-l(t2) '(E) , dont

la trivialisation classe

(E-F

6(t l)

• on par

montre

qu'il

de

(C)

en

d6duit

continuit~

ne

. Si une le

173

Cela @rant, s u p p o s o n s de vue le cycle @videmment

qu'on air

d I <...
¥(t) d@fini plus haut avec

y(t) = 0 . Lorsque

t

de voir que la classe d'homologie

f 6 FN]d~_l,d[ condition de

; au p a s s a g e

"contourner"

t 6 R-F

varie entre de

y(t)

varie par continuit@ au sens pr@c@dent,

, et e x a m i n o n s . Pout

d _1

et

t < d]

, on a

de , il est facile

(que nous noterons encore

rant qu'on ne p a s s e

d'un tel point

f , par e x e m p l e

de ce point

f , dans

y(t)),

pas un point

y ( t ) varie continuement le

demi-plan

sup@rieur

(cf.

figure i) fig.]

> /~ f

E n effet, soit P

D

un voisinage

ne se trivialise pas sur

suffisamment

P-I(D)

d'une mani@re compatible a v e c

>

petit de

f : guoique

l'application

, elle se trivialise au voisinage de

la conjugaison

complexe

,{(f)

et cela suffira pout ob-

tenir le r@sultat cherch@. Reste ~ examiner Soient

ce qui se p a s s e

t' , t" 6 I~ , a s s e z

~(t") 6 Hn_I(P-I(t"),(D) contournant

de

voisins de

du cot@

y£

/ ~ d

Nous

ne donnerons

essentiellement Milnov

"bien connu".

voisin de

ye(t")

Ii est c o n s @ q u e n c e

petit centr@e en

d~ , l'image de

y(de)-B

C e c i @tant, s o i t

xe

(autrement dit,

de support arbitrairement

soit a s s e z voisin de

ye(t")

est nulle, ce qui se fair en trivialisant un voisinage de

du type

.

au point

de ) .

dans

(%)

qui est

facile, des r@sultats de

sur les singularit@s isol@es d'hypersurfaces

assez

y(t') en

pas la d@monstration d@taill@e de ce l e m m e ,

boule ouverte de rayon a s s e z t"

t"

de .

, t" > d e , et soit

le long d'un c h e m i n

est un cycle ~ v a n e s c e n t

x e , pourvu q u e

la valeur

x t"

on peut trouver un repr@sentant de voisin de

t' < %

ye(t") = y(t") -~(t")

x t'

-

passe

pat continuit@ ~ partir de

I m t > 0 , c'est-~-dire

fig. 2

4.

t

de , a v e c

la classe obtenue

indiqu@ ~ la figure 2 . P o s o n s

LEMME

lorsque

[6]

; en gros,

soit

x e ; il faut montrer que, dans

B

une

pour

Hn_l(P-l(t'gmod P-](t")[3B,(E)

convenablement

la restriction de

P

(rn-B .

un chemin infini obtenu ~ partir de

]d~,+~[

en

174

contournant dans le demi-plan ye(t)

Im t > 0

les points de- F N ] d ,+o~[ ; soit

le cycle obtenu par continuit@ & partir de

l'int~grale au point

-(~%

dx

e iTt dt ~ Q ~ ) y~ (t)

y£

est absolument

le long de ce chemin ;

c o n v e r g e n t e pour

Im T > 0

t = d~ , cela r~sulte de la "positivit~" des exposants caract@ristiques

(volt [5] ) ; & l'infini, cela r@sulte des propri~t~s de croissance des int~grales dx yt(t

("th@or~me de r~gularit~")

d-"P

que nous rappellerons dans un instant.

A l o r s , la formule (3) et le lemme 4 n o u s d o n n e n t la formule s u i v a n t e

(5)

IQ(t) = Z ~(

III. -

Soit

Riemann")

Pl(~)

C£)

eiZtdt ~

O

Ye (t)

l'espace

dx d-'P"

p r o j e c t i f c o m p l e x e 8 une d i m e n s i o n ( " s p h e r e de

; le "th&or&me de r&gularit~" de N i l s s o n - G r i f f i t h s - K a t z - D e l i g n e

(cf. [ 3 ] ) nous affirme e n p a r t i c u l i e r c e c i : sur IPI(~) - F U { ~ ] , les dx Q ~ s e p r o l o n g e n t en des f o n c t i o n s m u l t i f o r m e s de d ~ t e r m i n a t i o n finie ~(t) (i.e. les diff~rentes d~terminations au voisinage d'un point forment un espace vectoriel de dimension finie), les diff&rentes d&terminations @rant ~ croissance mod&r@e dans tout secteur

de s o m m e t

un point singulier, i.e. un point de

F U {=] . Une autre mani~re de dire la m e m e tions multiformes satisfont, sur

chose est la suivante : ces fonc-

PI(~) , & une &quation diff@rentielle dont

tous les points singuliers sont "singuliers r~guliers", ou "du type de Fuchs". Ceci nous permet d'abord d'effectuer le prolongement analytique de Io(t)

pour

I r~el > 0 ; soit en effet (C~)

situ~e dans le demi-plan

une demi-droite d'origine

I m t > 0 , et faisant avec

[d£,+~[

l'angle

d~ ~;

, si

c~ est suffisamment petit on aura eiTtdt

dx

~

eiTtdt

q) pour

dx Yt (t)

Im T > 0

et

Im (Te i¢~) > 0 , et la premiere int~grale converge pour

Im(Te ic~) > 0 ; &off le prolongement cherch~.

C e t t e formule n ' e s t entre

(C~)

et

(C~)

plus e x a c t e

si

c~ e s t t e l que le s e c t e u r c o m p r i s

c o n t i e n t d e s p o i n t s de

c o n t i e n t p a s d ' a u t r e que

F

de ) ; il e s t n ~ c e s s a i r e

(on s u p p o s e que

(C~)

n'en

de la c o r r i g e r en a j o u t a n t

175

/J

des termes taires

correspondant

entourant

& des chemins

les points en question

compl~men(cf. f i g u r e 3).

Par ce proc~d&, en augmentant

d

ind~finiment,

prolonge analytiquement

en

une fonction mul-

tiforme sur

IQ(T)

on

(E- { 0 ]

fig. 3 PROPOSITION

5. - Pour

d~veloppement

r~el, t e n d a n t

T

asymptotique

+~

ver s

l'expression

a d m e t pour

, IQ(T)

~ eiTP(X)Q(x)dx

d&finie au

§I. Donnons suivant : soit d'autre part assez

une esquisse (C~)

~0

petit de

de la d~monstration.

un c h e m i n

comme

une fonction

C~

x ~ , ~gale ~

i

ei'[tdt

Ii suffit d'~tablir le r~sultat

ci-dessus,

avec

~ support c o m p a c t au voisinage de

dx

~n

0 < ct < 17 , et soit

contenu dans un voisinage

x e ; alors la difference

eiTP(x)

Yt(t) est & d~croissance de

rapide pour

T ~ +co • C e c i se voit en reprenant les calculs

[5] , § 6 et 7 et en montrant que le cycle

introduit dans cet article,

IV. -

§ 7 sous le nora de

Le but de cet e x p o s ~

ques precedents, exacte de

JQ(X)

y+

.

resomm~s,

asymptoti-

permettent d'obtenir la valeur

T >> 0 , ~ condition d'ajouter ~ventuellement

rections exponentiellement consid~r~s fig. 3. C e s

co1"ncide a v e c celui qui est

est de montrer que les d ~ v e l o p p e m e n t s

convenablement pour

y[

petites correspondant

aux

corrections sont e l l e - m ~ m e s

"chemins

des cor-

compl~mentaires"

susceptibles

du m e m e

trai-

tement que nous allons donner maintenant.

Tout ceci n'est en fait qu'un cas

particulier de la th~orie de la r e s o m m a t i o n

des

solutions des ~quations diff~ren-

tielles au voisinage d'un point singulier irr~gulier; voir ~ ce propos la conf~fence de T.-P. Ramis. Rappelons

le r~sultat suivant (NOrlund [7]

; nous nous

d'un ~ n o n c ~ relativement grossier ; des estimations chez divers auteurs ; cf. Ramis).

contenterons

ici

plus pr&cises se trouvent

176

THEOREME

6.

It l < A

et de

Soit

-

A > 0

, et

la demi-bande

:

soit

DAc

Re t > 0 ,

(E

la r~union

IIm t I < A . S o i t

dans

DA , e t t e l l e

soit born~e dans de L a p l a c e

pour un

k > 0

+co = 20 e - ° t f ( t ) d t

F(o)

a (~u) + ~2 O

est

convenable,

holof(t)e -kt

Tf

DAF~ {Ret > 1 } . Alors, pour

a

rielles

que,

f

i

q

morphe

du disque

tu >~-~ , la transform~e

d~veloppable

en

s~rie

de facto-

(®)n!

n O ~O (~+I)... (--O+n)

uniform~ment

converqente

dans

un

LU

demi-plan

Remarque o ~+Qo droite

Re o >> 0 .

7. - La c o n n a i s s a n c e

de d ~ v e l o p p e m e n t

, ou plus g~n~ralement

pour

g

asymptotique

de

F

pour

tendant vers l'infini sur une d e m i -

Im o = k Re o , Re o > 0 , permet de calculer de proche en proche les

a (w) ; par consequent, ce r~sultat peut s'interpr~ter e o m m e un proc~d~ de ten s o m m a t i o n de s~ries asymptotiques, d'ailleurs cas particulier du proc~d~ de sommation

de Borel ; si

variantes de la m ~ t h o d e

f

est holomorphe

dans un ouvert plus grand, d'autres

de Borel peuvent ~tre aussi envisag~es.

Pour pouvoir ~tre appliqu~ ici, ce th~or&me soit

f

une fonction multiforme sur

les hypotheses i)

suivantes

il existe born~es

ii) dans

:

k > 0 t e l que toutes les d~terminations de dans

croissance

0

, toute d~termination de

au voisinage

iii) n'importe quelle d~termination de voisinage de

on peut ~tablir que (avec

asymptotique remarque Si

les

d

restent

f

est

0 ,

f

sur

]0,+~o[

est int~grable au

s'~crit c o m m e

et les

k

et p o s o n s

une s o m m e

Fd, k

finie

F(o) = ~n

fo (t)e-Otdt;

~ c~CC(log o)kFd,k(g)

~tant justiciables du th~or&me

6 .

sont li4s de mani~re simple au c o m p o r t e m e n t

f quand t ~ 0 ; ce point est laiss4 au lecteur. Q u a n t o 7, elle s'appliquera encore ici sans modification. C~i

&

0 .

cL < 0 , k entier > 0) , les

Bien entendu,

de

une telle d~termination, F

e-ktf

D A N [Re t > ]] ;

mod~r~e

fo

:

DA-[0 ) , de d~termination finie ; on fait

tout secteur de s o m m e t

Soit alors

doit ~tre un peu g~n~ralis~

de

ne contient

pas

d'autre

point

de

F

que

d e , les

r~sultats

& la

177

pr@c~dents

s'appliquent au calcul des int~grales

moyennant

le changement

avec

de coordonn~es

[d~,+~o[) ; la connaissance

~ = JOe-I(~

suppos~e

dx

'

S( eITtdt S Q~C~) y%(t)

'

((~ @rant l'angle de

du d~veloppement

(C~)

asymptotique

de

cette int~grale permet donc d'en obtenir la valeur exacte pour les grandes va-

l e u r s de

r

•

Si l'on prend pour

T

r@el >> 0

e

assez petit, ceci donnera donc la valeur de

(je dois cette remarque

h L. Boutet de Monvel).

en pratique, il peut ~tre plus int~ressant de prendre

c~

ple voisin de

17/2 . II intervient alors des int~grales

pl6mentaires";

th~oriquement,

cependant,

sur des des m e m e s

Cependant, par e x e m -

"chemins

com-

m~thodes

;

leur calcul effectif pr@sente au moins une difficult~ suppl4mentaire :

re@me dans le cas o@ t o u s l e s complexes

elles sont justiciables

plus grand,

IQ(T)

points de

de nature assez simple

F

proviennent de points critiques

(par exemple

de points de Morse),

sance du cycle sur lequel on doit int~grer pose ici un probl~me qlobale, du syst@me p : (En_p-l(F)

local

t ~-~Hn_l(P-l(t),~)

associ~

la connais.

de monodromie

~ la fibration

-- (~-F .

A l'heure actuelle, il faut bien dire que l'on salt fort peu de choses sur ce sujet ; cette remarque,

un peu d~sabus~e, me

servira de conclusion.

Je remercie J.-P. Ramis de m'avoir signal~ que l'utilisation de la s o m mabilit~ de Borel et les s~ries de factorielles permettaient dans consid4r~,

d'arriver ~ des formules exactes

m'@tais content~,

comme

au sens des astronomes", s~rie asymptotique

;dans

l'exemple

une premiere version, je

[2] , d'examiner ee que Poincar~ appelle la "sommation c'est-~-dire,

le proc~d~ qui consiste ~ tronquer une

au plus petit terme.

BIBLIOGRAPHIE [I] [2] [3] [4] [5]

[6] [7]

V.I. ARNOL'D - Remarques sur la m~thode de la phase stationnaire (en russe), Uspekhi Mat. Nauk 28-5 (1973), pp.17-45. R. BALIAN, G. PARISI, A. VOROS - Quartic oscillator, Math. Problems in Feynman path integrals, Springer Lect. Notes in Math (1979). P. DELIGNE - Equations diff~rentielles ~ points singuliers r~guliers, Springer Lect. Notes in Math n°163 (1970). L. HORMANDER - Linear partial differential operators, Springer (1963). B. M A L G R A N G E - Int~grales asymptotiques et monodromie, Ann. E.N.S. 4-7 (1974) pp.405-430. J. MILNOR - Singular points of complex hypersurfaces, Ann. Math. Studies n°61~ Princeton (1968). N.E. NORLUND - Lemons sur les s~ries d'interpolation, Collection Borel, Gauthiers Villars (1926).

Reflection

of A n a l y t i c

Singularities

by Johannes

Let M be an n - d i m e n s i o n a l P a second

order

Sj3strand

real

differential

analytic

operator

and w i t h

real

principal

symbol

p.

and

near

any point

of the

boundary

that

nates

(Xl,...,Xn)

= (x',Xn)

P

where

dx,

:

such

Assume

that

~2m + r ( x , 6 ' )

n-i ~

~,r(x',0,~'), '

that

ZM,

with

analytic

boundary, coefficient

P is of p r i n c i p a l

there

are

local

M: x n _> 0

are

linearly

type

coordi-

(up to a n o n - v a n i s h i n g

,

6jdxj

manifold

on M w i t h

factor)

,

independent.

This

i

condition

can be

formulated

the wave

operator

invariantly

in a c y l i n d e r ,

~t

([5]),

x ~x

'

and

is

satisfied

by

~ c ~n-l.

We d e c o m p o s e T*~M\O where

and

&,

G, ~

ro(X',~')

=

are g i v e n

~ u G u

by

r

= r(x',O,~').

0

> 0 ,

r

= 0 ,

0

At a p o i n t

of

r

0

G ,

< 0

the

respectively,

complement

CM

i

is s t r i c t l y we

let

this

convex

inequality

Definition. I • I~

i°

If

along

define

An a n a l y t i c

is an i n t e r v a l ,

¥(t °

the b i c h a r a c t e r i s t i c s

ray

such

• p-l(o)Lo

,

a subset

G+ c G.

is a c o n t i n u o u s that

for

then

¥

iff

every

O

n

¥:I ÷ Z b

and

,

where

• I:

is d i f f e r e n t i a b l e

at

IM

~(t °

< 0

Let

curve t

~r/~x

t

and o

= Hp(y(to) ] ,

where

Hp

is the H a m i l t o n

field

of

2 P = 6n + r ( x , ~ ' ) . 2°

If

y(t °

• ~

then

¥(t)

• p-l(o)l~ enough.

for

It-tol

> 0

small

201

0

If

Y(to) • G

(x(t),~'(t))

and we write

y(t) = (x(t),~(t))

is d i f f e r e n t i a b l e at

(x'(to),@'(to))

to

and

,

then

Xn(to)

= 0 ,

= Hr O

Recall from M e l r o s e - S j ~ s t r a n d for

[6], that the c o r r e s p o n d i n g d e f i n i t i o n

C~-rays is obtained by r e p l a c i n g ~ by

~ u G+

in 2 ° .

Thus

contrary to a C -ray, an analytic ray may glide along a strictly convex part of an obstacle

(thinking of the wave operator in a

cylinder as our m a i n example). We write o

u • D'(M)

,

u • a(M)

,

v • a(~M) If

u • D'(M)

if u is an extendable d i s t r i b u t i o n on M,

,

,

WFba(U)

if u is r e a l - a n a l y t i c

near M,

if v is r e a l - a n a l y t i c

on ~M.

Pu • a(M)

=

WFa[Ul~]

we define u

[WFa(Ul~M]

u W F a [ ~ x n U l ~ M ]]

where WF a denotes the analytic w a v e f r o n t If

u • D'(M)

(i)

and

Pu • a(M)

it follows from results conic subset of

,

uI • a(~M) i ~M

of S c h a p i r a

[i0] that

WFba(U)

is a closed

Z b.

Our m a i n result

Theorem i.

set.

([ii].)

is,

If

u • D'(M)

then there exists a m a x i m a l l y

satisfies

(i) and

p • WFba(U)

,

extended analytic ray p a s s i n g t h r o u g h p

and contained in W F b ~ u). o

In

T'M\0

this is due to S a t o - K a w a i - K a s h i w a r a

and near ~ the result

is due to S c h a p i r a

[i0].

[9], H ~ r m a n d e r

[3]

202

This r e s u l t rays

cannot

is q u i t e

satisfactory

split there•

Near

G+

of a n a l y t i c

rays

can be c a r r i e r s

solution

(i).

The f o l l o w i n g

to

gliding

ray in

Theorem

2.

G+

6 > 0

with

y(0)

y(6),

y(-~)~

of the a n a l y t i c result

u e ~'(M)

, since a n a l y t i c

to ask what u n i o n s singularities

eliminates

let

satisfy

¥:[-6,6]

¥(t) • p - l ( 0 ) l o IM

WFa(Ul~],

A f t e r this proving

Let

small e n o u g h = p ,

6+

the case

of a

of a single

:

([12].)

for

away from

it is n a t u r a l

then

conference

essentially

,

(i), let

÷ Zb

• G+

p

be the u n i q u e

t # 0 .

,

and

C~

ray

If

p ~ WFba(U).

we r e c e i v e d

also T h e o r e m

a preprint

of K i y 6 m i

Kataoka,

2.

In the case of P =

n ~ 2

~2 2 ~xj

~2 2 ~x I

(i + Xn)

Friedlander-Melrose

[2] s h o w e d for a p a r t i c u l a r

gation

r a y s takes place.

along non-C

by J. R a u c h Let plicity be the

[7] in the f o l l o w i n g

~ c in that

is compact •

~u

u(0,x) a simple

theorem that

support

expect

are c o n t a i n e d Rauch's

=

0 ,

boundary

x° • ~

of

and

and a s s u m e let

for sim-

u • D'(~ t

~)

x

argument

if

WFba(U)

=

6(X-Xo)

based

on H o l m g r e n ' s

belong

is the d i s t a n c e

However,

but there

in ~ f r o m

rays,

in the passing

is a d i f f i c u l t y

rays whose

x-space

using Theorem

Rauch proved

to the a n a l y t i c

singularities

that all a n a l y t i c ,

uniqueness

speed of p r o p a g a t i o n , (±~,x I)

are a n a l y t i c

for a n a l y t i c

length minimizing.

~

0

x a~

~tu(0,x)

then

fr o m this

in

=

I

of f i n i t e

u , there

u I

argument

xI ~ x°

In p a r t i c u l a r

One m i g h t

Let

0 ,

and e l e g a n t

xI • ~ ,

singular x I.

=

and the p r o p e r t y

if

that p r o p a was o b t a i n e d

to the m i x e d p r o b l e m :

(2)

Using

result

situation:

be open w i t h a n a l y t i c

C~

solution

solution

A similar

x°

to

C~-shadow. over

(0,x o)

in c a r r y i n g

projection

is not

i and the t e c h n i q u e s

of

out

203

Rauch [7] we were able to show,

Theorem 3.

(Rauch-SJistrand

ary and assume that of (2), then

C2

WFba(U')

[8]).

Let

2 c ~2

have analytic bound-

is convex and bounded.

If u is the s o l u t i o n

is the union of all analytic rays passing over

(0,Xo). The idea of the proof is to introduce the u n i v e r s a l c o v e r i n g space

w: ~ ÷ 2.

Then over

Rx ~ ,

length-minimizing projections

I × ~

(~)

^^

where

~(Xo ) = x °

in ~.

all rays w h i c h hit

'

, ~tG(°'

- x o) ,

Then using T h e o r e m i, the t e c h n i q u e s

[9], we can first d e t e r m i n e

Then we recover

u(t,x)

have

Let G solve the "lifted" p r o b l e m

[7] and a refined version of H o l m g r e n ' s uniqueness Kashiwara

G+

WFba(~ )

of Rauch

theorem due to

completely.

by the formula

u(t,x)

=

~

~(~)=x

u(t ,x)

,

and using the convexity a s s u m p t i o n and T h e o r e m i we can verify that no c a n c e l l a t i o n of analytic

singularities

takes place.

The proofs of Theorems i, 2 combine the ideas of energy estimate~ used in [5], [6] and by A n d e r s s o n phase functions,

resolutions

[i], w i t h the t e c h n i q u e s

of the identity,

etc.

of complex

In a way,

the

proof of T h e o r e m i involves a complex canonical t r a n s f o r m a t i o n ,

which

J

transforms

(i) into an elliptic boundary value problem.

Mo~e recent

work based on this point of view seems to give c o n s i d e r a b l e g e n e r a l i zations of T h e o r e m i, to higher order equations and m o r e general boundary conditions.

In p a r t i c u l a r in the interior we would recover

a result of K a s h i w a r a - K a w a i larities for m i c r o h y p e r b o l i c

[4] on the p r o p a g a t i o n of analytic operators.

singu-

204

REFERENCES i. K.G. Andersson, R~gularit6 jusqu'au bord pour des probl~mes mixtes. C.R.A.S. 282 (2 fSvrier 1976), 275-277. 2. F.G. Friedlander, R.B. Melrose, The wavefront set of the solution of a simple initial-boundary problem with glancing rays, II. M a t h Proc. Camb. Phil. Soc. (1977), 81, 97-120. 3. L. H~rmander, Uniqueness theorems and wavefront sets for solutions of linear differential equations with analytic c o e f f i c i e n t s C.P.A.M., 23 (1970), 329-358. 4. M. Kashiwara, T. Kawai, Microhyperbolic pseudo-differential operators I. Journal of Math. Soc. Japan, 27 (1975), 359-404. 5. R.B. Melrose, J. Sj~strand, Singularities problems I. C.P.A.M., 31 (1978), 593-617.

of boundary value

6. R.B. Melrose, J. Sj~strand, Singularities of boundary value problems II. In preparation, but see S~m. Goulaouic-Schwartz, 1977-78, no. 15 for the essential results. 7. J. Rauch, The leading wavefront for hyperbolic mixed problems. Bull. Soc. R. Sci. de Liege, 46, no. 5-8 (1977), 156-161. 8. J. Rauch, J. Sj~strand, along diffracted rays.

Propagation of analytic singularities To appear.

9. M. Sato, T. Kawai, M. Kashiwara, Microfunctions and pseudodifferential equations. Springer Lecture Notes in Mathematics, no. 287. i0. P. Schapira, Propagation at the boundary and reflection of analytic singularities of solutions of linear partial differential equations, I. Publ. RIMS, Kyoto Univ., 12 Suppl. (1977), 441-453. ii. J. Sj~strand, Propagation of analytic singularities for second order Dirichlet problems, I. Comm. P.D.E., to appear. 12. J. Sj~strand, Propagation of analytic singularities for second order Dirichlet problems, II. Comm. P.D.E., to appear.

LES OPERATEURS METADIFFERENTIELS A. Unterberger D~partement de Math4matiques Universit~ de Reims B.P. 347~ 51062-Reims-Cedex

Ala

suite de Maslov,

lagrangienne

le professeur Leray a insist~

dens sen travaux sur l'Analyse

[8]~ pour que les outils d~finis en vue d'applications

~ la m~canique

quantique solent invariants par les changements de rep~res sympleetiques

de l'espaee

de phase R 2~. Noun adoptons ici le point de rue qu'un rep~re w = (Y~@) est constitu~ par la donn~e d'un point Y = (y,~) de l'espace de phase et d'une norme symplectique II II~ sur celui-ci, entendant par I~ toute norme euclidienne se d~duisant de la norme canonique de

R 2~ par une transformation

symplectique

; il revient au m~me de se donner

la "transformation de Fourier" ~i~ d~finie (OP½ ~tant la quantification par

de Weyl-Wigner)

i~ = 0P½(2 ~/2 e-~- exp - 2i~ iiX-YII2@) ,

ou encore l'oseillateur harmonique %

= 0P½(~ liX-Yli~) - ~ , on encore l'espace propre

de niveau d'~nergie z4ro de eelui-ci~ ou enfin le point Y e t l'espace de phase pour laquelle la forme

R-bilin~aire

B

la structure complexe sur d4finie par

B(XI,X2) = (X1,X2) ~ + i[X1,X2] ([ ,7

4tent la forme symplectlque)

est sesquilin4aire.

permet d'expliciter un 4tat fondamental ~m (~ # N~)

s'obtiennent ~ partir de

~ ~

ce proc~d4 est bien connu des physiciens Etant donn~ un ensemble

Q

de

~

La representation

de Siegel

, les autres ~tats propres

en faisant agir des op~rateurs de creation: [5].

de reputes muni d'une mesure d ~

pri4e permettra d'~crire les ~l~ments de L2( R 9)

une condition appro-

conmie sormnes convenables

de fonctions

~•~ (w ~ ~), avec i~l~ n ne d~pendant que de Q : on parvient ainsi ~ la notion d'espaces de rep~res~ le proc~d~ que noun venons de d~crire pouvant ~tre rattach4 ~ la repr4sensentation de Fock. Signalons qu'une d~composition de ce genre~ main ne faisant intervenir que des gaussiennes et par cons4quent entach~e d'un terme d'erreur, par Cordoba et Fefferman

a ~t4 d4crite

[6] dens une situation adapt~e aux op4rateurs pseudo-diff~-

rentiels classiques. Les op~rateurs m~tadiff~rentiels

associ~s ~ un couple ( ~ ' )

d'espaces de rep~res sont enfin d~finis comme ceux susceptibles de s'~crire comme som~es de "projecteurs-obliques"

u

~

(u,~)~ , oO

~

et

~ s0nt des gaussiennes,

le noyau de cette d~composition ~tant eoncentr4 de telle sorte que le couple ( ~ # ) reste en moyenne proche de couples du genre (~ ,,c0 ) (~'6 ~', m ~ Q). Des op4rateurs de ce type sont ~ventuellement tion de Schr~dinger

susceptlbles d'intervenir dens la discussion de l'4qua-

; des versions particuli~res

comprennent des classes tr~s g~n~rales

206

d'op~rateurs pseudo-diff~rentlels,

ou des op~rateurs proches d'op~rateurs Int~graux

de Fourier g~n~ralls~s. Pour facillter l'utillsatlon de ces op4rateurs~

il convlent d'exhlber des condi-

tions sur un symbole permettant de fournlr une repr~sentatlon m4tadlff4rentlelle

de

l~op~rateur associ~ : cecl n4cesslte de d~crlre compl~tement les fonctlons de Wigner H(~ W

, ~,)

d'~tats propres de deux oscillateurs harmonlques dlstlncts

: c'est la

w

partle la plus technique de ce travail. Bien entendu, il y a une quantlt~ de travaux qul utillsent l'oscillateur harmoO

nlqUe ~ cltons par exemple le traltement par Melln [ii] de l'In~gallt~ de Gardlng ; slgnalons ~galement que Ralston [12] a r~cerm~ent utills~ des gausslennes dans des probl~mes de propagation des slngularlt~s.

Le falt que les normes symplectlques

jouent un rSle dans la description des op~rateurs pseudo-dlff4rentlels

(les symboles

de ceux-cl 4tant d~flnls con~ne ayant des d4rlv~es normalls~es born~es,

la normallsa-

tlon d~pendant du point et dtune norme attach4e ~ ce point) a ~

mls en ~vldence par

Beals [2], HSrmander [7] ; dans [16], nous avons abandonn~ la technique tradltlonnelle d'~tude des op~rateurs pseudo-dlff~rentlels (l'Int4gratlon par parties) pour la remplacer par celle~ plus efflcace~ de ramolllsseurs harmonlques

qul conslste ~ d~composer l'op~rateur cormne somme

~ la d~composltlon

comme somme de projecteurs-obllques

a ~t4 Introdulte dans [17] ; ces deux derni~res m~thodes sont toutes deux employees dans le present travail. Plan de itartlcle I. G~n~ralit~s

:

sur le calcul de Weyl-Wigner

2. Extensions du l~mne de Cotlar 3. Sur l'~quatlon de Schr~dlnger 4. Un oscillateur harmonlque 5. Deux oscillateurs harmoniques 6. Ramolllsseursharmonlques 7. Normes de cr~abilit~-annlhilabillt~ d'oscillateurs 8. Espaces de rep~res 9. Op~rateurs m~tadlff~rentlels.

relatives ~ un oscillateur ou une palre

207

I. G~n~ralit~s sur le calcul de Weyl-Wigner On consid~re l'espace

R~

l'espace de phase

R 2~ muni de la forme symplectique

d~finie par (1.1)

[(x,~),(y,~)] = - < x,~ > + < y , ~ > ,

et la quantification de Weyl OP½ d~finie, pour a E £ ( R 2 v ) (1.2)

et u E ~ ( R ~ ) ,

par

OP½(a) u(x) = 77 a(X2-~,~) e2i~< x'Y'~>u(Y ) dy d~ .

On salt que, pour tout a E £ ( R 2 ~ ) , tout a E p ( R 2 ~ ) ,

Opt(a) op~re de ~' ( R ~) dans ~

R~), et, pour

OP½(a) op~re de ~ ( R ~) dans ~t( R ~) ; a est le symbole de OP½(a).

On a aussi~ pour u, v E ~ ( R ~) : (1.3) o~

(OP½(a)u,v) = ~F a(x,{) H(u,v,x,~) dx d{,

H(U~V) est la fonction de Wigner d4finie par

(I.4)

H(u,v,x,~) = 2 ~ 7 u(x+z) ~(x-z) e -4i~< z,~ > dz.

La fonctlon H(u,v) appartlent ~ ~ deux ~ ~( R ~) (rasp. ~ ( R ~ ) ) . Wigner H ( ~ )

R 2~) (resp.~l(R2~))

si u, v appartiennent toutes

II est classique, at fondamental, qua la fonction de

peut aussi s'interpr~ter cormne le symbole du projecteur oblique ~ ( ~ , % )

d4fini par (1.5)

~(~,,)u

= (u,,)~ .

On conservera par la suite cette notation : (1.6)

~(~,%) = OP½(H(~,,)). Les grandes lettres X~ Y~ Z~... d4signeront toujours des points de l'espace de

phase

R 29. On convient dans cet article~ dans le cas de symboles d~pendant de para-

m~tres appartenant ~

R2W~ d'attribuer toujours ~ X le rSle de variable op~ratoirej

c'est-~-dire que~ par convention~ (1.7)

OP½(a(X,Y,Z .... )) = 0P½(XI

> a(X,Y,Z,...)).

Rappelons la formule de composition des symboles : si a et b appartiennent ~( R2~)~ on a

(1.8)

OP½(a) OP½(b) = Op%(a 9b),

avec (1.9)

(a #b)(X) = 22 ~ , ~ a(Y) b(Z) e "4i~[Y-X'Z-X] dY dZ. Nous aurons fr~quemment ~ consid~rer des op~rateurs agissant sur les fonctions

d4finies sur l'espace de phase. Le symbole d'un tel op~rateur est donc une fonction

208 dSflnle sur l'ensemble des points (X,-~) de (R2w) * ~ sur

R 2~

(i.Io)

R 2v X ( R2~5". On ~vitera d'Identifier

R2w s mals on utlllsera souvent~ en revanche~ l'isomorphisme

~

de (R2~) *

d~fini par l'Identit$ < x, ~ > = -[x,oz].

SI a, b, f E ~ ( R 2 V ) ,

on a, pour tout X E

( a ~ f:@b)(X) =

R 2~ :

24~fffa(ZISf(YSD(T2)e-4i~[ZI-TI'Y'TI]e-4i~[TI-X'T2"X] d T 1 dT 2 dY dZ

= 22v; la(Z 15f(Y)b (X+Y-Z 15e "4i~ [ZI 'Y]e4i~ [X'Y-Zl ]dZ 1 dY

=

7;

a(~-~Sf(Y)b(~5

d'o~ il r~sulte que le symbole de l'op~rateur (1.115

(X,~_ - )~

b(X-

> a(X +

e-2iR[X-Y'Z] dY dZ,

fl > a ~

f ~b

est la fonction

~)

Rappelons qu'il existe tm homomorphlsme unique

M ~-+M~

de noyau r~dult ~ deux

~l~aents, du groupe m~taplectique Mp(~5 sur le groupe symplectique Sp(w) tel que~ pour tout a E ~ ( R 2 ~ ) , (1.125

on air

M Op½(a) M -I = OP½(a o M-l).

Si B e s t une matrice r~elle sym~trlque de format

w X ~, si A est r~elle~ inver-

sible de format v X w~ les transformations M e t N d~finles ci-dessous sont m~taplectlques : (1.135

(Mu)(x5 = el~(Bx'X)u(x). (NuS(x5 = Idet Ai-~u(A-ixS.

Leurs images dans Sp(~) sont donn~es s e n matrices-blocs~ par

1/

B

A'

Pour permettre au groupe symplectlque affine d'op~rer sur les symboles, introduisons ~galement les translations de phase ~Z definiess pour Z = (z~5 E z

(1.15)

(TzU)(X5 = u(x-z5 e 2i~< x - ~,~ >

Ces transformations v~rlfient les identlt~s

(1.165

(~Z)* :

(i.17)

Zl Tz2

[lZ

= ~zl

'

= ei"[Zi,Z2 ] ~ZI+ Z 2

R2~

par

209

et (1.18)

H(TzU,~zV,X ) = H(u,v,X-Z).

Cette derniare identit~ est ~qulvalente h (1.197

TzOP½ (a) T-Z = OP½(a(X-Z))

(cf. (1.7)). On se servira 4galement de la formule~ dont la v~rlfication est irmn~diate : (1.20)

H(T_zU,TzV,X) = e 4in[X'Z] H(u,v,X).

En eombinant (1.18) et (1.20), on obtient (1.21)

H(TyU,~y,V,X) = e -i~[Y'Y'] e 2i~[X'Y'-Y] H(u,v,X - Y2~'),

autrement dit~ en Introduisant des translations de phase sur l'espace de phase : (1.22)

H(~yu, Ty, V) = ~y+y, -I H(u,v). -~-,~ (Y-Y')

2. Extensions du lermme de Cotlar Ces deux lemmes sont des formes assez pr~cises du lermme de Cotlar-Stein sur les "sonmles de presque-projecteurs-orthogonaux" dont l'emplol dans la discussion de la continult~ des op~rateurs pseudo-diff4rentlels est d~clsif depuls le travail de Calderon-Vaillaneourt

[4]

; il sont un peu plus g4n~raux que les deux lermnes que

nous avions prouv~s dans [17]. II faut auparavant pr~clser la d4flnitlon suivante relative aux noyaux : solent Q et ~' deux espaces munis de mesures positives @-flnlesd~ et d~'~ et solt k(wp~') une fonctlon mesurable sur ~ X ~' ; nous dirons que k est le noyau d'un op4rateur born4 de L2(~ ' ) d a n s L2(~) sl les conditions suivantes sont remplies : (it pour toute f 6 L2(Q'), on a (il) la fonetlon (Ill) on a

llKfll ~

~Ik(w,w')~ ~f(w')~ d~' < ~

(Kf)(~) = 7 k ( ~ W ) C lifil

f(w') dw'

avee une constante

pour presque tout w.

alnsl d4flnie appartlent ~ L2(~). C >0

ind~pendante de f.

Naturellement, K s'appelle l'op4rateur de noyau k. La composition des noyaux marche bien dans le cas o~ les valeurs absolues des noyaux fournissent 4galement des op~rateurs born~s : elle marche aussl dans le cas suivant (non comparable au pr~c4dent) dont nous aurons davantage l'usage : Lermne 2.1 :

Solent ~, ~ I

Q,, trois espaees munis de mesures positives o-finies dw~ dW'

et d~" ; solent k(w~m') et %(~",m') deux noyaux d4finissant des op4rateurs born~s (entre les espaces L 2 correspondants) K et L ; on suppose de plus que pour presque tout w on a ~ I k(~,w')~2 dw' < ~

et que pour presque tout ~" on a ~l%(w",w') 1 2 d ~ ' < ~ .

210

Si l'on pose (pour presque tout (w,w")) : m(~,~") = ,~ k(w,~') 7(W",W') d00', la fonction m v~rifie la condition elle

eonstitue

~ Im(m,~")l 2 d~" < ~

l e noyau de l ' o p ~ r a t e u r

born~

KL

pour presque tout w~ et

: L2(~ '') ~ L2(~).

Preuve : posons~ pour presque tout m~ k (m') = k(m,m') ; la fonction k appartient w w l'on a~ pour presque tout w" :

L2(~ ') pour presque tout m e t

(L k )(~") = 1%(~",~') k(w,~') d~' w d'o~

m(~,~") = L k (~")~ ce qui prouve le premier point. On a par ailleurs, pour w toute fonction g ~ L2(~")~ et presque tout m : (KL*g)(w) = .I k(~,~')(L * g)(~') d~' = J g(m") L k (m") dm" = r m(m,m") g(~") d~". w

Lermne 2.2 : Soient

~

et

et d00'~ et soient % , ~ M

~'

deux espaces munis de mesures positives o-finies d~

et A

, des op~rateurs lin4alres born4s sur L2( R~)~ d4w w~w pendant mesurablement des variables sp4cifi4es. On pose k(~,W' ) =IIAw,(~,Ii , kl(W,~l)= IIMwlM~'I et k2(w',w ~) = II%~ L~,I1 , toutes normes repr~sentant la norme dans l'espace des endomorphismes born~s de L2(R~) On suppose que k~ k I e t k 2 sont les noyaux d'op~rateurs born~s de L2(~ ')dans L2(fl)~ de L2(~) dans L2(~) et de-L2(~ ')dans L2(~ ') respectivement, de normes respectives C, C 1 et C 2. Alors l'op~rateur A sur L2( R ~) d~finl par A = $~X~' M~ % , w ' % '

d~ d~'

et born~ sur L2( R~)~ de norme moindre que C(CIC2 )½. Preuve : d'apr~s [17], on a, pour tout u E L2(R~),

les in~galit~s

filM uiI2 d~ ~ C 1 lluil2 W et 7 Ii%, uli2 d~o' ~ C 2 liu]l2. Alors

!Z k(~,w') II%,uilllM vll dw dw'

< cCIllL,uil2 ~,)~ (r fin vli2 ~)~

211

c(cic2)½ Ilulilfvli . Lermne 2.3 : Solent

Q

et

Q'

deux espaces munis de mesures positives ~-finies d~ et

d~'. Soit k(m,~') un noyau mesurable sur ~ X ~', et ~ I

~ #m (resp. w ' i - - ~ , )

une fonction mesurable sur ~ (resp. ~') ~ valeurs dans L2(R~). On pose kl(w,~ I) = ( ~ 1 , 9 )

et

k2(w',~' I) = ( ~ { , ~ , ) .

noyaux respeetifs k~ k I e t

Solent K, K 1 et K 2 les op4rateurs de

k 2. Alors s i c e s op~rateurs sont born~s entre les espaces

L 2 correspondants~ K 1 et K 2 sont autoadjoints positifs, et l'op~rateur A sur L2( R 9) d4fini faiblement par (Au,v) = !!~x~,k(~,~')(u,~w,)(~ est

et

v

rifle

,v) d~ ¢:~'

II All =

•

Preuve • Soit (En) une suite croissante de parties de mesure finie de ~, telle que [~ ----U En, et II~011 < n pour w ~ E nSi l'on d~finit Mn : L 2 ( R ~ ) -~ L2(En) (M*f)(X)n = ~E

f(00) ~w(x) dw

par (MnU)(w) = ( u ~ w ) ~ on a

pour toute f n L2(En ), et

n (MnMnf)(w) = fE

kl(~'~l) f(Wl) dWl ' n

d'o~

fiNn M*ii ~ C

que

IIMnuil ~ C [iuli, et par suite l'op~rateur

(Mu)(~) = ( u , ~ )

avec

C

ind~pendante de n puisque K. est borne. II en r~sulte M ". L2(iR ~) -~ L2(Q) d~finl par

est born~.

Q u e l l e s que s o l e n t f e t g a p p a r t e n a n t ~ L2(f))~, on peut a l o r s ~ c r i r e (MM*f,g) = I (M*f,~) g(w)dw = ,~ (f,M~ O) g(~) dw = ~I g(w)dw f f(wl)(*Wl'* m) d~, d'o~

MM

= K I.

On d~finit de m~me L : L2( R ~) -~ L2(~ ') par (Lu)(w') = (u,~,),

d'oO LL* = K 2.

On a enfinp quelles que soient u et v E L2( R~)~ l'identit$ (M*KLu,v) = (KLu,Mv) = I (~ ,v) do)~ k(~,w')(u,~,) D'oO

d~'.

A = M*K L, et (puisque ilM*fll2 = (Klf,f) pour toute f E L2(Q)) : ~" ½ On ~erlra aussi Au = ~f k(w)~')(u,~,)

ou m~me (cf°(l.5)) • A

=

f2

,~ dw dw'

½

K ½ II

212

3. Sur l'~quation de Schr~din$er Soit A = OP½(a5 un op~rateur lin~aire sym~trique sur L2(R~), a est donc r~el, et l'on supposera que A op~re de fondamental

~( R ~) dans

de domaine ~ ( R~):

~( RVS. Un probl~me

est la recherche de conditions pour qu,il existe un groupe fortement

continu (3.15

R

t

= e

2i~tA

d'op~rateurs unitaires de L2( R ~) dont le g~n~rateur infinitesimal de 2i~A ; le cas ~ch~ant~ il faut voir ~galement s i c e

soit une extension

groupe est unique.

On sait~ en vertu d'un th~or~me de Stonej que le premier probl~me ~quivaut ~ la recherche d'extenslons autoadjointes

de A ; la seconde question posse est de savoir

si A, de domaine ~ ( R~)~ est essentiellement

autoadjoint

: on y r~pond par l'affir-

mative si l'on salt construire un groupe R t qui v~rifie la propri~t~ suppl~mentaire que, pour tout t, R t

op~re de ~ ( R ~ )

Jans lui-mSme. On peut, en pensant aux repre-

sentations de Schr~dinger et d'Heisenberg de la m~canique ondulatoire, construction d'un groupe R t

~tudier la

en deux temps. Le premier (sur lequel le present papier

ne contient pas d'information5 va consister en la construction du groupe

R t d~finl

par

(3.2)

Op~(~tb) = R t OP½(b5 R~ I

pour tout symbole b E L2( R2~5 ; compte-tenu du fait que les ~l~ments de L2( R2~5 sont exactement les symboles des op~rateurs de Hilbert-Schmidt~

on volt ais~ment

que (3.2) d~finit effectivement un groupe unitaire fortement continu si R t a d~j~ ~t~ construit (mais c'est le passage dans l'autre sens qui nous int~resse5 ; le g~n~rateur infinitesimal (3.3)

2i~ ~ =

d~fini pour b ~ ~ ( (3.4)

de ee groupe dolt ~tre une extension de l'op~rateur

ad 2i~ a R 2~) par

(ad 2i~ ash = 2 i ~ ( a ~

b-b ~ a5 :

cette formule r~sulte naturellement (3,55

d~(Rt u)

de (3.2)et de la formule

= 2i~ Au t=o

valable si

u ~ ~(

R~5.

D'apr~s (I.II), on peut done ~crire (3.6)

~ = OP½(a~

avec

(3.75

a~(X,~ ') = a(X + g'~5 - a(X - ~ - 5 .

213

Remarquons d'ailleurs que

~

op~re toujours de ~ ( R 2w) dans ~ ( R 2~) pulsque,

alnsl qu'il a ~t~ remarqu~ dans ~15] (preuve du th~or~me 4.2)~ chacun des deux termes de sa d~composltlon est le conjugu~, par une certalne transformation m~taplectlque, de iVop~rateur A ® I operant sur les fonctlons d~flnies sur

R 2w X R2w ~ ce r~sultat

est ~galement annonc~ dans ~i]. A tltre de curloslt~, ces symboles

~

v~rlflent la

propri~t~ que l'op~rateur OP½(a~ co~nclde avec l'op~rateur Op(~) obtenu en falsant aglr les convolutions avant les multiplications : plus g~n~ralement, ~ par le groupe

est invarlant 1 d~flnl dans l'exerclce 1.3 de [14], dont la valeur en t =

jt

eonnecte les deux calculs symbollques consld~r~s. Le probl~me ~tudi~ Icl consiste ~ remonter du grouoe

R~t = e21~t ~ , suppos~

constrult, ~ un groupe R t. Naturellement, si l'on est optlmlste~ on peut penser, apr~s d~veloppement de Taylor au premier ordre dans (3.7), que l'op~rateur bi-->b o ~t' o~

~t

est le flot hamiltonien de a~ peut dans certalns cas ~tre le premier terme

dVune approximation convenable de m~me que l'on a

Rt = ~t

~t ; sl l'on est pesslmlste~ on retlendra quand

exactement sl a est un polynSme d'ordre 2, le th~or~me qul

suit fournlssant alors une nouvelle construction des op~rateurs du groupe m~taplectlque~ o~ l'on n'~prouve pas les dlfficult~s usuelles relatives aux singularlt~s de Maslov des phases (deux gausslennes n'~tant jamais orthogonales ) : le but essentlel de ce th~or~rae est pour nous de montrer l'Int~r~t qul peut s'attacher d'une part aux op~rateurs d~flnls cormne sommes de projecteurs-obllques, d'autre part aux d~composltlons de l'identlt~ d'un genre qul sera ~tudi~ plus loln. Th~or~me 3.1 : solt a un symbole r~el tel que Opt(a) d~flnisse un op~rateur ~( R ~) dans ~( R ~) ~ solt (3.6) et (3.7). Solent D u n

~

de

sous-ensemble de ~ ( R 2w) et to > O. On suppose que pour

tout b ~ D il exlste une unique fonctlon de classe C 1 : t ~ L2( R2w)~ prenant en falt ses valeurs dans ~ ( R 2 W ) , Cauchy

A

l'op~rateur de ~ ( R 2~) dans ~O( R 2~) d~flni par

{

d~ b t = 21~ ~ b t

>

b t de I-to,to[ dans

qul solt solution du probl~me de

(it[ < to )

b ° = b.

Solent par ailleurs

~

un ensemble munl d'une mesure positive ~-flnle d ~

> ~ une application mesurable de f2 dans lasph~re unlt~ de L2( R~)~ et k t~ une fonetlon ~ valeurs complexes sur f~ X ~ v~rlflant les propri~t~s suivantes :

tOI

(1)

pour tout

W E f2, ~tU = H(~m'~m) ~ D

(ll)

l'op~rateur de noyau

w

~

(~ ,~t~,) sur

L2(~) et ceux de noyaux k(~,tO') et k(~',w)

sont b o m b s et v~rlflent l'hypoth~se suppl~nentaire du lemme 2.1 qul permet la composition des noyaux. (Ill) l'op~rateur

~

k(~,~l)~(t,,~ ~ , ) dw d~' (born~ sur L2( R ~) d'apr~s le

lermme 2.3) co~nclde avec l'Identlt~.

214

On suppose enfln la condition suivante v4rifi4e :

(~v)

pour t o u t Posons 9 pour

ot tout t c ]-to,%[,

~ E ~

: o t ( ~ s ~ H(2i~A ~ , ~ W ) , ~ )

O.

et It i < t

o et

on a

(s

ds ,

, ~)

(t) ~t = e W

OP½(~t) ~

•

La famille (R t) (Itl< to ) d'op~rateurs d~finis par R t = $~ k(~,~') ~ ( ~ ,

%,7

dw dw'

s'~tend alors en un groupe fortement continu d'op~rateurs unitaires sur L2( R ~) ; le domaine du g~n~rateur infinitesimal de ce groupe contient l'espace vectoriel engendr~ par les fonctions ~

(~ C ~)~ et sur ce dernier espace le g~n~rateur infinl-

t~simal coYncide avec 2i~ A. Enfin, si l'on a

Rt(~(R~))

c ~ ( R ~) pour I tl < to ,

l'op~rateur A est essentiellement autoadjoint sur ~ ( R ~ ) . Preuve ". elle sera d~compos~e en lem~nes pour plus de clart~ ; signalons avant tout que, pour tout ~, % Lemme 3.1 : pour

E ~ ( R v) puisque

~

E SO(R2~).

I tl < to~ et tout ~ E ~, ~t

est le symbole d'un projecteur ortho-

gonal de rang I. Preuve ." posant

c(X,~_ ) = 2i~ a~(X,~-), on a

cette propri~t~

eomme le montre un calcul ~l~mentaire, signifie que l'op~rateur

2i~ ~ (3.8)

c(X,_~-) = c(X,- -~--) ."

cormmute avec la conjugaison complexe ) l'~quation diff~rentielle d ~t = 2i~ ~ t d--{ ~

montre alors que

~

t

est r~el pour tout t.

Cette ~quation fournit ~galement (3.9)

ii,~til

= 1

pour tout t.

Par ailleurs~

= 2i~[(a ~= t

t#a

) # t

+ ~t ~ = ( a ~

t - t

#at7

= 2i~ ~( t # t ) . Jointe ~ la condition initiale ~t ~

~

~= o = o ~ cette ~quation diff~rentielle entra~ne ~ t ~ est le symbole d'un projecteur o r t h o -

~t = ~t • Nous avons donc prouv~ que

gonal de norme de Hilbert-Schmidt ~gale ~ i~ ce qui ~tablit le lem=ne.

215

t ,W v~rifle le probl~me de Cauchy

Lemme 3°2 : Id

%t = 21~ A %t

(iti < %5

*~= % Preuve : le second point est 4vident. Utilisant le lemme 3.1~ posons provisolrement (3.105

Ht( q 0 t , ~

@ ).t

On a alors, pour itl < t : o

(3.115

(%,i5 ~ o,

(3.125

~ (t5 =

(%,~5

et

(3.13)

,~=e°

(t)

(%,15~

t

Partant de la d~flnitlon origlnale de t

d ~ (2i~~ , 1 5 = (2i~ A % , < 5 t

= 2in A~t +

t

~ on obtient

~(Op~(¢ %5

t + 21~ e W #W

(t)

(A

t OP½(~)

- Op~(~t)A) #~ "2 W

(2i~ A#w,
ce qul fournlt l'~quatlon dlff4rentlelle en se servant ~ nouveau de (3.13). =.<

t

Preuve : c'est vrai lorsque t = 0 ; par ailleurs~ d'apr~s le lerm~e I0.2~ on a

Lemme 3.4 ". les op~rateurs

Rt

(itl < to ) sont unltaires sur

L2(Rg).

Preuve : en utilisant le lermne 3.2, on volt que (%wt, t , ) = (~W,#W,) ' et le lemme 2.3 montre alors que les op~rateurs R t sont born~s.

216

.On peut en outre ~crire, compte tenu du lemme 2.1 : * t t R t R t = ,I k(~l,W:)k(w,W')(%~,%~l)~9~,l,:~W,)

d~ d~' d~ 1 d~:

et

Rt R t

dVo:

=

I.

Lemme 3.5 : Les conditions

isi< tO , it[ < t o

et Is+tl < t o

entratnent RtR s = Rs+ t

Preuve ! & l'aide du lemme 3.2~ on volt que, quels que solent d . t+s t ~{#W l , %)=

0

, d'o5

et

~i

' on a

. t+s t. ~@~i ' % ) = (~t~l'~!'W)

et R: ,t+s = ~ ~t+s t ~l k(w'W')(%l '%W) % '

s

Rt

~W 1

Lemme 3.6 : si

Rt

Par suite

=

# t+s

Wl

(3.14)

; le lemme en r4sulte irmn4diatement.

conserve

Preuve : quels que soient (A % t1 , t )

d~' d~

W

~ ( R W) et

pour

it I < to , on a

AR t = RtA.

~I~ on a

= (A ~ i,~0~) :

c'est une cons4quence du lemme 3.2. On peut alors 4crire t

t

= .~,I:(W,~ ')(A % 1 , % ) =A

d~,

*~0' d~

dw'

%1"

II en r4sulte que (3.15)

A % t1

= Rt(A ~~I ) •

Remarquons maintenant que l'hypoth~se (iii) impllque que les fonctions

~

(wCQ)

forment un syst&me total dans L2( R ~) (en effet, si u et v c L2( R~))~ la condition V J__%

pour tout

W

entra~ne ( u , v ) = O ) ;

~galement, R t % 1

= ~

~ ~( R~)"

Pour prouver le lerm~e, il suffit donc de montrer que les conditions u G ~ ( R ~ ) , v 6 ~( R ~) et

R* v C ~ ( R ~ t

entratnent (Rtu,Av) = (RtA u, v).

217

Or on a

= j~ k(~,cu')(u,%~,)(RtA9 ,v) d~ d~'

= (u,ARtv) = (Au,Rtv) = (RtAu,v). Preuve du th~or~me 3.1 : Pour tout W E Q~ et tout t 6 ~,to[ ~ on a d'apr~s le lemme 3.2 et la preuve du lemme 3.5 "

(3.16)

Rt ~ - ~ W t

=

t_ ~ t

1 = ~ 2i~ A %kt dk 1 = $ 2i~ Rkt(A ~ ) o

d'o~

d%,

liNt *W" ~w II~ t ii2i~ A ~WlI. II en r~sulte que le groupe (R t) est fortement continu (puisque les ~I~ forment

un syst~me total) ; par suite liE# ~ - A #WII tend vers 0 lorsque

t -~0 pour tout W,

et le g~n~rateur infinitesimal du groupe (R t) coincide bien avec 2i~ A sur les ~ . Enfin, pour prouver la derni~re assertion, il reste ~ montrer (cf. [13], th. Vlll.10) que moyennant l'hypoth~se suppl~mentaire que

Rt(~(R~))

c ~(R~),

on a

Ru-u t

---> 2i~ Au

pour tout

u 6

R ~)

lorsque

t -~ O.

Or, pour tout u 6 ~ ( R ~) et tout v E ~ ( R ~) tel que Rtv E ~ ( R ~) et livU ~ I, on peut ~crire Rtu_ u t (~,v) = Ill k(~,W')(u,*w,)( w t w, v) d~ d~' = ~

d~ ~! k(~,~')(u,~,)(2i~

R~tA~,V)

d~ d~'

1 o 1 = 21~ ~

(Rktu,Av) dk o

= 2i~ ~I (Rkt Au,v) dk. o Par suite Rtu-u

(--6-- - 21~ Au,v)l = i ~ ((R%t-17 21" Au,v)d% i

#

il(R%t- It 21~ AUll dk o tend vers 0 quand t -. O, ce qui termine la preuve du th~or~me 3.1.

218

4. Un oscillateur harmonique Soit phase

B

une forme bilin~alre r~elle sym~trlque non d~g~n~r~e sur l'espace de

R 2v, et solt par ailleurs une structure complexe sur

morphlsme J de

R 2~ de carr~ -I ; nous dlrons que la forme et la structure complexe

sont compatibles complexe

~

(4.1)

R 2~ d~finie par un endo-

(relativement ~ la structure symplectique de

R 2~) si la forme

d~flnle par ~(XI,X 2) = B(XI,X 2) + i[Xl,X2]

est sesquilin~aire. Soit

B°

la matrlce sym~trique telle que

B(XI~X 2) = < XI,BoX 2 > .

La condition de compatlbillt~ est l'identlt~ B(XI,X2) = [JXI,X2] ~ et s'exprime donc par l'~galit~ (4.2)

J = ~ B

, O

qui montre que la donn~e de D~finition 4.1 : soit

J

ou celle de

B

sont ~quivalentes.

X i---->IIXIi une norme euclidienne

sur l'espace de phase

nous dirons que cette norme est symplectlque si la forme quadratlque polarisation est compatible a v e c l a de

R 2~

B

R2~;

associ~e par

structure complexe d~finie par un endomorphisme J

symplectique de carr~ -i.

Posant IIXII2= < X , B X > a v e c

B ° sym~trique~

tique. Cormne~ dans la base canonique de pr~sent~e par la matrice repr~sentant

B -I

".1 (0

R 2v

on demande que J = a Bo soit symplecet la base duale de ( R 2 V )* , ~ est re-

-i) O ' il revient au m~me d'exiger que la matrice @

dans ces bases soit symplectique. On peut alors choisir V symplec-

O

tique avec

VV' = ~, d'o~ llXl]2=iV-iXl2,o~ X ~---~-X

est la norme canonique de

Nous reviendrons incessament ~ cette d4composltionp

bas~e sur la representation de

Siegel. Ce qui precede montre qu'une norme euclidienne est symplectlque ment si elle admet une base orthonormale la base canonique de

R 2~.

si et seule-

dont la matrice de passage~ relativement

R2v~ est symplectique. Autrement dit~ l'ensemble des normes

symplectiqum s'identifie ~ l'espace homog~ne des classes ~ gauche Sp(v)/Sp(~)~O(2~). Nous l'identifierons

de preference $ l'espace

sym~triques positives sur o~ (

,

P

des matrices

R 2~, par la correspondance

) d~signe le prodult sealaire canonique sur

ce qui concerne l'espace P

prlvil~gi~ de

symplectiques

R 2~. Dans ce qUl suit, tout

et la repr~sentatlon de Siegel est emprunt~ au livre de

Maa~ [IO]. II convient d'observer~ normes symplectiques par P

@

d~finle par IIXli~ = (@'Ix,x),

cependant~

que la repr~sentatlon de l'espaee des

~ puls celle de Siegel~ d~pendent du choix d'un rep~re

R 2~, commode pour effectuer les calculs ; mals les r~sultats finaux

de cet article, comme on pourra s~en convaincre, ne d~pendent que de la donn~e de l'espace de phase sous la forme du produit d'un espace vectoriel r~el de dimension par son dual,

219

D~finition 4.2 : solt

W = (Y,@) G

R 29 X P

•

Nous appellerons oscillateur harmonique associ4 l'op4rateur posera ~galement Avec

%

L

=

Op½(~lix-Ylil).~ On

= L~ - -~2 "

@ = VV', V ~ Sp(~), on a done

(4.3)

L

= OP½(~IV-I(x-y)i2),

et les formules (1.12) et (1.19) montrent que L lateur harmonique canonlque par

1 = ~

Pj , L ° = ~

j ~o

est unitairement ~quivalent ~ l'oscil-

L ° = OP½(Wl X 12) dont la r~solution spectrale est donn~e

(~ + j)Pj ~ o~ l'image du projeeteur P

j>o

est engendr~epar ~0(x)= o

29/4e-~Ix12, celle de P par les fonctions d'Hermlte ~0~(I~i = j) ; la description 3 de celles-ci ~ l'aide des op~rateurs de creation est bien connue (cf. par exemple [5]), mais nous allons la rappeler pour l'oscillateur g~n4ral L . Solent Y = (y~), ; le gait que tions

@

appartient ~

P

et

s'exprlme (ef. [i0]) par les condi-

B~ C B'C = CB, AB' = BA, AC - B 2 = I, et l'on a @ = VV', avec

o IA' o)

A-IB

1

0

A -½

"

c'est une formule de Siegel. Un ~tat fondamental de l'oscillateur OP½(~ IIxiI~) est alors, d'apr~s (1.12), (1.13), et (1.14), donn~ par (4.5)

~0@(x) = (det A) -~ 2 ~/4 e -n(A'Ix'x) e i~(A'IBx'x).

Enfin, d'apr~s (I.19)~ un 4tat fondamental de l'oscillateur L (4.6)

est donn~ par

~gw = TY ~@ "

Cette formule a d~j~ ~t~ donn4e donn4e dans [16], D~finltion 4.3 : On appellera

~

itapplication de

ainsi que la d~finition suivante : P9

dans l'espace

~)

des

matrices sym4trlques complexes de format ~ X ~ dont la partie r~elle est d~finie positive, donn~e par (A Bl (4.7) ~( ) = A "l- i A'IB. B' C La formule (.4.5) exprime donc que, $ normallsatlon pros, un ~tat fondamental de L@

est donn~ par

exp-~(~(@)x~x). Pour construire les autres ~tats propres, on a

besoin des op~rateurs de cr~atlon et d'annlhilation : D4flnltlon 4.4 : lorsque

f

parcourt l'espaee des formes lln4alres complexes sur R29 associ~e

R 29, antlholomorphes (i.e. antilln~aires) pour la structure complexe sur

8, l'op~rateur Op%(f(X-Y)) d4crlt l'espace des op~rateurs de cr4atlon assoei4s = (Y~),

et l'op~rateur Op%(~(X-Y)) d~crlt l'espace des op4rateurs d'annlhilatlon

associ~s ~ W.

220

Plus s~cifiquement,

toujours avee @ = V V ' ,

o3 V e s t

bases d'op~rateurs de cr4ation et d'annihilation

d4finie par (4.4), voici des

: lorsque ~ = (O,I)~ on d~finit

(i ~ k g ~) a~k)(x,~) = ~%(Xk-i ~k )-

(4.8)

= (Y,8), on pose

Pour

a(k)(x) = a(k)(v-l(X_y)), o

(4.9) et

A (k) = Op%(a(k)),

(4.~0)

(A(k)) * = ~~p % ~ ,a(k), = ~ ~ •

Les A (k) (resp. ( A ((k))*) constituent donc une base de l'espace des op~rateurs de ~) crdatlon (resp. annihilation)

associ~s h ~. Soit

1.12)). En posant N = vyM~ on a donc

M ~ Mp(~) telle que

~=

V (cf.

(k) A (k) = N OP½(a ° ) N -1, ce qul permet de

r~dulre au cas de l'oscillateur canonlque L

la v~riflcatlon,

alors Irmn~dlate, des

O

r~sultats qul suivent : Proposition 4.1 : On a l e s et non forme symplectique %=

5~ A •(k) (A(k)~ • .* ;

%+

relations

[A (j), A (k)] = 0

(crochet de deux opdrateurs

• ,-, ~(A(k))*] ~ =- 0 ; [A(j ) (A(k))*q !) ; [(A (j)) t~ (~ t ~ ~ ~ J -- - jk '

~ = L

+ ~v = ~(A(k)) * ~

A(k); A(k)L • L~ A(k)= • • • + A •(k) .

Cette propositlon montre d'une part que, pour tout multl-lndice ~ = (~1,~2,...,~) ~ la fonction

A~w ~

= (A(1))~Iw "'" (A~ ~ ) ) ~ ~w

est un ~tat propre de nlveau d'~nergle

~I (i.e. correspondant ~ la valeur propre ~ + I ~

de l'oscillateur L

, d'autre part

que~ dans l'espace engendr~ par les op4rateurs de cr~atlon et d'annlhilatlonj

les

op~rateurs d'annlhilatlon peuvent 8tre caract~rls~s cormne eeux qul annlhilent l'~tat fondamental ~ ee qul prouve que tions A ~ W (3.11)

@

. Sl ~ = I, on a (A~,A~) = (A A<0,@) = ((A+l~0,@) pour tout co, ~AJ@II2 = j!. Dans le cas g~n~ral, on normallsera donc les fonc-

en posant ~

= (~!)-½ A •~ ~

•

En d~slgnant par 0 le point prlvil~gi~ (0~I) de plus haut, les relations

~

R 2~ X P

= N ~o ~ ~ condition que M a l t

~ on a alors~ avec N d4fini ~t4 d~flnl, conform4ment

au proc~d4 qui a conduit ~ (4.5)~ par la formule (4.12)

(Mu)(x) = (det A) -% ei~(A'iBx'X)u(A-~x).

Pour ~ fix~, les dlverses fonctions le cas o3

~

~ = (0,I)~ on se famine au cas

sont orthogonales entre elles pulsque, dans ~ = I~ o3 ces diverses fonctions corres-

pondent alors ~ des nlveaux d'4nergle dlff~rents.

221

Ii est utile de transporter sur l'espace des op4rateurs d'annihilation la structure hermitienne h~rlt4e, par passage aux symboles~ de eelle de dual complexe de l'espace de phase : se ramenant au cas o~ ~ = (O,I) on v~rifle que l'on a, pour deux tels op~rateurs (4.13)

F

et

G~

(F,G) = ~(G m

, F ~ ).

5. Deux oscillateurs harmoniques D~finition 5.1 : Solent

~ = (Y~@)

d4flnlt ~ o ~' appartenant ~

et

m' = (Y',@')

appartenant ~

R 2~ X ( R 2 9 ) * X P2~ par ~ o ~

R 2V X P

. On

= " 2 ~ -l(y_y,),®)~

avec

=llx + 5=li ÷ tlx %=Ife On d~finit alnsi une composition "ext~rleure", mais il convient de v~rifier que la norme d~flnie par ® est symplectique .. c'est bien le cas lorsque

~ = @' = I

(on esp~re que~ @ ~tant par d~finition sym~trique~ le leeteur tol~rera la notation @' pour une autre matrice sym~trique ind~pendante de @) ; posant

JlXlI~ = iV'IX1 2

et

Ilxll~ = lwlxl 2 , et r~solvant les ~quations y

+ ~ =v-l(x + ~-~-)

Y - - ~oH = w-1(x . ~ ) , on volt qu'il faut v4rifier le caractare symplectique de la matrice

2 ~-i (v-l- w'l)

¢

. 1 4 _ 1 + W" 1 2

= C

D

autrement dit les 4quations AIC = C'A~ BID = D'B, AID - C'B = I. Tenant compte des ~quations et

W

A' = ~I - I ( v + w ) ~ Proposition 5.1 : X 6

et

V ''I = o'Ivo

et

W '-I = ¢-Iw~

qui expriment que

sont symplectiques~ cette v4rification r4sulte ir~n~diatement des formules , B' = - I(V-W)o, C' = - o - I ( v - w ) ,

D' = I(V+W)o

avec les notations de la d4finition 5.1, appelons, pour tout

R2V, f(X) i tunique Y tel que lJX-YII~

+ nYli28, soit minimum, et posons

QS,@'(X) = Fx,f(x)].

Au sens de la d~finltion 4.3~ ~(®) est alors donn4 par (~(e)X,X) = 2 IIXIIZ~,@,~ - 41 QS,@, (x).

V

222

Preuve ." slgnalons d'abord que la norme

II If@,@, diff~re par le facteur 2 de la

d~flnltion que nous en avlons donn~e dans [16]

II II~),(~= I

et

[17~ (cecl pour assurer

II(~).

En utillsant,

corinne dans ([16], lermne A 2), une base symplectique dans laquelle

on alt

llx%

÷

2

_~_

kjl

2

on a alors

llx,de, e,=2~I+~j et

x2 +

j

1

2

~j

1-xj Qe,@'(x'1) = Z ~ Avee

X = (x,~), et

xj ~j .

(7~ = Z = (z,~), on a, en n4gllgeant l'indlce j :

E2)2 ,

z 2+ ({ + ~2)2+ X-I(~ = (x + 2)2+ 3k(x- ~) dto~

=

-o et par suite

® =

,

4

I+X!

x-._.£ 2~.

o

- 4~. -

C

4X

0

o

avec B'

A =

x+_!

et

B =

I+

. l-k

-/-I

2

Ii en r~sulte que ~(®)) = A "I- i A-IB =

4k I+---~

0

O

O

-i

21-~

I-%

i

21+ i

o

/

ce qui termine la preuve de la proposition 5.1~ Proposition 5.2 : avec les notations de la d~finltlon 5.1 , la fonction ~ , = H ( ~ , ~ 0 w , ) est un ~tat fondamental normalis~ de l'oscillateur Preuve :

l'~galit~

il ~,w,ll = 1

L

w, .

r~sulte de (1.6) et de l'~galit~ entre la norme L 2

d'un symbole et la norme de Hilbert-Schmidt

de l'op~rateur associ~. Le cas o~ ~ = (O,~)

(on ~crira ~ = e par abus d'~criture commode) r~sulte de la proposition 5.1 et du

223

lermae A2 de [16]. Dans le cas g~n~ral, grace $ (1.22), on a

H(.,~0 ,) = H(~y ~@,~r,~,) = ~Y+Z'

-l(y.y,) ~,~'

'

2 ~ ce qui termine la preuve de la proposition. On passe maintenant ~ la description des ~tats propres non fondamentaux de l'oscillateur L~o~' " Pour tout w = (Y~@) E R 2~ X P~ ~ introduisons les coordonn~es complexe s (holomorphes pour la structure complexe d4finie par 8) X (k) d~finies par (5.1) o5

~½ X (k) = ~(k)(x),

a (k) est d~fini par (4.9).

Ceci permet de d~finir ~galement les op4rateurs D~finltion 5.2 : Pour tout w E R 2~ X P

Dx(k )

et

5~(k)

"

, on d4finlt les 4~ op~rateurs dlff~rentlels

suivants operant sur les fonctions d~finies sur l'espace de phase : ~k)

=

½ ~(k). I_/__

w

~

~k)

= ½

~

2~% b--~

x(k)_ 1_!_

( ~ ( k ) ) * = w~x(k) +

W

1

2 ½ ~ x(k)

"

Rappelons que

X (k) = ~(k)(v~I(X-Y)) (cf. 4.9)), avec V t~ o Le symbole de l'op~rateur 21~ 1 ~x(k) est la fonction

d4flni par (4.4).

I ~ (e2i~< X'~->)(X = O) = - - ~ < X,~- > 2in .x(k) bx'-"

~ ) < VX,~ >= bx(k o Avec

-bx(k / _ ) [x,v-lo~].

v - l o ~ = (z,~), cela s'4crit _

i

2n½

a ( k ) (V-I

o •

i

i(~ k + i zk) = ~(z k- i
_~).

o

II en r4sulte que le symbole de ~ k )

est la fonction

a(k)(v-I(x-Y))o + 21 a(k)(v-i o G ~)

= a(k)(x + ~). t~

224

Avec trois autres calculs analogues, on obtlent : Proposition 5.3 • avec les notations de la definition 5.2, on a l e s IdentitEs

( ff~(k))*= o~ ~(~(~)(x + ~)) (~k))*=OP½(a(k)(x-

~-)).

Proposition 5.4 : quelles que soient ~

et

E ~(R~),

on a l e s identltEs (o~ A (k)

a ~tE dEflnl en 4. I0) • H (A(k)~o,~) =~k)H(~0, $) W H (~,A(k),)= ~(k)H(~o, ,) g(A(k)* ,$} = ( ~ k ) ) * W

H(~0,~)

Preuve : dEmontrons par exemple l a seeonde i d e n t i t ~ • d ' a p r ~ s ( 1 . 6 ) , H(~,A~k)~)-- e s t l e symbole de l ' o p ~ r a t e u r u

,

-

(u,A(k)~) cp -----((A(k))*u,~) ~0 ,

¢'est-~-dlre que l'on a ~(~0,A(k),) = ~(e0,,) o (A(k)) *. Corame (A(k))*0~= O P ~ ( ~(k))' il rEsulte alors de (I.Ii) que l'on a H(~o,A(k),) =0P½(a(k)(x-

~))H(~,,),

ce qul prouve la proposition. DEfinition 5.3 : avec les notations des definitions 5.1 et 5.2, on pose

~'~

= H(~, ~0~ ) = (~:1"~(~!1"~ ~(A ~

P r o p o s i t i o n 5.5 • on a l ' i d e n t l t ~

A~

%,)

225

Preuve : quelles que solent (5.2)

z~k)(~k))*

¢0 et

# 6 ~ ( Rg)~ on a

H(~,*) = ~ H(A(k)(A(k))*W W ~0,#) =H(( L -

~) ~ ~0,*),

o3 l'on a utills4 la proposition 4.1. D'apr~s (i. II), le symbole de l'op~rateur on verifie" de mSme qua le symbole de

~ ~(k)(~k))*

7~,k)(~k))*

est done ~ilX+°--~--Yli2

est ~ llX- ~

- yv 2_e, -~2 •

Enfin~ d'apr~s la proposition 5.1~ le symbole de l'op4rateur L (5.3)

~)IX--~-- + ~ ~ ( ~

o lJx + ~

- o'I(Y-Y'))I I +~I{X

~ ~I~+~ ix

o-I(Y-Y'))

--o~ ~ 11~ 2

e' "

Proposition 5.6 : les fonctions

~8

propres de iVoscillateur

la fonctlon

L o~,

Y+Y'_ 1 o ( ~ 2 2

est

constituent une base orthonormmle d'~tats ~0~@ w,Wi

ayant le nlveau d'~nergle

J~i + 181.

Preuve : d'apras (1.3) et (1.6), on a

= (~o~ , ,~0~ , ) (cp~,m ~)

Par ailleurs~ compte tenu de la proposition 4.1, on a (pour (P, # ~ ~ ( R ~ ) )

les Iden-

tit4s k)H

) =

H(A (k)

-

est l'identlt~ analogue relative ~ ~ Ces Identit4s prouvent que d'4nergle

~,W, •~'8

k)

=H(%A(k)m,9)

+

~

e'%'~)

.

est une fonctlon propre de

%o~'

de niveau

I~I 'i~I-

C_ompte tenu des propositions5.4 et 5.6~ il est clair que les 2~ op~rateurs ~ k ) et

v

et

(

ooot

oe

uoo

teurs de cr~atlon (resp. annihilation) relatlfs ~ l'oscillateur

%o~'" Rappelons

que sur l~espaee des op~rateurs d'annlhilatlon exlste une norme euclldlenne canonlque. la formule (4.13) fournlt (5.4)

((~j))*,(~k))*)=~(~k)~

•~

,,$~j)~ •

~

. ,

,.(k),O ~(j),O) = ~'

8jk

e t deux ~gallt~s analogues, qui montrent que la base constltu~e des op~rateurs

(~(i))*

.,(~))*

( -- i)7" ,.

(

, )

est orthonormale.

226

6. Ramollisseurs harmoniques D~finition 6.1 • nous appellerons ramolllsseur harmonique attach~ ~ w=(Y,~)6 R2~X P le semi-groupe e-t~ = e -t(l~ " ~) (t E R+5. On pose ~galement e

-t A W = OP½ (kt5 •

D'apr~s [16], on a

t %t(x) = 2v(l+e't5 "~ exp-2rT(th ~)

(6.15

ilx Yli~

et en partlculier (6.25

%°°(X5 = 2Vexp-2~ IIX-YIi~ .

Proposition 6.1 : avec les notations de la d~finition 5.1~ on a, pour tout a E ~ ( et tout t >_ O, -t %o~' %t (6.35 e a = w#

t a ~ ~, .

Preuve : d'apr~s (I.II), le symbole de l'op~ratlon (6

5

t(x +

R2~5,

t

a ~ + ~ ~

a ~ ~Wt,

est

=

22V(l+e't)'2Vexp-2~(th 251''X + ~

- Y'i~ + iiX - ~

-Y'ii2@,},

ee qul prouve la proposition $ l'aide de (5.3) et (6.1). -i -I Proposition 6.2 • avee les notations de la d~finition 5.1, soit I%1,...,% ,~,...,% ~ 1 l'ensemble des valeurs propres de la forme quadratique i(XII~ par rapport ~ la forme

~, ; posons

quadratique i X 2 La norme sur (6.55

~j = %~ + %?½ (I ~ I ~ vS.

L2( Rv5 de l'op~rateur

e

-~

e -t

%'

est alors donn~e par

il e - t % e-t %'11 = 2~(l+e "2t) " v ~(BJ th t + (~j2 th2t + ch-4t)½)-½ exp - ~(th tsII Y,-YII~,@ , .

Preuve : d'apr~s des ealculs assez laborieux effectu~s dans [17], on a, Si e = 28(1+82) -I et B~ = OP½(2~/2(l+625v/2 e "2~8 IIX-Y~), l'~galit~ (6.65

IIB~ Be~,}i = 2 ~ ~(~je + (~j2 2 + ( 2_l)2)½exp_ ~Iiy,_yl i 2@,@, •

Ii suffit alors de poser 6 = th 2' s = th t, d'o~ e - t % = (l+e'2ts'V[2Bg . On se contentera dans les applications de l'in~galit~ (qui r~sulte de (6.55 et de l'in~galit~

~j ~ 2) •

227 (6.7)

II e - t %

e-t%'ll ~ 2 ~ U(lqq~jth t) -½ exp- ~(th t) liY,-YII2~,£, .

7. Normes de cr~abilit4-annlhilabilit~ relatives ~ un oscillateur ou une paire dVoscillateurs. Dans ce paragraphe s on 4tudie la possibilit4 de representer une fonction u comme sormae de translatSes de phase de l'~tat fondamental ~

d'un oscillateur harmonique ; w remontant sur l'espace de phase~ on ~tudiera ensuite la representation d'un symbole comme sormne de translat~es de phase de l'~tat On posera

~ = (Y~)

et

@ =VV'

tie ; dans la seconde, on posera Lergne 7.1 : pour tout

avec

V

~~ w t relatif ~ une paire d'oscillateurs d~fini par (4.4) dans la premiere par-

W' = (Y',~')

et (~ o ~') =(~---" -l(y,y,), ®).

u ~ L2(R~), on a

IIull2 = ~ i(u,Tz (0w)I 2 dZ . Preuve ." .I I(U'Tz ~ ) I 2 HZ = (Au,u), avec

(7oi)

A = ~

(VE~W , TZ ~W) dZ = 0P½(a),

o~

(7.2)

a(X) = ,~ H(~ z r0W, Tz r0w,X) dZ = ,[ ~,~(X-Z)dZ = Tr~(¢0 ,~w) = i.

Lergne 7.2 : sl

u = ~I f(Z) TZ q°W dZ~ avec f E L2( R29)~ on a

l]uli2 g ~if(Z)i 2 dZ Preuve : liul~ = ~ f(Z)(T z ~0 ,u) dZ (~If(Z))2dZ) ½ (71(T Z ~0 ,U)I 2 dZ)½ = (7if(Z)I2 dZ)½ llull . Len~ne 7.3 • on a l e s identit~s (A(k)) * ~

,rZ cow

= ½

(7.3)

(A(k)) * ~

~Z ~0 ,

.(k) est d~finie par (5.1). m8

o~ la eoordonn~e complexe Preuve : avee

(k)

Z@

a(k)(x'o ~) -- ~½ (Xk + i~k)' on a = 0P½ (a~k)(v-l(x-Y))),

d'o~ d'apr~s (1.19) (7.4)

-

v z(A(k)) * TZ

=

^

(-(k)..

uP½ a °

Iv

-i

(X+Z-Y))),

228

et (A(k55*~ Z ~0 = Cz((A(k)) * ~

(7.55

+ 7~k)(v-iz))~

= a(k)(v-IZ)o ~Z W~ = a~k)(z5 ~Z ~m =

D4finition 7.1 : Soit

k = 0,I~2,...~ et soit u ~ ' ( R g ) .

½

.(k) m~

Tz <°[D •

On pose

Iiull > = (f <~+ ilzi~ ~ l(u,~z ,%51 ~ az5½ On appelle ~ k l'espace veetoriel (ind~pendant de ~ ainsi qu'il r~sulte des lermmes qui suivent5 des

u 6~'(R~)

pour lesquels cette norme est finie.

Lemme 7.4 ". il existe une constante

C > 0 , ne d4pendant que de k et de ~, telle que

I

l'on alt pour tout u ~

:

c-liiu!l~,~k K

Preuve :

~7 I[A~ ull2 I~I ~ k

~ c llu/I2 ~,~

d'apras le lemme 7.1~

I~l~ k

IIA~ uil2 = Z ~I(A~u,T z <00)i 2 dZ = Z ~ ta' I~ k ~ I~l~ k

=

~

,f I(u,(~)* ~z %)12

~!~If {z~;2 l(U,~z,0)l 2 dZ ~IIul,2

dZ

k"

Lemme 7.5 : il existe une constante C~ ne d~pendant que de k et de u~ telle que l'on alt~ pour

tout

u E

I~l< k Preuve :

i~l~ k

W

l~I~ k

cormnen~ons par remarquer que lyon a k! ~ ±~! l~)=k

A~ * = % < A I>...
W

En effet~ pour k=l~ ce falt a ~t~ remarqu~

dans la proposition 4.1 ; on proc&de

ensulte par r~currence, en utillsant la relation [(A~ j ) ' *)- ~ (kS.| = 6jk , et en remarA

quant que la relation ~ prouver peut s'~crlre aussl %(%-15...(%-k+15 = Z A

(Jo~

. A

(Jk-I) Jo * (AJk-I * (A)'... ) ,

la somme ~tant 4tendue ~ toutes les suites (jo~...~Jk_l) de k entiers > O. On montre de m~me que 1 (A~ 5" A ~ = ( % + ~ ) ( % + ~ ~ I)...(A + ~ qc k-l). Z ~. iffl= k W Ii suffit alors de remarquer qu'll existe (Co~...gGk) tels que l'on alt l'Identltg k!

entre polynSmes ~

R[A] :

229

k

(7.6)

(A+,a)(A+V+I)...(A+'~+k-I) =C

+ ~ Cj A(A-I)...(A-j+I). o J 1

Les lemmes qui precedent montrent que u ~ K

si et seulement sl l'on a x~DBu 6 L 2

pour tout couple (~,8) tel que I~I +!81 < k. Proposition 7.1 : la norme

llul| ~/~k reste dens un rapport born~ (par une constante

ne d4pendant que de ~ et de k) avec la borne inf~rieure des expresssions (7.7)

(f (I+ iZilg)k lf(Z)l 2dZ)½

~tendue aux fonctions f m L2( R 2~) (7.8)

u = f f(Z) wz ~0 dZ.

Preuve : Si llul ~

(7.9)

v~rlfiant

< i, on peut ~crlre,d'apr~s (7.2) :

u = I (u, ~Z e0 ) wz vow dZ,

et l'on a alors

F

(1+

zfje)k i(u z 00D 2 dZ

1

R4ciproq;~ement, si l'on a (7.7) avec (7.10)

~ (l+IlZll2@)klf(Z)12dZ < I,

on peut alors ~crire, pour l~I~ k : (7.11)

(A~)*u~= ~ f(Z)(A~)* ~Z ~w dZ = ~ i ~

f Z8fff(Z) wz ¢0w dZ ,

d'o~

}l(A~)*u]i2g rr Io~lfiZ~12 If(Z)i 2 dZ d'apr~s le lemme 7.2. Les lemmes 7.4 et 7.5 permettent alors de conclure. On passe maintenant sur l'espece de phase, en eppliquant les r4sultets qui pr4c~dent au cas de l'oscillateur associ~ ~ ~ o w'. Proposition 7.2 ". pour

a 6 k~ o_w ,_

, la norme j[all o~,9~ k,~

born4 avecla borne inf~rieure des expressions (7.12)

( ~ ( I + [iZ~ 4 ilZ'l[2@,)k iK(Z,Z')I 2 dZ dZ') ½

~tendue eux fonctions K E L2( R 4~) v4rifiant l'identit4 (7.13)

a(X) = f l K(Z,Z') H(TZ~ , TZ,%,,X) dZ HZ' .

reste dens un rapport

230

Preuve : d'apr&s (1.22), on a

H(Tzq0~ , ~Z,~w,) = TZ+Z, 2

-1. '~

- H(9°W'~°~')'

(Y "Y ')

ce qui permet d'4erire (7.135 sous la forme

D ' a p r a s la p r o p o s i t i o n 7 . 1 , [lall~o~, ~ k~ borne i n f g r i e u r e ,

r e s t e donc dans un r a p p o r t borng avec la

pour l e s K v & r i f i a n t (7.13)~ de l ' e x p r e s s i o n

o7 ~Z 2 ½ (J; (i+ ll(S,c)ii )k iK(S +__~ , S - -~)I ms dC)

(7.15)

2

= (J~r I (1I +lls s +~ ~12 2 e + =

(1 +

lzll

+

- -f~r~ le,)

k

Uz'li e,)kI

K(S+

~--~ , S - g-~)l 2 dS d~)½

az z)½.

8. Espaces de rep~res Dans cette section nous allons consid~rer un ensemble Q et une application P ' ~ Pour tout ~ ~ f] on posera toujours, tacitement, pour simplifier~

on d~signera par LW ' %

R 2~ X p ~-

P(w) ~ (Y,@5 (et de m~me p(m,5=(Y,,e,55;

~ ~ o w'~ etc...,

ce qui, conform~ment aux

notations employees jusqu'~ maintenant,

devrait ~tre not~ LO(w) , Ap(~0), P(w)o p(w'5,

etc ..., et ceci bien que l'application

p ne soit pas n~cessairement

injective.

Quels que solent @ et @' E P~ , on d~signe par det@,@, le d~terminant de la forme 2 I[XIle,~, par rapport $ une base symplectique de R 2~ ; les nombres ~j

quadratique

d~finis dans la proposition 6.2 permettent d'~crire (8.15

det@,@, = 22~(U ~j5

-2

.

D~finition 8.1 : On appellera espace de rep&res la donn~e d'un espace ~ muni d'une mesure positive ~-finie d~, et d'une application mesurable les propri~tgs suivantes (i)

p : Q -~ R 2v X P

, v~rifi~t

•

il existe N > 0 (W,W')F->det~,6,

tel que la fonction (i +IIY-Y'!Ie,~,2 )-N

soit le noyau d'un opgrateur born4 sur L2(Q,dw). (ii)

il existe une application mesurab]e wl---)" a de ~ dans ~ R 2~) telle w que d'une part la norme iia il o~ ~ok (cf. section 75 soit born~e pour tout k, et que d'autre part on ait l'identitg soient

u

et

v ~ L2 ( R ~ ) ,

1 = ~ a (X5 d~

on a

(u,v) = ~ d~ (.! a (X) H(u,v,X) dX).

au sens faible suivant • quels que

231

Remarque : il convient~ pour justifier cette d~finition~ de prouver la convergence de l'int4grale

i; %(x5 .(u vx5

= I C°P½(a2 u vSi

D'apr~s la proposition 7.2, pour tout k = O~ I, 2,..., il existe C > 0 6 Q, une d4composition de a (8.2)

et, pour tout

de la forme

a~(X) = ~7 K(~,Z,Z'5 H(~ z ~0 , ~Z' ~ , X) dZ dZ'

avec (7~iK(m,Z,Z,512(l~ +I[Z[I~ + ]iZ,l]205k dZ dE') ½ < C.

(8.3)

D'apr~s (8.25 et (8.3)~ on a (8.4)

+llZ'II20)'kl(U,Tz,
l(OP½(%)U,V)}K C ( ~ ( l + ~ Z ~

et l'on est ramen~ ~ prouver la convergence, pour u 6 L2( R ~) et k assez grand, de l'int~grale I~ (l+iiZll~5-k/2 I(u,T~0w)I 2 dZ d~ . Ce r~sultat est consequence de la proposition plus pr4clse sulvante : Proposition 8.1 : Solt (Qjdg0,05 v4rifiant la condition (i5 de la d4finition 8.1. Soit

N

la borne inf6rieure des

o

N >_ ~ z

~N(W,= ') = (i +iiY-Y'II ,0,5

tels que la fonction

~ (~j j=l

(o5 les ~j = ~j(ege') ont 4t4 d~flnls dans la proposition 6.2) soit le noyau d'un op~rateur born4 sur

L2(Q).

Alors~ quel que solt

N' > N

o

j il existe C > 0

tel que~ quel que solt u 6 L2( R~Sp

on alt 2 -N ' 17 (i +IIZI105 l(U,¢z <0 51 2 dZ d~ ~ C ]lull2 . Remarque : d'apr~s(8.15~ on a -N

kN(~,~v5 ~ 2-~;/2 det~,0 ' (lq_iiy_y,},20,e,5 ee qui justifie la d~flnltion de N

o

+-~ 2

,

.

Preuve de la proposition 8.1 : C'est essentlellement la preuve du thdor~me 3.1 de [17], dans un cadre dlff~rent : on cholslt (8.5)

N 6 ]No,N' [ e t l'on pose

IIAlull2 = ~

(I+ ]IZ 12 S )'N ' }(U,~z,aw512 dZ d~

C ~i cN'-I (Kcu,u5 de, o

232

avec (8.6)

(Keu,u) = ~y e-2~¢

L'op~rateur

(8.7

II zII~ l(U,~z%)I 2 dE ~

K

a pour symbole

e

e

dZ d~ =

et il s'agit de prouver que la norme de C o m p t e - t e n u de ( 6 . 1 ) ,

(8.8)

K' c

on p e u t ,

N

K

(l+e -v

,I

exp- ~ + E

II

d~

est born4e quand g ~ O,

e

darts ce p r o b l ~ m e ,

remplacer

K

par

= ,~ e-2¢71~0 d~.

D'apr~s le lemme 2.1 de [17] (d~j~ utilis~ dans la preuve du lemme 2.2 du present papier), il suffit de prouver que la fonction

(8,9) est,

ks(~,~,) = N ii e-¢% e'e%'l i pour

0 < s N 1, l e n o y a u d ' u n o p g r a t e u r

b o r n ~ e n norrae s u r L2(Q) p a r u n e

constante ind~pendante de ¢. D'apr~s (6.7), on a (8.I0)

k¢(m,m') g C N ~ ( I

+ ~j th

e) -½ exp - 2(th ¢)i' Y'-

Yi120,e,

+ (th e) IIY' -YII,2@, 8, )-N

C sN~(I + ~j th s)-½(l

2

1

C ¢N{[[(1 + ~j th e)(l+ (th e)IIY'-YII6),e,)}-~(l+(th C (~j

+llY'-yJ129,@,) -½

(I

Nq e)~Y'-Yli2(9,@, )

2 +IIY'-Y~e,e,)-N+-i .

Ceci termine la preuve de la proposition 8.1. Th~or&me 8.1 : exlste

C >0

Soit (Q,d~,p) un espace de rep&res. Pour tout n = 0, I, 2,..., il telle que~ quel que soit

E

j

(8.11)

on ait

f I(u,~o~) I 2 d(jU < C llull2

De plusj pour n assez grand, il exlste

Pr euve :

u ~ L2(Rv),

I(u,£0c~)12 c~ ~

Quel que s o i t

~ j>_o

e >0

2

C >0

telle que Iron ait

C -1 IIuJi2

e t q u e l que s o i t

u E L2(Rv), -2¢A

e-~J~ ~

l(u, ~)I ~ ~ : 7 (~

H =J

= (K'u,u) ~ C ¢-N g

]lull2,

~u,u)

on a

233

oO l'on a utilis~ les notations employees dans la preuve de la proposition 8.1 : la premiere partie du th~or~me 8.1 en r4sulte. Par ailleurs, d'apr~s (8.4), on a (8.12)

Ilull2 N C liAlull2 ,

d'oh, d'apr~s (8.5) et la preuve de la proposition 8.1 : (8.13) pour

llUll2 ~ C ~i eN'-I(K u,u) d~ g 6 6 >0

assez petit ind~pendant de u.

Comme (Keu,u) est une fonction d~crolssante de e, on a aussl, pour

assez petit

-2~81iX-Yil~ (8.14)

llu!i2 ~ C ~ (OP½ (2 ~ e

)u,u) d~ .

La deuxi~me partle du th~or~me 8.1 se d~nontre alors exactement comme le th~or~me 4.1 de [16]. Th4or~me 8.2 : Soit (Q,dw,p) un espace de reputes. Ii existe un entler n > 0 et des fonctlons rateurs

k

8

K

(I~I< n, I~I ~ n) d~flnles sur

born4s sur

Q × Q, constltuant les noyaux d'op4-

L2(~), telles que l'op~rateur identlt~ de L2( R ~) pulsse

s ' ~crire

=

!j

I~1~ n 181~ n Preuve : remarquer d'abord que le membre de drolte est blen d4flnl en vertu du I er~ne 2.3. Posons

Qn =

.I t~((P.~,(P,?)

~

I~I <

tU

d~

:

uu

n

c'est, d'apr~s le th~or~me 8.1, un op4rateur born~ et Inverslble sur L2( R 2~) pour n assez grand. On 4crlt alors

B

avec

k

On I

Remarquons 4galement que d'une part (cf. len~ne 2.3) on peut ~crlre l'op4rateur associ~ ~ k ,~

sous la forme

et que d'autre part k

ML

avec (Mu)(w) = (u,~n I

) et (Lu)(~')=(u,

e0 ,),

v4rlfie l'hypoth~se du lemme 2.1 relatif ~ la composition

234

des noyaux. F. Bruyant [3] a donn~ des conditions pour que Q~I soit un op~rateur pseudodifferential ; des applications sont donn~es dans ce mSme travail, ainsi que dans celui de N. Lerner ~9]. Bien qu'il soit quelquefois techniquement n~cessaire de sortir de ce cadre, l'exempie de loin le plus important d'espaces de rep~res est constitu4 par un triple (R2~,dX,p), o~ dX est la mesure de Lebesgue sur l'espace de phase R2~ ~

R 29, et p :

R29 X P

a pour premiere coordonn~e l'identit4 ; autrement dit, un tel espace est d~fini par un champ YI--~II fly de normes symplectiques sur R2~ ; l'hypoth~se (ii) de la d~finition solent X,Y et Z ~

8.1 est v~rifi~e lorsqu'il existe C > 0

R 2~, l'in~galit~

IIY-XIIX ~ C "I

telle que, quels que

entra~ne IIZiiy ~ CIIZIIX , cormne on

le volt en utilisant la partition de l'unit~ d~finie par (3.27 dans [16]. Dans le cas o~ il existe une transformation canonique

K

telle qu'on ait l'identit~

llXlly =I~'(Y)XI, la structure quasi-complexe d~finie par (4.2) est int~grable, et l'on obtient une structure k~hl~rienne sur R 2~ : cette structure tr~s riche sera ~tudi~e ailleurs. Signalons enfin que si l'on pose (avec Y = (y,~), 0 ~ 8 < i) (8.15)

IIZ~ = (I +I~I) 28 Izl 2 + (i +I~I)-281~I 2 ,

ce qu'il est convenable de faire si l'on s'int~resse aux classes S la condition (it de la d~finition 8.1 est v~rifi~e d~s que Proposition 8.2 : Soit ( R2~,dY, o ) l'espace de champ

P : Y~-->II fly

soit

de HSrmander,

rep~res d~fini par la donn~e d'un

de normes symplectiques sur l'espace de phase. Pour tout M ~ R,

l'espace des fonctions a mesurables sur

EM

8,8

N > ~.

R 2~ telles que ~I (l+iYl)2M la(y)l2
On suppose la condition sulvante v~rifi~e : (iii) pour tout MI~ il existe k et M 2 tels que l'op~rateur de noyau K(Y,Y')=(I+IIY-Y~) "k op~re de

~2

dans

Alors~ quel que soit

EMI. MI ~

R, il existe

k >0

tel que l'int~grale

M1 ~,~ (l+llZily)-k (l+iYl) reste born~e lorsque Preuve :

u

l(U'~z ~f)l 2 dY dZ

pareourt une partie born~e de f ( R ~ ) .

cette int~grale s'~crit (Au, u), o~ le symbole a de A est donn~e par a(X) = 2~ ~ (l+ilZlty)-k(l+IYI)MI e -2n~X'Y'Z~2 dY dZ MI (l+llX_y ny)-k

L'hypoth~se (ili), tradulte en termes relatifs au transpos~ de l'op~rateur de noyau K, montre que cette int~grale est major~e par C(I+IXI) M2+2v61, ce qui prouve que

a

est temp~r~e et Ermine la preuve de la proposition.

235

On peut volr facilement que (iii) est v~rifi~e si d'une part la plus grande des 2 valeurs propres de la forme II fly est une fonction $ croissance lente de Y, et si d'autre part il ex£ste

M >O

et

C >O

tels que~ quels que soient X et Y ~

R2~

on ait l'in~gallt~ + liX-YIIx

~

C(l+~X-rIIr)M.

Proposition 8.3 : Soit (R2~,dY~o) l'espace de rep~res d~fini par la donn~e d'un champ " Y~ >II

fly de normes symplectiques sur l'espace de phase. Supposons que la plus

grande des valeurs propres de la forme II liy 2 solt une fonction g croissanee lente de Y. Soit

R > O.

Si u E ~ ' ( R ~) est telle que~ quel que soit M 1 > O , ~ al0rs

u E~

e

-2T~RIIZ 112

(l+iyl)

R ~) ; sl E c ~ (

MI

on ait

i(U,Vz ~f~2 dY dZ < oo,

R ~) est telle que, pour tout M 1 > 0 ,

cette int~grale

reste born~e lorsque u parcourt E~ alors E est une partie born~e de ~

R~).

Preuve ." il suffit de prouver le second point. D'apras le th~or~me 8.1, on a~ pour n assez grand :

ilug N

II u, 12 )~I ~n

D'apr~s ithypoth~se faltej les op~rateurs de cr~atlon et d'annihilation canoniques sont combinalsons lin~aires, $ coefficients moindres que C(I+IYI )M pour un certain M > I~ de l'Identit~ et des op~rateurs de creation et d'annihilation relatifs l'oscillateur associ~ ~ (Y,II ily). Ii en r~sulte que pour tout entier N >_ O, on a iluil~N ~ C ~ (I+IYI)2MN

~ l(u,~) i 2 dY. j~t~ n+N

Pour tout r < i~ on a done i}Ui~N ~ C 7 (I+IYI)2MN (OP½(e-2~r,IX-yIl2)u,u) dY. Enfln~ d'apr~s la formule (8.7)~ eette expression s'~crit aussi - 2 ~ r ~I Z~2

liu~ ~ c ~ e ~-r

(~+iYI)2MNi(U,,z~)I 2 ~ NZ.

Proposition 8.4 : supposons les hypotheses de la proposition 8.3 satisfaites, et solt F

une partle born~e de ~ ( R ~ ) .

II existe

7J' (l+iIZlly)-k(l+IYl)-M 1 est born~e lorsque Preuve : u ~(Rg),

solent

v C > O

M1 6

R

et

k >O

tels que l'int~grale

I(V, VZ ~y)} 2 dY dZ

pareourt F. et

N

un entier >_ 0 tels que 9 pour tout v E F et toute

on ait i(v,u)l ~ Cliu~ , norme de creation-annihilation canonique.

236

L'argument utilis4 dans la preuve de la proposition 8.3 permet d'4crire II~Z % I ~ N ~ C(I+I Zl +irl M)N ~ C(1+iiZliy)N(l+iY~ )M', -M 1 lj(l+llZlIy)'k(l+Iyl ) l(V,~z~)l 2 dY dZ ~ C

d'o~

~(l+ilZIIy)-k+2N(l+Yl)

-MI+2M~ dYda

9. Op~rateurs m4tadlff4rentiels Solent (fl,d~p) et (Q'~d~',p') deux espaces de rep~res ; soit dans la proposition 8.1~ et solt N'

o

N O le nombre d~flni

le nombre d~fini de la fa~on analogue relativement

au deuxiame espace de rep~res. D~finitlon 9.1 :

soit k > 2(No+N'o). On appellera op~rateur m~tadlff~rentiel d'ordre

0 et de classe ~

(relatif ~ (Q~Q')) tout op~rateur A pouvant s'~crire

A = ~7 0P½(a ,~,) d~ d~', o~ ( ~ ' ) I

>a,

est une application mesurable de Q × Q' $ valeurs dans ~ k ( R2~)~

telle que ~[a ,~,~o~0,,~ k soit le noyau d'un op~rateur born~ de L2(Q ')dans L2(Q). II convient de pr~clser le sens de l'op~rateur A~ ce qui est l'objet du th~or~me suivant : rappelons que les normes [iallo~, ~ k ont ~t~ d~finiesdans la section 7 ; on pose

k = 2(NI+ N'I) ~ avec

N1 >N o

et

N' 1 > N ' o.

Th4or~me 9.1 : supposons les hypotheses de la d~flnition 9.1 v~rifi~es. Ii existe alors

C >0

telle que, quelles que solent (Au,v) =

f,~

u

et

v E L2( R~)~ l'int~grale

(OP½(%,w,)u,v) d~ d~'

solt absolument convergente~ et de plus v~rifie

l(Au,v)l ~ C lluil 11v11. Preuve : d'apr~s la proposition 7.2~ on peut pour tout ( ~ ' )

~ Q × Q'~ trouver

une d~composltlon aw,w,(X) = I~ %,w'(Z'Z') H ( T Z % ,TZ,~0 ,,X) dZ dZ' avec

t l+nZ

2 dz

0,1a

on a alors

L[ l+llz, "NI l V, z% l

2

-N'

2

dz'

237

II suffit alors d'utiliser la proposition 8.1. Remarque 9.1 : symplectiques

consid~rons le cas oO (Q, dw) = ( R29, dY), et oO le champ de normes Y~-~I fly d~finissant l'espace v~rifle la proprietY, d4j~ mentionn4e,

que~ pour une certaine constante

C >0,

l'in~galit~ IIY-XIIX ~ C "I entra~ne

llZiIy K C)IZI~ ; alors les op~rateurs pseudo-dlff~rentiels d'ordre 0 d4finls dans ~16~ sont effectivement m4tadiff~rentiels au sens de la d~finition 9.1 : en effet, une partition de l'unit4 banale (celle d~finie par (3°2) dans [16])permet d'~crire l'op~rateur A sous la forme fonctlon remplacer ]-oo,c-l[

.[ OP½(%) d~, avec

11%11 ~ o ~ , ~ k ~ C ; pour obtenir une

a , ~ plutSt qu'une mesure concentr4e sur la diagonal% il suffit de ay(X)

par

et v~rifie

~ ay(X)f(llY-Y'~)dY'

Remarque 9°2 : si (~,d~) = ( R2v, dY) % ~ W , = ay~y,

, o~ f ~ ~ ( R )

est ~ support dans

I' f(IZi 2) dZ = I. et

(Q',¢~') = (R2~,dY'), et si de plus

est concentr~ en un certain sens, pros du graphe d'une certaine trans-

formation canonique~ alors l~op~rateur m~tadiff~rentiel A est proche d'une version g4n~ralis~e (il n'y a pas ici d'hypoth~se d'homog~n4it~) d'un op4rateur integral

de

Fourier. Lemme 9.1 :

quels que solent

N > ~, et N' < N~ il existe

que solent (Y,8) et (Y',8') E R 2~ X P ;; (l+',Z'l~)-N exp-~,' C(I+ ~ Y-Y Preuve : soit

ii

E

t '2

C >0

telle que~ quels

, on ait

2 -N dZ Y+Z-Y'-Z'I, 2•,@,(I+'[Z',,@,)

dZ'

- - N '+ 9

8,@,)

l'ensemble des couples (Z~Z') tels que

z-z,lle,e, > ½1iY-Y'~e,e'

"

L'int~grale sur le compl~nentalre de = 9 + N - N' > ~p l'Int4grale sur

E E

ne pose pas de dlffleult~ ; avec est major4e par

O[[E (l+iIZ~2@)'~(l+ItZ'll)-~(l+ilellg 2 2 ~-N'+~ dZ dZ' + llZ"le,, C(I+ Inf(i~Zll + IIZ'}} ,))-N'+~

C(l + g~Y-Yqe,~,.

E

(cf. prop. 5.1). Th4or~me 9.2 : Soient

(Q~dw~p), (Q',dw',p') et (Q",d~",p") trois espaces de rep~res;

solt N t tel que la fonetlon X(W~,W') = det~,@, (l+iiY'l-Y'

2

~½(-N'+~)

6~,8'"

soit le noyau d'un op~rateur born4 sur L2(~')(Cfo d~finition 8.1). Soit A un op~rateur m4tadiff~rentiel d'ordre 0 et de classe ~ kl

relativement ~ (Q~[~') et soit B

un op4rateur m~tadlff~rentiel d'ordre 0 et de classe ~ kl relativement ~ (~',~") ;

238 alors, sl

kl-k = N >max(N',v)

et si k v~rifie l'hypoth~se de la d4finitlon 9.1

relativement ~ (Q,Q"), AB est un op4rateur m~tadiff~rentiel d'ordre 0 et de classe ~

relativement A (~ ,Q").

Preuve : ~ l'aide de la proposition 7.2~ 4crivons A = ~ KI,w,~,(Z,Z')~(~Z~ ,TZ,~,)__

dZ dZ' dw d~'

et cObOl,, TZ., %,,) dZ; dZ" d ~

d~"

d'o~ AB = ; K , , ( Z , Z " ) ~ ( ~

Z %,

TZ,, loW,,) dZ dZ" d~ dw",

avec (9.1)

K~,W"(Z'Z") = 7 K~

,®,®

,(Z,Z')K . . . . (Z:,Z")( %,)dZ'dZ~ ~,w i,~ ~ %{%~'~z,

d~'

d~1

Posons

. .~o,,(z,~,,) = (,+lJzi,~ + t, z,,ll~,,) "/~ %..,,(z.z,,). kl

.,.o.o,(~..z,)

= <,+ Ilz,j~ + ,z,lf ~,) ~ ~,.~.o,(z.z,),

"~,7,~ ........ (z:,z,,)~

hi

: (~+, z~, ~; + uz,,,2e,,) 2 ~2,~;,~,,(z~,z,,),

ainsi que K(W,W") =IIMw,~,,II L2(d z dZ")' K I(~,~') = fiNl,w,~,il L2(d z dZ') et K2 (w;,~")=I~ 2 On peut

supposer que

L2(~) et que

K2

, ,,If 2 '~I 'w L (dZ; dZ") K1

est le noyau d'un op~rateur born~ de L2(Q ') dans

est le noyau d'un op4rateur born4 de L2(Q '')dans L2(Q'), et il

s'agit de montrer que

K

est le noyau d'un op4rateur born~ de

L2(~ '')dans L2(Q).

Pour estimer le prodult scalaire intervenant dans (9ol), on utilise la proposition 6.2 avec t = ~, ce qui donne

l(~z~%~, ~z %)I= Partant de (9.1)~ on obtient

1

det~, ~

!

!

2

, exp - ~IIY~ + Z~ - Y -Z llq,@, •

239 I det

,,

,

,@, exp- gilY 1

+

,-

ZI

1,8,

iMI,w,~,(Z,Z')I IM2,w~,~,,(Z~,Z")i HE'dZI d~' dw~

I (l+ilZ'Jl ~,)'N/2(I+IIZ'Iil 2 det

3-N/2

exp- ~IIY~ + Z~- Y'-Z'II2@~,@,

IMl,~,a~,(Z,Z') t IM2,~l,w,,(el,z") f dZ,dZ~ dw' d~l . A l'aide de iVin~galit~ de Schwarz

(seulement par rapport ~ dZ' dZ~) et du len~ne

9.1, on obtient

i~(zz)l , c I ~ ~

det~e<1+,y~,2~ e ~<"+~)

(~IMI,w,wv(Z'Z')I21M2,wl,~"(Z~'Z") 12 dZ' dZl) ½

(IIMI,w,~,(Z,Z')I 2 IM2,~I,~,..ZI, ( ' Z"~, 2 HZ' dZl) ½. A l'aide de l'in~galit$ de Minkowski intSgrale sur (L2(dZ dZ")), on obtient K(w,w") ~ C ~ X(~,~') du~' d¢l~. KI(~,~' ) K2(~,~"), soit enfin

K ~ C K 1 o % o K2

au sens de la composition des noyaux, ce qui termine la preuve du th$or~me 9.2. Th$or~me 9.3 • si A

est un op~rateur mStadiff~rentiel d~ordre 0 et de classe ~ k

relativement ~ (f~,~')~ A* est un opSrateur m~tadiffSrentiel d'ordre 0 et de classe ~ k relativement ~ (~',~). Preuve t c'est ~vident. Nous allons ~tudier pour termlner l'action sur ~( R v) d'op~rateurs mStadiffSrentlels d'ordre ~ventuellement non nul~ en nous limitant toutefois $ des espaces de rep~res dSfinis par des champs de normes symplectiques sur l'espace de phase. Proposition 9.1 • soient (R2~,dY,p) et ( R2~,dY'~p ') les espaces de rep~res d~finis par la donn~e de deux champs de normes symplectiques sur l'espace de phase ; on suppose que le premier v~rifie l'hypoth~se de la proposition 8.3, et que le second v~rifle

240

la propri~t~ (Ill) de la proposition 8.2. Solt (Y,Y')i

2~vay~y, une application mesurable de

~(R 2~)=n~k(R

R2V X R 2~) dans

>.

k On suppose les deux conditions suivantes remplles : a) il existe

C >0

et

M >0

tels que

ay,y, ne puisse 8tre non nul que sl l'on a

l+IYi~ c(1+!Y,l )M. b) il exlste

M 2 >0

tel que la fonction -M 2

(Y,Y')~---~(l+iY'l)

llay,y,Iy o y , , ~ k

soit, pour tout k, le noyau d'un op~rateur born~ sur L2( R2vt. Alors l'op~rateur

A =

7f Op½(ay,y,)

dY dY'

op:re de 9(R~)) Hans

Preuve : remarquons d'abord que les symboles pseudo-diff~rentlels

~( RUt. a d'ordre temp~r~

quelconque peuvent effectlvement s'~crlre sous la forme

a = .~~ ay,y, dY dY' de fa~on que les conditions at et b) solent remplies ; pour les op~rateurs m~tadlff~rentiels~ la condition at est une exigence raisonnable~ mals non vide. II s'agit, pour u C ~ ( R ~) et v ~ ~'(RV),

de donner une d~finitlon sesquili-

n~aire de (Au,vt telle que (Au,v) reste born~ quand u parcourt une partle born~e de ~

R~)

et

v

parcourt une partie born~e

F

de ~ ( R v ) .

my,y, = ,~ Ky,y,(Z,Z') H(T Z ~f,TZ, ~ , )

E

Posons

dZ dZ'

conform~nent ~ la proposition 7.2. On peut alors supposer que K(Y,Y')=(I+IY'I )

-M 2

(~fI~,y,(Z,Z')l 2(l+IIZIi2+llZ'IIy2,)k dZ dZ')½

est le noyau d'un op~rateur born~ sur L2( RVt(d~pendant de k, qui peut ~tre choisi arbitrairement grand). L'in~gallt~ de Schwarz fournit M2 I(Au,v)( ~ .~ (I+IY'I t K(Y,Y') dY dY' ,, 2. - k / 2

(f(l+ IIZ4,y)

I(v' ~Z ~f)I

2

dZ)½

(~ (l+i!Z'll2,)-k/2 l(U'~z ' ~f,)l 2 HZ,)½. Choisissons

MI

et k (on pourra augmenter ce dernier plus loin) eonform@~nent

la proposition 8.4. Ii suffit alors~ pour que (Au,v) reste borne, qu'il en solt ainsi de l'int~grale f~ (I+IY' I ) 2M2+MMI (l+IIZ'liy,)-k

(U,~z, ~f,)i 2 dY' dZ',

ce qui (M,M I e t M 2 ~tant connust est assur~ pour k assez grand par la proposition 8.2.

241

REFERENCES (I)

M.A. ANTONETS - The classical Physics 2 (1978), 241-245.

(2)

R. BEALS - A general calculus of pseudo-differential 42 (1975), 1-42.

(3)

F. BRUYANT - Estimations pour la composition d'un grand nombre d'op~rateurs pseudo-diff~rentiels et applications, Th~se de 3~me cycle, Univ. de Reims (1979).

(4)

A.P. CALDERON & R. VAILLANCOURT - A class of bounded pseudo-differential rators, Proc. Nat. Acad. Sci. USA 69 (1972), 1185-1187.

(5)

P. CARTIER - Quantum mechanical commutation relations Proc. Symp. Pure Math. IX, A.M.S. (1966).

(6)

A.

CORDOBA

& C.

FEFFERMAN

Comm. Part. Diff. Eq.3,

limit for Weyl quantization,

Letters

operators,

in Math.

Duke Math. J.

ope-

and theta functions,

- Wave

packets and Fourier integral operators, 1] (1978), 979-1006.

(7)

L. HORMANDER - The Weyl calculus of pseudo-differential (1979), 359-443.

operators,

(8)

J. LERAY - Analyse lagrangienne (1976-77).

au Coll~ge de France

(9)

N. LERNER - Sur les espaces de Sobolev g~n~raux associ~s aux classes r~centes d'op~rateurs pseudo-diff~rentiels, Th~se de 3~me cycle, Univ. de Reims (1979)

et m~canique,

S~minaire

(10) H. MAAS - Siegel's modular forms and Dirichlet 216 (1971), Springer-Verlag.

Series, Lecture Notes

(II) A. MELIN - Lower bounds for pseudo-differential 9, 1 ( 1 9 7 1 ) , 117-140.

operators,

(12) J. RALSTON - Gaussian beams and the propagation

of singularities

(13) M. REED & B. SIMON - Fourier Analysis, (14)

A.

UNTERBERGER

(15)

A.

UNTERBERGER

(16)

A.

A.

in Math.

Arkiv fSr Mathematik (preprint).

Ac. Press

operators and applications Lecture Notes, Aarhus (1976).

- Encore des classes de symboles, (1977-78), Ecole Polytechnique, Paris.

(1975).

: an

S~minaire Goulaouic-Schwartz

- Oscillateur harmonique et op~rateurs pseudo-diff~rentiels, de l'Institut Fourier 29, 3 (1979), 201-221.

UNTERBERGER

Annales (17)

Self-Adjointness,

- Pseudo-differential

introduction,

CPAM 23,3

- Sur la continuit~ sur L 2 des op~rateurs pseudo-diff~rentiels, S~minaire Kree sur les ~quations aux d~riv~es partielles en dimension infinie, 4~me annie, Institut Henri Poinear~ (1979).

UNTERBERGER

PART I I I GENERAL STRUCTURE OF GREEN FUNCTIONS AND COLLISION AMPLITUDES

ASYMPTOTIC BEHAVIOUR OF FEYNMAN INTEGRALS M.C. BERGERE

DPh-T, CEN Saclay, BP n°2, 91190 G i f - s ~ - Y v ~ e ,

France

I - INTRODUCTION The description of physical observables in various kinematic domains is one of the ultimate purpose of quantum field theory. Among these various domains, the asymptotic domains where some kinematic variables become large represent one of the best testing ground between theory and experiment (large energy-momentum behaviour of Green functions; scattering of two particles at large energy, large transfert and fixed angle; Regge phenomenology; deep inelastic scattering; etc...). The predictions of quantum field theory on this subject were for a long time rather small, although several attempts and various methods were constructed to achieve this goal ; let us for instance quote the original work of Gell-Mann and LowLljr~ on the renormalization group; let us also remind the enormous amount of work on the BetheSalpeter [2] kernel and its application to Regge theory. It is only in the 70's that a large step forward has been performed by the introduction of modern renormalization group techniques [3], differential equations [4], short distance and light-cone expansions

[5]

However, these techniques do not directly describe the asymptotic behaviour of Green functions and somewhere an approximation, according to what asymptotic behaviour We are looking for, has to be performed; such an approximation is obtained from a power counting theorem [6] which is proved to be valid in perturbation, on Feynman amplitudes, and the hypothesis is made that what is leading in perturbation remains leading on Green functions. In these lecture notes, we wish to describe how to obtain the asymptotic behaviour of Feynman amplitudes; this technique has been already applied in several cases

[7]

,

but the general solution for any kind of asymptotic behaviour has not yet been found. From the mathematical point of view, the problem to solve is close to the following problem : find the asymptotic expansion at large % of the integral f... f

{dx} e -%P{x}

where P{x} is a polynomial of several variables. Theoretically this problem is

(I.l)

243

solved [8] but first, Feynman amplitudes

are not as simple as the above integral,

second, we wish to compute the coefficients coefficients

of the expansion

and

so that we may sum these

over all Feynman amplitudes which describe a physical process. The first

part of these notes

(section II) introduces

the Feynman amplitudes

transform in regards to some asymptotic variable

and its Mellin

; in section III and IV we prove

the following theorem[9]: Given a Feynman amplitude IG{ q} attached to a Feynman graph G with external momenta N

qi

satisfying

(qi +q" +" 1 12

~ qi = 0 i=I

" "+qip

; we denote by {q} the set of all Lorentz

)2 with p < N. Let us now scale by a parameter

belong to a given subset

{q'} ; then IG[{Xq'},{q"}]

IG[{%q'},{q"} ] =

-co rmax(P) ~ ~P ~ p=f~ r=o

The above theorem is proved in Euclidean remains valid in Minkowski

invariants

% the invariants which

has the asymptotic behaviour

(Log%) r gpr{q}

(1.2)

space for any asymptotic behaviour

and it

space for a subclass of asymptotic behaviours.

Finally in section V we show how to compute the coefficients

gpr{q} in the simpler

case.

II - FEYNMAN AMPLITUDE AND MELLIN TRANSFORM For sake of simplicity, tudes, ultraviolet

we shall only introduce here Euclidean

and infrared convergent.

kind of Feynman amplitudes

encountered

scalar Feynman ampli-

What follows may be extended

to other

in quantum field theory.

Given a Feynman graph G with ~ internal lines, we define a Feynman amplitude

in

dimension D as -

I({q}) =

co S o

~ ~ d~ a=! a

e

~

a=l

~

m a

2

v(~,{q})

a

P

(~)

e p(~)D/2

(II.

where one variable ~ is attached to each inter.nal line and where the polynomials P(~) and V((~,{q}) are defined as follows

Defi~on

:

." - A one-tree T| is a connected

tree graph

(no loop) which contains

all the vertices of the graph G. - A two-tree T 2 is a tree graph with two connected components,

and

which contains all the vertices of the graph G. A two-tree is obtained from a one-tree by omiting one line. A two-tree splits the set of external momentum

{q|,...qr } into two sets {qi ''''qi } and {qi 1

s

"''qi } s+l

r

1)

244

such that

s

2

~

r

2

(II.2)

by energy-momentum

conservation.

Then, = ~ ~ ~ T1 aETI a

PG (~)

V(~,{q})

The notation

,

= T2~ f~aET 2H

~a(Zq)$2 }

(ll.3b)

{q} stands for the set of all possible squares

T 2 define the same square square can be performed.

(ll.3a)

(Zq) 2. Several two-trees

(Zq) 2 and partial summation of the two-trees

for a given

We define a cut c as a set of lines such that if we cut

these lines the graph G is split into two connected parts R and L and such that no subset of these lines have the same property. A cut defines a square V(~,{q})

(Zq)~. Then

may also be expressed as

c The polynomials when some ~'

ace

P(~) and V(~,{q})

are positive when all ~'

vanish (see l a t e r ) .

s

are positive and vanishes

The r a t i o V ( a , { q } ) / P i s r e g u l a r f o r a '

s

> O. s

Suppose now that some (not necessary all) invariants

-

are scaled to infinity

({q} + %{q}), then V becomes V ÷ V'+ % W and we study the large % behaviour

(11.5)

of I(%). We introduce the Mellin transform

oo M(x) : f o

d% %-x-I 1(%)

which is found to be _

oo £ M(x) = F ( - x ) f TI d~ e o a=l a for ~ < Rex < 0

~

a= 1

~

m 2

V'

a a

fW~ x e

P (II.6)

~P-J pD/2

where ~ is the largest singularity

in x of the integrals ~. We now

study these integrals. Definition of Hepp's

sectors

We define a sector S as an ordering of the ~' 0 < ~ -

< ~ a|

-

s

: (11.7)

< ... < ~ a2

-

-

a£

245

There a r e ~

sectors and the union over all sectors is the integration domain in &' s

A sector defines a set of nested subgraphs R i = {al,a2,...a i} , R i ~ Ri_l, R%= G. In each sector, we perform the change of variables

a.i

=

71 B. B i j=i+l J

(II.8) 0 ! Bi< £ < 1

so that,

the Jacobian

% % [ ~(Ri)-I ] II d~ ÷ 71 [B i dB ij a=l a i=i

PG(~) ÷

with

(%(R i) =i)

(II.9)

L(R i) ~ B. [l + O(B)] i=l l

(II. |0)

where L(Ri) are the number of independent loops of the subgraphs Ri, and Q(B) is a non-negative polynomial.

De~£n~on

: A function f(&a ) is said to be FINE if in each sector l

f(&a ) ÷

~

i=l

Bi

(II.||)

g(B i)

where g(Bi) has a simultaneous Taylor expansion in all B's around the point Bl= B2=... = B% = O. In general W(~,{q}) is NON-FINE and transforms into a sum of FINE functions. Nevertheless, many asymptotic behaviours have the property to possess a FINE W(&,{q}) function and we shall first study this simpler case.

III-

THE CASE W(&,{q}) FINE

The function W(~,{q}) satisfies the property that under the change' of variables (11.8), we get : W(~,{q}) ÷

£ ~. H Bi I W'(Bi,{q}) i=l

~ ~£ = L(G)+I ,

[

, (III.l)

Vi > L(Ri)

where W' has a simultaneous Taylor expansion in all ~'

around the point BI=B2 = ... = S

B~ = O, and where v. are non-negative integers. If we decompose M(x) over the sectors, l we obtain after integration over B~ , w(R i) I ~-I 2 + [~i-L(Ri )]x-I Ms(X) = F(-x) F(x- Gw~) - ) S ~ lab i B i 1 h(Bi,x,{q},m ~) o i=l J (III.2)

246

where, from the property of ultraviolet convergence, (III.3)

~(R i) = L(Ri)D - 2%(R i) < 0

(~(Ri) is called the superficial degree of convergence of Ri) , and where h(~i,x) is analytic in x and has a simultaneous Taylor expansion in all ~'s around B I =

~-i

B2 = ...

= 0.

The structure of the B-integrals and the Euler functions r show that Ms(x) is a meromorphic function. The poles of Ms(x) are generated from the snbgraphs R° such l that ~i > L(Ri)" Such subgraphs are called essential subgraphs. The essential subgraphs are those which determine the asymptotic behaviour of 1(%). The poles of M(x), apart from those at x = 0,1,2,... which comes from r(-x), are found at negative rational values of x (if the space-time dimension D is rational) S ~(S) - 2k Xk = 2 ~ S -L(S)]

for

{ k = 0,1,2,... S essential

(111.4)

where ~S is given by vS

W(P~aES,~a~s,{q})

~ p p÷o

[W°(~a,{q})+0(p)]

(III.5)

The leading pole is found to be

= sups

L~2[~s-L(S)]J

(111.6)

essential and we call leading subgraphs, those essential subgraphs such that xS = ~. o M(x) The function r--~-~

s can be written as an infinite sum of multiple poles at x = Xk

and from the inverse Mellin transform 1

1(%) = 2i~

o+i ~ S a-i~

d(Imx) Ix M(x)

~ < ~ < 0

we may push the contour o towards the left (the convergence at Ilmxl ÷ ~

(III.7)

is ensured

by the function r(-x)) and we obtain, by Cauchy's theorem, the asymptotic behaviour

IG(1) =

_~ ~ IP p=~

rmax(P) ~ r=o

(Log%) r gpr({q})

(III.8)

S where p runs over the rational values x k and where r, for a given p, runs over a finite number of non-negative integers.

247

IV - THE CASE W(~,(q}) NON FINE The form obtained for the asymptotic behaviour and given at the end of section III remains valid when W(~,{q})

is NON FINE. We now explain how to obtain this result.

The function W(~,{q}) can be decomposed in many ways as N W(~,{q})

=

Wj (a,{q})

j!l

(iV.l)

where each function W.(~,{q}) is FINE. Then we may transform a sum into a product by J using a representation of the type o+i~ f dlmz F(-z)F(z-x)AZB x-z

I

F(-x)(A+B)X = 2i~

(IV.2)

with Rex < ~ < O, ReA > 0, ReB > O. This leads directly to the introduction of a multiple Mellin representation.

From

W. _ _!

(~.+ioo

P

J

1

e

2i~

f ~.-i~ J we obtain the multiple Mellin representation as N

oo

F[ M(xj) = J=! F(-xj) of

9~ 1I a=|

x~

/Wj~ j dlmx r(-xj) k~-] J -

-~ do~

a

e

~am2a a=l

N

'

~. < 0 (IV.3) J

V' (~j)ix. pD/Z e- -~-

y[ j=!

--=,-=

(IV.4)

The above ~-integrand is FINE and the technique of section III may be applied. For each sector, we find that oM°(xj) is analytic provided o~j = Rex.j satisfies the inequalities ~. < 0 J

for all j

, (IV.5)

N

1

n..O.j=! ij j

~(Ri~_. > 0

for all i

where nij = ~ij- L(Ri) > 0 and

Wj(O~aERi,~a~Ri,{q})

N O lj [W~(~a,{q})+0(0) ] 0+o

(IV.6)

The above conditions over ~. define a closed convex polyhedron PS for each sector. J This polyhedron is not empty because the amplitude is ultraviolet convergent. The inverse multiple Mellin transform defines Oj IS(X) =

+i~

S ~.-i~ J

dlmx. J 2i~

N

~ X

J=!

x. J

M^(x.) ~ J

(IV.7)

248 with o~j E Peo " Because n PS ¢ ~ ' it is also possible to define an inverse multiple S Mellin transform for I(X). The behaviour of IS(X) at large

_ xj

X is given by the point (or points) such that

is minimum in PS ; IS(X)

~ X~

aS

X

up to powers of L o g X

,

(IV.8)

where (IV.9) PS If

a = sup a S , I(%) N %a up to powers of L o g % . S X-~o (it must be noted that a < Inf (~ Xj) ). np s

S of Ms(X j) are on the v a r i e t i e s

The s i n g u l a r i t i e s (7.

=

k.

J

3

N

j=l

(IV. lo) 1

nij(7j- ~ ~(Ri) = -k.

1

where k i or k.j are non-negative integers. The functions Ms(x j ) can be written (up to F(-xj) functions) as an infinite sum of terms of the type

R({q)) %

[

i=l

_

N ~(Ri) ] ~ nijx----~---+ k i j" = I J

(IV.l|)

We may now use the inverse multiple Mellin transform, and push the contour along a /

N

\

path where

\ _ ( ~ oj)/ is monotonically decreasing ; if we use Cauchy's theorem as j-I many times as needed we obtain for IS(1) and consequently for I(%) an asymptotic expansion of the type described before in (111.8).

Although, there are many ways of splitting W(~,{q}) into FINE functions W.(~,{q}), J two ways seem to be topologically interesting : a) split according to the cuts c. which define the large invariants ; then J W.(~,{q}) = ~ J aEc. J

~

PR(~)FL(~)(Eq)~ a

(IV.12) j

In this case, the coefficients n.. corresponding to a graph R. and a cut c. are equal lj l 3 to the difference between the number of connected parts when Rii is cut by c.j and the

249

original number of connected parts of R. . 1

b) split according to all two-trees ; we decompose the function V according to

(II.3b), Vj(~,{q}) =

2

~I ~a (Eq) , a£T~ TJ

(IV.13)

In this case we may write N

- ~ xj-D/2 [p(e)] j=l

Ok+i°° dlmy k N M f 2i~ £(_yk) [pk(~)]Yk 6[j!iXj+k~lYk+ D]

1

=

(IV. 14) where we split P(~) over its one-tree components Pk(~) = ble to integrate over all ~'

~ ~ . a~T~ a

It is then possi-

and we obtain the complete Mellin representation of s

Feynman amplitudes as obtained by C.de Calan and A.P.C.Malbouisson [lO] N

dimxJ l({q}) =

F(-xj) 2i-----~

f o.-i~j

f

~

(m~

IYk+D/2 ) kk~l_ Yk

]

[(Eq)2j] x.

{ M

2i~ ~k-i ~

x

M

dimyj F(-yj) [J lxj+k r

)-~aF(¢a)

J

T2

(IV.15)

a=l where the formS~a are linear N

~a =

~

M

najXj +

.=

j 1

~

makYk ÷ 1

(IV.16)

k=l

and the coefficients n . ,

m , are 0 or 1 depending whether the line a belongs to the • aj k am two-tree T J or one-tree T 1 , or not. The point {oj ,ok} must be chosen on the intersection of the variety M n

M

D

]

with the closed convex polyhedron P

a. < 0 J ~a > 0

,

o~ < 0 (IV. 17)

and any asymptotic behaviour in {q} may be obtained by minimizing in M n P a given linear form of the x'.s. J Let us end this section by mentioning that the asymptotic behaviour obtained (in 111.8) for I(X) is valid for amplitudes with spinor and coupling derivatives, for renormalized amplitudes and also for infrared (zero mass) asymptotic behaviour. If

250

we try to extend the theorem to Minkowski rotation transform the functions Feynman amplitude) functions

space, two cases appear

gpr{q} into distributions

: either the Wick

(as it does for the

and the asymptotic behaviour remains valid,

gpr become infinite because the Landau singularities

in this sense, or the perturb deeply the

asymptotic behaviour.

V - THE COEFFICIENTS

gpr{q} IN THE CASE W(~,{q})

A way to obtain the coefficients

FINE

gpr{q} is to compute the residues of the poles in

each Hepp's sector. The problem in doing so is, first, that it is very tedious to calculate over

~: sectors,

structure of the subgraphs analytic

form of gpr{q}

second,

that it completely destroys

and reduced graphs,

the topological

third, that it does not lead to an

which may be easily summed over all graphs contributing

to

the Green function. We present here an alternative method which can be applied, e-integrals

We first give a one-dimensional

Pg~i~On

step by step, on the

in its general form and which avoids the three above disadvantages. example.

: We introduce the Taylor operators acting over a function f(x) infinitely

differentiable

at x = 0 ,

as xP

T n f(x)

f(P)(0)

for n >

0

p=o T n f(x) x

=

0

We now extend this definition infinitely differentiable

n

x

f (x)

(v. I )

for n < 0 to functions

at X = 0

; then, we define the operator T n x

= x ~) T n-E'(~))

where E'(~) is the smallest integer

(I-T) and consequently, It is important

if

f(x) such that x -~ f(x)

(~ complex) as

{x -~ f(x)}

(V.2)

> Re~ . One property of this operator

~ f(x) X÷O

X

n+~

,

f dx f(x) is not integrable, o

e > 0

is that

(V.3)

S dx (l-Tx I) f(x) is integrable o

to note that the number of subtractions

change by one whenever v increases and becomes

is

is dependent with ~ and

integer. As an application we may

obtain analytic continuation of functions defined from an integral. Example

: F(x) = S d% (l-T~ I) {%x-I e-l) o

is defined for all x # -k, k = 0,1,2, . . . .

(V.4)

251

We now generalize to several variables. Given a function f(~) which is FINE, and given a subgraph S , we define

[

~ f(~) =

n

]

f(~)l

0

l~a+P~aVa6 S

(v.5)

Jo= 1

Then, if (l-T n) f(e) = g(~)

(v.6) g[P~a6S'~agS ]

p~o

Pn+g

c>O

'

We define the operator

R

=

(1-To£(S))-

]I

(v.7)

0

Sc:G

where i(S) is the number of lines of the subgraph

S.

Although the operators TS do not commute when relative to overlapping subgraphs, in Rf(e), the product which defines R may be taken in any order if f(~) is FINE.

Theorem 1 [11] Given the meromorphic function i M(x) = F(-x) f

H

o a=l

~ d~

a

e

~ m2

a=;

a a

V' W x

(g)

P

~

(v.8)

where W(<~) is FINE , the above integral is convergent for Rex > ~ and M(x) has multiple poles at x = x~ for k = 0 , | , 2 . . ,

and S essential. The s-integrals can be

analytically continued in x for Rex < ~ and away from the poles. We obtain the absolutely convergent integral representation ~

-

~ 2 ~ama

a=! M(x) = F ( - x ) f o

][ d<~ R a= | a

e

V'

w x e- -~(~)

~

f

(V.9)

Again, the number of subtractions performed by R depends of the band (between two consecutive poles) where stands the point x in the complex x plane.

Theorem 2 [|2] : In the function _

oo ~ M(x) = F ( - x ) S ]l d~ e o a=l a

~ ~m 2 a=l a a

{W x R (~) e ~

_V-P

1 f

(V.10)

the s-integrals are absolutely convergent Vx. The function M(x) is not continuous on the right of the lines x~ + iy (y from -~ to +~).

252

To compute the coefficients

gpr we use these theorems

in the following manners

:

we are looking for a multiple pole of M(x) at x = p. On each side of the line Rex = p, we define the bands B_ and B+ which separate coefficient

the pole x = p

from its neighbours.

The

gpr may be obtained by taking a Cauchy contour around Rex= p taking into

account the definition of M(x) in B

and in B --

B_

as given in V.9. Another way of cal+

culating

is to define the function M (x) which is equal to M(x) in the band B gpr and which is analytic for all x away from the poles which are at the left of B

Then,

all the meromorphic

structure at Rex = p.

In both cases, we are led to compute the action of TS operators

the difference

[M(x)-MB-(x)]

over the ~-integrand.

In doing so, factorization

contains

properties

occur between a function attached to the sub-

graph S and a function attached to the reduced graph [G/S] are found to be a sum of expressions, overlapping

subgraphs)

of essential

zation of coefficients graphs

each being attached

subgraphs.

; the coefficients

gpr

to a forest F (set of non-

For each forest, we obtain a factori-

(defined by their integral representation)

attached to reduced

[S] F where S E F and where [S] F means the graph obtained from S when all sub-

graphs inside S and which belong to F are shrinked into points. It is this quasi-geometric

structure of the coefficients

gpr which explains why, in

some cases, the asymptotic behaviour of IG(%) for a given graph G may be summed over all graphs of a given Green function.

Let us mention that we have applied efficiently scaling [7] and the deep-inelastic

behaviour

integrand of the Mellin transform is FINE adapted to renormalized viour

(planar graphs)

to determine

the

(and where of course, we use a formalism We also determined

the Regge pole beha-

of scalar field theories [7] where the ~-integrand

but where nevertheless, coefficients

Feynman amplitude).

this technique

of scalar field theories where the ~-

the operator R has been shown to determine

is not FINE

the leading

g~r Vr.

On the other hand, the construction the coefficients

of a generalized

R operator which could determine

gpr in the case NON FINE is definitely needed,

wish to handle the infrared behaviour of quantum electro-(chromo)

for instance, dynamics.

if we

253

REFERENCES [I]

A.Peterman, E.C.G.St~ckelberg,

Helv.Phys.Acta, 26, 499 (1953).

M.Gell-Mann, F.E.Low, Phys.Rev. 95, 1300 (1954). [2]

N.Nakanishi, Suppl.Prog.Theor.Phys.

[3]

K.G.Wilson, Phys.Rev. B.4, 3174 (1971) ; 3184 (1971). K.G.Wilson, J.B.Kogut, Phys.Reports

[4]

43, I (1969).

12C, 75 (1974).

C.G.Callan, Phys.Rev. D.2, 1541 (1970). K.Symanzik, Comm.Math.Phys. 18, 227 (1970) ; 23, 49 (1971).

[5]

K.G.Wilson, W.Zimmermann, Comm.Math.Phys. 24, 87 (1972). W.Zimmermann, Ann.Physics 77 (1973) 570. S.A.Anikin and O.I.Zavialov, Ann.Physics

[6]

116, 135 (1978).

S.Weinberg, Phys.Rev. 118, 838 (1960). P.Cvitanovic, T.Kinoshita, Phys.Rev. D.10, 3978, 3991 (1974).

[7]

M.C.Berg~re, Y.M.P.Lam, Comm.Math.Phys. 39, | (1974) ; "Asymptotic expansion of Feynman amplitudes , part.II. The divergent case". Freie Universit~t Berlin HEP (May 74/9). M.C.Berg~re, C.Gilain, J.Math.Phys. 19, 1495 (1978). / M.C.Berg~re, C.de Calan, "Regge pole behaviour from perturbative scalar field theories" DPh-T preprint n°79/7, to be published in Phys.Rev.

[8]

A.N.Vartchenko, Functional analysis and its applications IO, 13 (1976). A.G.Kouchnirenko,

Inventiones Mathematicae 32, ] (1976).

[9]

M.C.Berg~re, C.de Calan, A.P.C.Malbouisson,

[10]

A.P.C.Malbouisson,

Comm.Math.Phys. 62, 137 (1978).

"Comportement Asymptotique des Amplitudes de Feynman",

Doctoral Thesis, Paris VI (1978). C.de Calan, A.P.C.Malbouisson,

"Complete Mellin representation and asymptotic

behaviours of Feynman amplitudes", Ecole Polytechnique preprint A.318 (Sept. 1979). [11]

M.C.Berg~re, F.David, J.Math.Phys. 20, 1244 (1979).

[12]

M.C.Berg~re, Y.M.P.Lam, J.Math.Phys. 17, 1546 (1976).

INTEGRAL RELATIONS IN COMPLEX SPACE AND THE CLOBAL ANALYTIC AND MONODROMIC STRUCTURE OF GREEN'S FUNCTIONS IN QUANTUM FIELD THEORY : SOME GENERAL IDEAS AND RECENT RESULTS

J. BROS

DPh-T, CEN Saclay, BP 2, 91190 Gif-sur-Vv~%te, France

In this lecture, we wish to present some of the ideas of a global consistent approach to the analytic and monodromic structure of Green's functions and scattering amplitudes of elementary particles on the basis of general quantum field theory. This approach, which originates in Symanzik's pioneer program [ I] called "Many-Particle Structure Analysis" (in brief M.P.S.A.) has already been presented by us in its complex momentum space formulation at several places [2][3] It integrates in an organized scheme concepts which pertain to three kinds of standard investigations concerning analyticity in particle physics namely : (i) Analyticity properties of Green's functions H (n) and scattering amplitudes* on the basis of the axioms of local fields [4][5] (ii) Analytic structure of scattering amplitudes in axiomatic S-matrix theory [

6

(iii) Analytic structure of the Feynman amplitudes in the perturbative treatment of Ouantum Field Theory[ 7 ][ 8 ][ 9 ][101 The foundations of the M.P.S.A. approach are the axior:s used in (i), supplemented by an extra-axiom called "asymptotic completeness" (A.C):the latterisactually equivalent to the property of "unitarity of the S-matrix" which is one Of the basic ingredients of (ii). The method which we use in M.P.S.A. is based on the study of certain integral relations in complex momentum space which have similarities and closed links with the Feynman integrals ; the technique which we apply to studyin~ the generation of complex singularities through the integration process and the results which we obtain are comparable to those of a standard approach of (iii) [8

]

From the framework of (i), the M.P.S.A. approach keeps the benefit of welldefined global analyticity domains for the Green functions H(n)(~ I .... ,kn) in C 4(n-l) = {k=(k I .... kn) ; kl+ .°. +k = 0} ; these domains - called "primitive n ~ analyticity domains" and denoted by Dn--Can be enlarged by holo-convex completion (i.e. by computing their holomorphy envelopes or parts of the latter)

, but they

Let us recall that the scattering amplitudes for the nr+n-m particle processes are appropriate real boundary values of the restriction of Green's function H(n)to a certain complex manifold M(c) called the "complex mass shell". n For example, for n=4, this yields the domains of "dispersion relations" for the two-particle scattering amplitudes of certain reactions.

]

255

obey strong limitations : they are "schlicht" somains in C 4(n-l) which are bounded by certain "cuts" accross which it would be impossible to expect analytic continuation for the H (n), were the A.C. axiom not postulated. Exploitation of the A.C. axiom allows one to investigate the monodromic structure of Green's functions around well defined singular sets, called "Landau varieties". Investigations of this type are in some sense analogous to those which can be carried out in the S-matrix framework (ii) by exploiting the unitarity integral relations. In (ii) however, no primitive ~lobal analytic structure based on physical princiDles has been proposed up to now (except the one which was based "a priori" on requirements of "maximal analyticity ''[I| ] ) so that the monodromic structure of the scattering amplitudes can only be studied there in the framework of microlocal analysis, the latter having being justified on physical grounds by the macrocausality assumptions and through the method of local Fourier transformation[6r

]

The integral relations of M.P.S.A. through which the A.C. axiom can be efficiently exploited are exact counterparts of certain identities which relate aDpropriate formal series of Feynman amplitudes

in the framework of perturbation theory.

The various terms of these integral relations can be represented by graphs which obey similar integration prescriptions to those of Feynman amplitudes, but whose vertices

are associated with "general n-point functions" instead of constants or

monomials. The "general n-point functions" which occur in these relations are not only Green's functions H (n) themselves, but other quantities which are called "~-particle irreducible n-point functions" (~=|,2,..., ~) and which have an intuitive interpretation in perturbation theory [ ! ]. These functions, whose definition involve a recursive algorithm starting from the lowest values of the integer ~, enjoy the following important property : for increasin~ values of ~ they are analytic in increasingly larger domains of ¢4(n-1) which all contain the primitive domain of H (n) ; in the axiomatic M.P.S.A. approach which we consider, these analyticity properties of the k-particle irreducible functions are expected to be obtained as consequences of the A.C. axiom which supplement the primitive analyticity properties entailed by (i) . At the present stage of our program these properties have only been established for ~=1,2,3 but some steps have also been taken in the treatment of the general case. From considering the integral relations of M.P.S.A., it emerges that the global analytic and monodromic structure of Green's functions should be completely governed by the two following operations :

Here, it is interesting to notice that in some tractable models of Quantum Fields which have been studied in the constructivist approach (P(~)2 at small coupling), the required analyticity proDerties of %-particle irreducible functions can be checked directly [12][13].

256

(I) Integration of analytic functions over prescribed classes of cycles in definite complex domains. (II) Fredholm inversion of analytic kernels depending analytically on parameters over prescribed classes of cycles in definite complex domains. In the recursive process of M.P.S.A., one can indeed expect that to each Green function H (n) and integer ~ , there

corresponds a definite domain D (k) in comDlex n four-momentum space C 4(n-I) in which the above operations (I) and (II) should determine exhaustively the Landau singularities and the associated monodromic structure of H (n) implied by the completeness of the 1,2,...,k-1, k-particle asymptotic states of the theory (in brief : the "],2,...,k-I, k-particle structure"). The domains D (k) should be determined by the primitive analyticity domains of the corresn

ponding k-particle irreducible n-point functions,

; their size, governed by the pro-

duction threshold of (k+1)-particle states should increase with %, and to the extent that in the limit k ÷ ~, D (k) fills up the whole complex four-momentum space t 4(n-I), n the M.P.S.A. program may have - in principle - the ambition of characterizin~ at a global level in ~4(n-I) the singularities and the organization of the complete monodromic structure of Green's functions. At the present stage, the obtained results

(for k=|,2,3) are

indications

in favour of this wishful description. Before coming to a brief account of some results which have been recently obtained for the two and three-particle structure of the six-point Green function H (6) and of the three,particle scattering amplitude, we shall make a few comments on the way in which the mathematical operations (I) and (II) are acting in the M.P. S,A. approach. (I) Integration in complex space : t=(t],...,tq) C cq, z=(z l,...,z n) E cn .

I(t)

=

fFtf(t,z) dz I A ....A dz n

,

(I)

f is analytic in a domain D = U (t,D), and F c D is an n-dimensional cycle ~. t t t • t.~, which has to "vary contlnuously,, wlth t(see [8 ]). Atypical

example which illustrates the relevance of this operation to M.P.S.Ao

is the "triangle term" which contributes to the r.h.s, of the structural equation (9) k6 (given b e l o w ) f o r H (6)

namely : '

k5

k| ~

i

~ k4

~ = ~

~ i ~ - - k 2 =

k3

in the latter the bubble vertices represent the Green function * H(4) , and the associated contribution to the six-point function is the integral : * More precisely, a suitably modified form of H(4)(klk2kBk4)2whi~h i~ regularized at infinity and "amputated" from its two initial poles k I =m ,k2 = m 2 (see [ 2],[3]).

257

H(6)=~D= = (k I ..... k6)= S H(4)(kyk6,kFk-k5-k6)H(4)(kl~2,-k,k-kl-k2)x

...

~kl...k 6) • .. x H(4)(k3,k4,k5+k6+k,kl+k2-k) F(kl...k6 ) being a well-defined class of four-dimensional At the preliminary step of

(2)

d4k cycles.

M.P.S.A. - at which H (4) is only known to be ana-

lytic in its primitive domain D 4 - the integral

(2) can be shown to define a func-

tion which is analytic in the primitive domain D 6 : this is in fact the content of the "G -convolution theorem" proved in [14], in the case when G is the triangle graph, here considered. At the step %=2 (namely, the exploitatlon of the two-particle

structure) H (4)

can be shown to be analytic in a larger domain which is a two-sheeted Riemann surface with branching singularity s=(k|+k2)2=(2m) 2 (m being a fundamental mass parameter in the theory). Correspondingly,

the integral (2) can be shown to have an analytic con-

tinuation on a certain Riemann surface which contains the so-called "three-particle elastic scattering region on the mass shell" (defined by k 2i = m 2 , i=I,...,6 ; (3m) 2 < (kl+k2+k3)2 < 4m2). This Riemann surface has common features with that of ~ J the Feynman amplitude corresponding to the Feynman graph ~ _ ~ ' ~ , but it is confined over a certain definite domain : this restriction expresses the fact that On£y the two-particle

structure of the theory has been taken into account at

this stage. As in the perturbative case, the Landau singularities of H (6) are produced[ 1[ by]the pinches of the integration cycle F(k]...k6 ) by certain "vanishing cells ''LSJ~15][17 6f the singular set S of the integrand; here S is the union of the three polar manifolds manifold

k2=m2, (kl+k2-k)2=m 2, (k5+k6+k) 2

= m 2 , and of the branching

(k]+k2+k3-k)2=(2m) 2 (notice that the latter is absent for the corresponding

Feynman integrand). (II) Fredholm inversion in complex space or "Bethe-Salpeter One is interested in the analyticity properties

inversion"

:

(in(t,z,z',%~of theFredholm resol-

vent F of a given analytic kernel G, satisfying the equation : V~E

C, F(t,z,z',X)

= G(t,z,z')+% fF(t,z,z",~)G(t,z",z') dz~ ^...^dZ"n ; (3) Ft G is analytic in a domain D = U (t,DtxD~) and F t c D t fl D't is an n-dimensional cycle tE~ which has to vary continuously with t (see [16 ] for a systematic study of the analytic and monodromic structure of F, under suitable simple assumptions). A typical example which illustrates the relevance of this operation to M.P.S.A. is the Bethe-Salpeter two-particle

equation which expresses H (4) as the Fredholm resolvent of the

irreducible four-point function L (4) •

H(4)(K,z,z ') = L(4)(K,z,z')+ ~I fF k H(4) (K,z,z") L(4)(K,z",z

)dz"

(4)

258

or graphically : k =

+

k4

k 2

2

K=k3+k 4 =-(k|+k2) , z =

w h e r e

--

-

k2-k ! 2

,

z v =

k3-k 4 2

k2

~k~

k4

kc~-k ~ '

z"=

2

'

a~d FK is defined as

the "euclidean subspace ''~ ( i ~ ) × ~ 3 . Equation (4) is actually the starting point of the step %=2 ** of M.P. S .~2],since it accounts for the generation of the two-sheeted Riemann surface of H (4) (quoted in the previous paragraph), once the primitive "schlicht" analyticity domain of L (4) is given. The latter includes the two-particle elastic scattering region on the < 2 . . (4m)), and the branchlng slngu-

mass shell (k~ = m 2, i = ] ..... 4, (2m) 2 ! (k1+k2)2

larity s=K2=(k]+k2 )2 = (2m) 2 of H (4) is produced by the pinch of the integration cycle FK by the vanishing cell of the polar set S of L (4), namely : S = {(K,z") ; ( ~K+ z,,)2 = m 2} U {(K,z");( ~K - z,,)2 = m 2 } Let us point out that if the primitive ("schlicht") domain of H (4) is distinguished (as usual) as the "physical sheet" of H (4), the second sheet or "unphysical sheet" is a domain of

m~omorphy

for H (4) , because the solution of (4) can be written

in the whole domain of H (4) • H(4)(K,z,z ')

N(4)(K,z,z ')

=

D(2) (K) where N (4) is a general four-point function and where D (2) is a general two-point function [ 2 ] which may have zeros in the considered second.sheet domain : such poles of H (4) have the physical interpretation of

new particles which are either

bound states or unstable particles according to whether their position lies on the real axis of the s-plane or not. At a further ste~Bethe-SalDeter equations of the form

h a v e t o be c o n s i d e r e d ,

and for

as far

of the Landau singularities

as the analysis

such equations

ficulties

are linked

separable

components" with respect

with the occurrence

of the form 1/(zrz~)2-m2). whose elements

of singular

to z and z'

The l a t t e r

qf type

generate

(3) new d i f f i c u l t i e s

of F are concerned. sets

- variables infinite

arise

These dif-

S for G involving (for

instance

sequences

"non-

: Doles

of Landau varieties

are singularities for the successive terms of the Neumann series of

equation (3). ^ Here~ a Lorentz frame has to be chosen : K = (K(°),K), and the Minkowski metric KZ=K (°)2- K 2 defines a euclidean metric in ( i ~ ) x ~ 3 . ~*Equation (4), which we quote~here, actually corresponds to the case of even quantum field theories : in the latter the step %=I (... % odd) does not exist for H(4).

25g In the physical case of equations of the type (5), such singularities accumulate to the threshold sinEularity (kl+k2+k3)2 = (3m) 2, and a careful analysis of the various sheets of the reso[w~nt has to be done. The results for H (6) which we shall now state are valid in a domain which is bounded by the cut : s=(k]+k2+k3 )2= (3m)2+p

(p > 0) ; in other words the monodromic structure around the three-particle

threshold has not yet been explored there. In order to describe properly these results, we first need to specify some notations. We consider the case of even Quantum Field Theories with a single fundamental mass m. We put H(6)(kl ..... k 6) = H(6)(K,z,z ') where K=k +k +k =-(k +k +k ), K ' | 2 3 4 5 K6 z=(k4,k5,k 6) with : -ki=ki + 7 ' i:4,5,6 ; z' = (k.,k~,k~)_, --z --~ with --i ik'=k"-7 ' i=1,2,3. K is kept in the manifold K = (K(°),~), and K (°) varies in a domain AMcut=A~{KO~3m}, K"

03~ lmKI°1~

ReK(°i

"////A~".~ M

%

cut : The domain ~M

/Q(K)

(~ -.,,..,~~

IL

/~I °, :o/" Fig.1

02 ~

k(3O): 0

//k~Ol=0/ / l \

\

Fioo.2b. K (°) < 3m

Fig.2a. K(O) > 3m

where A M = {K(°);0 < Re K ° < M , with 3m < M ! 5m} (Fig.]). Fig.2represents

-

"X

the situa-

2

tion in Re z'(°) plane ; ~h is the polar manifold k h = m 2, o h is the branch manifold sij=(ki+kj)2 = 4m 2 ({i,j,h} = {1,2,3}) ; the dark disk at the center of fig.2a) represer~ts in projection the physical region of the reaction in which the incoming (resp. outgoing) particles are labelled by the set {4,5,6} (resp.{i,2,3}). For every K (°) in A M , one defines in z-space a domain D K

of the following

form : D K = {z= (z(°),7) ; Im z (°) E ~ 2, Re z(°) E~(K), Re 7 E~6;ilm~I < c

(Re z(°))}

where the domain ~(K) is represented in light hatchings on Fig.2. The following domains DM,Vocut M are used in the results below : DM(resp.~Mcut ) = {(K(°),z, z,);K(O)E AM(resp. AMcut~, ; z E DK; z' E D K}

One-and-two p a r t i c l e d ~ s i n ~

of H(6) :

The aim of this step is to express

H (6) in terms of a suitable irreducible part

G (6) and of auxiliary kernels which only involve H (4) and L (4) . We first define the *Actually, H (6), H(4), H (2) are replaced here by the correspondin~ regularized and amputated forms(see one of our previous footnotes).

260

latter, namely : Tin is the tree-graph product

Tou t

=

h

~

n

h

~

- 61 hn 1

A V = ~

L(4)•

n and similarly

n ; they admit B.S. inverses (in the sense of Eq.(5))

~

which we call respectively A =

h.~n , h

U. and U . We also introduce the operators in out

h ~ n

and

V = ~ + ~ h~n

h

~

: which n satisfy

I 1 - ~ Tin , V A = ~ - ~ Tou t .

The B.S. inversion of Tin, Tou t allows Us to define the inverse of A as : I 1 A -I = V.( | + ~ Uin) (I + ~ Uout).V. We now introduce the six-point function G (6) through the equation : G (6) = AHI6)A - ATou t

(6)

(note that ATout = Tin A ) ,where HI6) =

H (6)

-

~

(7)

Suitable one and two-particle irreducibility properties can be proved for G (6), 6

which imply in particular the following analyticity property of ~(6)= ~ (k -m2)G (6). i=l

Lemma : ~(6) ~

a n a l y t i c i n a domain ~N~cut" such t h a t ~4 = 11m/3.

Now the equation (6) which defines G (6) can be inverted and yields the following "two-particle dressing equation" I = 1 G(6) 1 H 6) (I + ~ Uou t) V V(I + ~ Uin) + Uou t V

(8)

By using the analyticity properties of H (4) and G (6), and by exploiting in complex space some standard combinatorics of Fredholm theory, one can prove

f, ccct (with M = 11m/31 the following decomposition of ~(61 holds : Theorem = In ~M

H1

=

h~rt ~

"'" + h,n~ h

i n the l a t t e r ,

n

h,n

~ @hn + h,n

a l l the "bubbles" of the G - c o n v o l ~ o n

product~ represent H (4) and

each a n a l y t i c function ~hn of the r e s i d u a l sum (h E {4,5,6}n E {1,2,3} h ~ the following p r o p e r t i ~ : i) ~hn i s a n a l y t i c in a " f i r s t - s h e e t domain" which %s D~!~ cut)- (~h U a-n) ; here the s e t s ~h" ~n denote the following "cut~" : ~h = { ( K ' z ' z ' ) ; s i j ( K ' z )

= 4m2+p;o>- 0} and~n = { ( K , z , z ' ) ; Slm(K,z') = 4m2+p;o >_0} .

261

i i ) ~hn admits a l o c a l a n a l y t i c continuation across t h e s ~ t s ~ h and ~n on t h e Riemann s u r f a c e associated with t h e Feynmann graph i cut

Besides each G-convolution term of t ~ minus t h e Landau s i n g u l a r s ~

decomposition Zs a n a l y t i c i n D~

associated w i t h t h e corresponding graph G ; i t more-

over admits a l o c a l a n a l y t i c c o n t i n u ~ o n ^

on t h e Riemann s u r f a c e associated with G. 6

Remark • The only singularities of H(6)=i~ I (k~-m ~ 2) H (6) which are produced in DMCUt_ U (~h U ~n ), considered as a first sheet defined by the extension of the prih,n mitive analyticity domain, are the Landau singularities of the O-loop, l-loop and two-loop "truss-bridge

graphs" ; the %-loop graphs (% > 2) ~ - - _ -

are potentially present in the~Neumann expansions of Uin, Uou t and illustrate the two-particle dressing of H~ 6) from both sides of G (6) in formula (8) ; however, it is a consequence of the previous theorem that their Landau singularities

are

e f f e c t i v e o ~ y i n other sheet~. By taking the restriction of ~(6) to the complex mass-shell M~~ c) , one obtains the following corollary of the previous theorem :

~c~

Corollary : In ~M

.(c)

N ~s6

, the 3 ÷ 3 p a r t i c l e s c a t t e r i n g am~lZtude admZ~s t h e f o l l o -

wing decomposition as a sum of a n a l y t i c functions :

S(6)_ ~ ~ IJ - h,n h

+ h,n~

h

~

n

+ h,n~ ~hn'

^ c ~ ~ ~6(c) - (F h ~ ~n ) ; across each c ~ where each function ~hn i~ a n a l y t i c i n DM

~h'~n

~hn admZt~ a l o c ~

analytic con~nu~on

which spreads i n a t w o . s h e e t e d

Riemann s u r f a c e around each threshold ~h,~n. As far as the t h r e e - p a r t i c l e s t r u c t u r e of H (6) is concerned, we can only say here that : (i) a three-particle

irreducible kernel L (6) has been introduced on the basis

of threeoparticle asymptotic completeness (ii) By using a Bethe-Salpeter

(see [ 3h],[ 17]).

equation which relates G (6) to L (6), it is then

possible to show that G (6) , and H (6) can be analytically continued at least locally across the cut s=(3m)2+O

(0 > 0). The global study of this monodromic

structure re-

mains to be done.

Remark : This result contains as a subproduct the local analyticity properties which are sufficient to ensure macrocausality conditions

for the 3 + 3

in the corresponding low-energy region.

in the sense of Stapp-lagolnitzer's

S.matrix theory [ 6 ].

scattering process

262

REFERENCES [I]

K. SYMANZIK, J. Math. Phys. ~, 249 (1960) and in "Symposium on Theoretical Physics", 3,121, New York, Plenum Press (1967)

i[2]

J. BROS in "Analytic Methods in Math. Physics", p.85, Gordon and Breach, New York (1970) J. BROS, M. LASSALLE, Cofnm. Math. Phys. 54, 33 (1977)

[3]a)J. BROS, M. LASSALLE, in "Structural Analysis of Collision Amplitudes" p.97 North Holland, Amsterdam (1976) b)J. BROS, in A.M.P. Conference, Lausanne, August 1979 [4]

R. OMNES in "Relations de Dispersion et Particules El~mentaires Paris (1960), and references quoted therein

, Hermann,

[5]

H. EPSTEIN in "Axiomatic Field Theory", vol.1, p3-133, Gordon and Breach, New York (1965) and references quoted therein

[6]

D. IAGOLNITZER, The S-Matrix, North Holland, Amsterdam (1978) and references quoted therein ; and D. Iagolnitzer in this volume.

[7]

L.D. LANDAU, Nuclear Physics, 13, 181 (1959) ; R.J. EDEN, P.V. LANDSHOFF, D.I.--OLIVE, J.C. POLKINGHORNE, "The analytic S-matrix", Cambridge Univ. Press (1966) and references quoted therein

[8]

D. FOTIADI, M. FROISSART, J. LASCOUX, F. PHAM, Topology (Gr. Brit) 4, 159 (1965); F. PHAM : "Introduction g l'~tude topologique des singularit~s de Landau", Gauthier-Villars, M~morial des Sc. Math. 164 (1967)

[9]

G. PONZANO, T. REGGE, E.R. SPEER, M.J. WESTWATER,

Comm. Math. Phys. 15, 83 (,1969)

[10] M. KASHIWARA, T. KAWAI, T. OSHIM~, Comm. Math. Phys. 60, 97 (1978) and references quoted therein. A. VAN DEN ESSEN, in this volume [11] G.F. CHEW, The Analytic S-matrix, W.A. Benjamin, New York (1966) and references quoted therein [12] T. SPENCER, F. ZIRILLI, Comm. Math. Phys. 49, I (1976) [13] M. COMBESCURE, F. DUNLOP, "n-particle irreducible functions in euclidean Q.F.T." Preprint IHES, Bures-sur-Yvette (1978) H. KOCH, Thesis, Gen~ve Univ. [14] M. LASSALLE,

(1979)

Comm. Math. Phys. 34,

185 (1974)

[15] J. LERAY, Bull. Soc. Math. de France 87, 81 (1959) [16] J. BROS, D. PESENTI, "Fredholm theory in complex manifolds ..." to be published in Journal de Math. Pures et Appliqu~es (Preprint Orsay University, 1979) [17] J. BROS, "Cours de 3gme cycle", Lausanne University,

May 1979

MICROCAUSALITY, MACROCAUSALITY AND THE PHYSICAL REGION (MICRO)ANALYTIC S-MATRIX

D. IAGOLNITZER

DPh-T, CEN Saclay, BP 2, 91190 Gif-su~-Yvette, France

ABSTRACT Recent works on the physical region analytic structure of multiparticle collision amplitudes in relativistic quantum theory are presented. First, the structure that can be expected and which is the expression, in terms of general essential support or microanalyticity properties, of macrocausality and macrocausal factorization, is described. It is shown that, taken together, these properties are equivalent to decompositions of the S-matrix, in bounded parts of the physical region, in terms of generalized Feynman integrals. Derivations of this structure obtained recently for 3 ÷ 3 processes below-the four-particle threshold both in S,matrix theory (without recourse to the crucial ad hoc assumption of "separation of singularities" in unitarity equations used previously) and in axiomatic field theory are then reviewed. It is finally explained how this structure applies in two-dimensional space-time and yields factorization of the multiparticle S-matrix itself for a class of models.

CONTENTS I. INTRODUCTION

263

2. THE S MATRIX

267 269

3. THE MACROCAUSAL S MATRIX 3.1. Macrocausality

269

3.2. Macrocausal factorization

272

4. THE THREE-BODY S MATRIX BELOW THE FOUR-PARTICLE THRESHOLD

277

5. S-MATRIX THEORY

279

6. AXIOMATIC FIELD THEORY

284

7. TWO-DIMENSIONAL SPACE-TIME

289

APPENDIX I

290

Essential support, structure theorems and related topics APPENDIX 2

293

Complements on macrocausality and macrocausal factorization REFERENCES

294

264

1. INTRODUCTION The purpose of these lectures is to present recent results on the derivation, in relativistic quantum theory, of the physical-region analytic structure of multiparticle collision amplitudes between sets of massive particles with short-range interactions. The unitary collision operator S, usually called the S matrix, and collision amplitudes are introduced in Sect.2 (see Ch.I of [I] for details). Analyticity was recognized long ago as an important notion in particle physic~ 2], particularly in the study of strong (= short-range) interactions, and important progress was made in the sixties in the understanding and derivation of general momentum space analyticity properties, on the complex mass-shell,for two-body processes (two initial and two final particles). In the multiparticle case, however, progress has been made mainly in the study of anaiyticity properties in the neighborhood of the real

physical regions.

This study has appeared to have its own physical and mathe-

matical interest and to be possibly a starting point for further developments. A satisfactory understanding of the structure that can be expected in the physical region, and which is in fact equivalent to physical properties of macrocausality and

macrocau~al factorization

(= asymptotic properties of transition amplitudes bet-

ween appropriate sequences of initial and final wave functions, corresponding respectively to non causal and causal configurations of displaced particles or subBroups of particles), was first achieved at the end of the sixties [3'4] and in a more precise and complete way in the seventies (see Ch. II of [I]). The above properties are stated mathematically in the form of essential support properties which, by the general results of essential support theory (see App.1), are equivale~Jd, for each given physical process, to

physical-region analyticity, or more generally micro-

analyticity, properties of the scattering function f of the process considered (= its momentum-space connected collision amplitude, after factorization of its energymomentum conservation ~4-function : see Sect.l). Away from certain +s-Landau points, macrocausality is in particular equivalent to decompositions of f, in bounded parts of the physical region, as a sum of contributions fB associated with the various +s-Landau surfaces L+(GB) of connected graphs G B encountered, each f~ being analytic outside L+(GB) and being along L + (G B) the boundary value of an analytic function from the "plus is" directions dual, at each point P of L+(GB), to the causal directionsu+(P;GB)

. Macrocausality by itself yields no further information on the struc-

ture of each fB " Macrocausal factorization then yields (under some conditions), and is in fact equivalent in the re~ions considered to similar decompositions in which each fB is a "generalized Feynman integral" associated with G B . Equivalently, in usual cases (and for "simple" graphs : see Sect.3), L+(GB) is the integral,

the discontinuity of fB around

over internal on-mass-shell four-momenta,of the product of

connected S-matrix kernels associated with each vertex of G8 .

265

These results are described in Sect.3, with emphasis on aspects relevant later or not treated earlier, and illustrated in Sect.4 in the case, of interest below, of the three-body (3+3) S-matrix below the 4-particle threshold. A brief introduction to the notion of essential support, microanalyticity and related results, which play a crucial role throughout these lectures, is given in App.l. Many efforts have been devoted on the other hand to the derivation of the above physical-region structure from basic principles. However, as recalled below, the results obtained up to a few years ago either made use in S-matrix theory of a crucial, ad hoc assumption, or were in axiomatic field theory, very far from the desired ones. Progress has then been achieved recently in both theories for the three-body S matrix below the four-particle threshold. It is presented in Sect.5 and 6 respectively. The various approaches proposed in S-matrix theory are based on th e study of the possible singularities of the "bubble diagram functions" that occur in equations derived from unitarity and the decomposition of S into connected components, and of the way these singularities have to cancel among themselves.

A

first approach,

proposed in the sixties [5], starts from the principle of "maximal analyticity" which, applied to the physical region, is interpreted as providing analyticity of individual scattering functions outside the singularities of phase-space integrals (= Landau surfaces, or in a more useful way modified Landau surfaces[6]), and the plus iE rule at +s-Landau points. The aim was then to establish analyticity outside the +s-parts of the Landau surfaces and discontinuity formulae around the +s-Landau singularities. The "proofs" of [5] rely and ,

crucially,however, on some assertions that were unproved

as recognized recently, cannot be expected to be correct in any reasonable

framework (see Remark4 in Sect.5).Results were obtained on the other hand at the end of the sixties [7] and in a more complete and satisfactory way in the seventies

(see

[8,9] and Ch.lll of [I]) in the related approach that starts, besides unitarity, from macrocausality.

In view of the results described in Sect.3, macrocausality

can be considered as a precise and strengthtened version of the analyticity properties associated previously in the physical region with "maximal analyticity" • As recalled above, the results that one aims to derive are then equivalent to macrocausal factorization. However, while this latter property has a satisfactory physical interpretation, it already contains detailed information, and does not seem, in the quantum case, to be independent from macrocausality and unitarity. It is therefore desirable, to derive it, if possible, from these more general and basic physical principles. Besides macrocausality and unitarity, the above results are based, however, in a crucial way (see Remark 4 in Sect.5) on an ad hoc assumption of'~eparation of singularities" in unitarity equations according to their underlyin~ "topological structure" (see App.|). "Separation of singularities" is believed to hold, but it has no

266

priori basis and should thus itself be derived. Proofs that do not use this assumption have then been achieved partly in [10], and in a complemway in [11], for the three-body S-matrix below the four-particle threshold ~nd away from #4o points such that some initial, or some final, four-momenta are colinear). Macrocausality orovides (see above) decompositions of the three-body scattering function f, in the region considered, as a sum of "plus i~" boundary values of analytic functions associated with the surfaces L+(GB) encountered (= here surfaces of ~raphs G8 with one internal line and triangle graphs). Possible pathologies of these boundary values are first eliminated by a "no sprout assumption", which slightly completes the analyticity properties derived for f from macrocausality (*). The use of unitarity equations then allows one, as explained in Sect.5, to show that they necessarily coincide with the generalized Feynman integrals introduced in Sect.3 ."Separation of singularities" is established when needed in the course of the derivation, and can then be derived in other cases (in the region considered) from the results obtained, following [9] and more generally App. B of Ch.lll of [1]. In the course of the derivation

a refined version of macrocausality (see the

end of Sect.3.1), which provides the regularity property R (see mathematical lecture) for the connected S matrix is also used to solve [12] the crucial "u=0 '' problem (see Sect.5) encountered for some of the bubble diagram functions. For the alternative approach to the u=0 problem developed in the framework of the theory of holonomic microfunctions, see the lectures by Kashiwara-Kawa[. The approach of axiomatic field theory presented here (Sect.6) is that of [|3]. The alternative approach developed by J. Bros [14] and coworkers , with related resuits, is described in his lectures. As first recalled in Sect.6, and the

SpecYt~alc o n ~ o n

miUtogau~y

(on the spectrum of the energy-momentum operator) yield

general information on the essential support of collision amplitudes in their physical region [|5]. This information is, however, much weaker than that associated with macrocausality. We emphasize that the properties of microcausality, which applies to the

fieZc~ on

the microscopic space-time level, and of macrocausality,

which is expressed in terms of pcV~c~£6 at the macroscopic level, are of a different nature and that microeausality therefore does not imply hy itself macrocausality. We shall then explain, following [13],how the essential support properties associated with macrocausality and macrocausal factorization do follow for the three-body S matrix below the four-particle threshold (and up to some limitations) from a further exploitation of the "off-shell" structure of the theory, together with the axiom of

a~ymptotic completeness

and regularity assumptions needed to avoid

priori possible ~ la Martin pathologies. (*)It is a weak aspect of the idea of "maximal analyticity", or of holonomy, and can possibly be derived from the refined macrocausality condition mentioned below.

267

We conclude these lectures by outlining in Sect.7 how the general structure described in Sect.2 applies in ~o-cLO~e~ion~ space-time. Whereas and

macrocausal

macrocausality

factorization are, as mentioned above, asymptotic properties of

transition amplitudes between initial and final wave functions,

for non causal or

causal configurations of displaced particles, we shall see how they yield in this case[1~]7]an important factorization property of the momentum-space multiparticle

S

matrix itself for a class of models. Sect.2 to 6 are presented for simplicity for a theory with only one type of particle, a "boson" of mass D > O, and in four-dimensional and results extend, however, to more general theories,

space-time.

The statement~

including different types of

particles with strictly positive masses, and to other space-time dimensions, with only minor changes. The

results of sect.5 apply, however, only for space-time

dimensions strictly larger than two. Needed modifications dimensional

spaceitime,

to Sect.2 and 3 in two-

and for theories with several equal-mass particles,

are

outlined in Sect.7.

2. THE S-MATRIX As announced in Sect.| we consider below, for simplicity, a theory with only one type of particle, a "boson" of spin zero and mass ~ > 0. The Hilbert space of free-particle

states is then defined as : ~=

where']

Q m>|

~m

= L2(M) is the space of "wave" functions (0 of an energy-momentum variable +

p = (po,~)

(where Po is the energy and p = { p~},~ = 1,2,3, is the momentum), which 2 +2 are defined on the positive-energy, real mass-shell manifold M (D ~ 2 =Pop = ~ 2 ' Po >0) and are square integrable with respect to the invariant measure d~(p)=6(p2_ 2)@(po) d4p ~

d3~_+ ' ~(~) = (~2+ 2) 1/2. ~ m 2~(p)

is the sym~etrized

tensorial

product of m s p a c e s ~ I. Each physical system, which is considered

here

with

all

its

evolution,

is assumed to be characterized before and after interactions by well defined rays i n ~ ( = non zero vectors up to multiplication (= conservation of probabilities)

op~or

S from~to~,

by a complex % # 0). General principles

then ensure the existence of a unitary c o ~ i o ~

usually called the S matrix, which, to each vector

I~> re-

presenting the state of a system before interactions associates the vector SI(0> , which

represents

its state after interactions.

For unit norm I~> and l~>,I<~IsIcp>I 2

is the transition probability from I(p> to I~>, i.e. the probability of detecting the final state of the system in the state I@> , if its initial state is I~>.<~IsI(p> is called the transition amplitude from I(0> to I@>. If I(0> E % if I~>and I~> are both symmetrized tensorial products

, I~> E ~ n , and

(over the values k=]

, ...,m

268

and k = m+l,...,m+n) particle,

of wave functions ~

associated with each initial and final

the transition amplitude S({~pk}) = <~ISl~p> can be written formally as : S({~})

=

f Sm,n (Pl ..... Pm ; Pm+l ..... Pm+n ) m m+n m+n xTl-i=l ~Pi(Pi)

jgl=

~j(pj) x

J=

d~ (pk) ,

(I)

where ~ is the complex conjugate of ~ and d~(Pk ) = ~(p~-~2)e((Pk)o ) d4pk . The kernel Sm,n(Pl,...,Pm+n) and the final, variables)

(which is symmetric with respect to the initial,

is called the momentum-space

transition,

or collision am-

plitude of the process m + n. Since S is unitary, it is the kernel of a bounded operator Sm, n f r o m ~

m to ~ n , and is thus in particular

(by Schwar~'s nuclear

theorem) a well defined tempered distribution on the space M of all real points 2 2 m,n p=(pl,...,Pm+n ) such that Pk = N ' (Pk)o > 0, Vk. From general principles,

the operator S is known to satisfy

en~gy-momentum

conserv#Jzion. Correspondingly, one has (away from points p such that all Pk are colinear)

: m+n

Sm'n(Pl ..... Pm+n) = Sm'n × 64(~1 P i - m ! l

pj)

(2)

is a distribution defined on the physicS-region ~ of the process m+n m,n 2 2 m,n (={p=(pl...Pm+n), p k = ~ ,(Pk)o > O, Yk, EPi = Z ~ } ) "

where s

The connected parts of the S matrix, which will correspond physically to "interactions the formulae

involving all particles together", are defined by induction through : m,n

m,n

K E K

mK'nK

K

K,N(K)>I where the sum ~ runs over all partitions

K

of the sets of initial indices

and final indices m+1,...,m+n into subsets

(IK,JK), K=I,...,N(K),

numbers of indices is IK and JK ' and (~S~,nK)K~. operators S c

I,... m

mK, n K are the

is a tensorial product of

associated with the partition K in the appropriate way. We recall

mK,~ K that SI, I

=

Sc

1,1 = 11,I ' S],n ~ 0 if n # I, Sm, I ~ 0 if m # 1 . In view of (3),

Sc is, like S a bounded operator which satisfies also energy-momentum conserm,n m,n vation and whose kernel can thus be written in the same form as (2) (away from points p such that all Pk are colinear)

:

(4)

scm,n (Pl ''" "'Pm+n ) = f m,n x ~4(Epi - Epj) The distribution f

m,n

is again defined on ~4 and is usually called the m,n

scattering function of the process m ÷ n. s

General arguments allow one to conclude that the connected S-matrix kernels,

269

in contrast to the non connected ones, include

no contribution containinR 6-func-

tions of energy-momentum conserva£ion between subgroups of initial and final particles. This is, in particular, a consequence of the macrocausality condition introduced in Sect.3 (in the form of the factorization property for non causal configurations of subgroups of displaced particles), which will give, moreover, further information on the analytic structure of f

m,n

3. THE MACROCAUSAL S-MATRIX We first mention some geometrical facts that will be useful below. The variables dual to the energy-momentum variables are identified physically with space-time variables. Being given a point P=(PI'''''Pm+n ) in R 4(m+n) and a point u=(ul,...,Um+ n) ÷

in the dual space, where u k = ((Uk)o, Uk) is a point with time component (Uk) ° and ÷ space component u k , the scalar product will be defined as

=

~ k

~k Pk " Uk

' ÷

where ck = +I if k = l,...,m,

gk=-|

if k=m+l,...,m+n,

÷

Pk.Uk = (Pk)o(Uk)o-Pk.Uk .

A point in the cotangent space Tp Mm, n , resp. Tp Mm, n , at a point P={Pk } of Mm,n is thus a point u=(u I .... ,Um+n) defined modulo addition for each k of %kPk where %k is any real scalar, resp. is a point u defined modulo addition for each k of %k Pk + a, where a is an arbitrary space-time vector independent of k. It can be characterized

by the configuration of trajectories

(= straight lines) (Pk,Uk)

in space-time which, for each k, are parallel to Pk and pass through Uk, resp. by the relative configuration, defined modulo global space-time translations, of these trajectories. 3.1. Macrocausality [3'I] It will be convenient, in this section and in Sect.3.2, to consider sequences of initial and final wave functions of +the + form :

-y~(pk-Pk)2 ~k,T(pk) = Xk(Pk ) e

- i pk. TUk e

(5)

÷

÷

where Pk is a g i v e n momentum, ×k i s a n a l y t i c ÷ 2 1/2 (Pk)o = (Pk+~)

in the neighborhood of Pk=((Pk)o,Pk ) ,

, u k is a space-time vector, y > 0 and T is a Dositive Scalar

that will tend to infinity. The factor e ipk'Tuk

corresponds to space-time trans-

lation of particle k by Tu k . These wave functions correspond to narticles that are asymptotically localized, in the T ÷ ~ limit and in a space-time coordinate system scaled to T (i.e. each point Tx in space-time is replaced by x), along the classical trajectories (Pk,Uk), modulo well specified exponential fall-off properties. Apart from exceptional cases (not encountered later) in which further limiting ,

proceaures

[6,12]

..

might have to be considered (see App.2)

, macrocausality states the

exponential fall off (in a well specified sense) in the T + ~ limit of transition

270

probabilities

between initial and final

wave

functions of the form (5), for

non

causal configurations of the initial and final trajectories, i.e. if there exists no classical multiple scattering diagram D+(P,u)

[6, 12 ] (4) (with possibly vertices at infinity)

whose external trajectories are those determined by (Pk,Uk) for each k. This statement corresponds physically to the above mentioned localization properties of the initial and final particles and to the short-range character of the interactions, which is interpreted as inducing exponential fall-off with distance in actual space-time) of all transfers of energy-momentum to actual stable particles.

The absence of pathologies,

(i.e. with T

that cannot be attributed such as the existence of

sequences of unstable particles with arbitrarily small rates of exponential decay, by which energy-momentum might be transferred,

is also assumed.

The above property yields trivially a similar exponential fall-off property of the transition amplitude S({~k,T}) , which can be stated, in terms of essential supports as : u ¢ ESp(S)

if there exists no D+(P,u)

,

(6)

where u is defined here modulo addition of XkP k for each k. An analogous(and valent

: see App.l) statement has been proposed

equi-

independently as a "microanalyti-

city postulate" by Pham and Sato [18] in terms of the notion of singular spectrum. A slightly stronger form of macrocausality,

which will be that needed in Sect.5

(see Remark 3 in that section) says this : if the only possible diagrams D+(P,u) are composed of several disconnected parts connecting together given subgroups K of initial and final particles,

then the overall transition probability is equal,

modulo a remainder to which the previous exponential fall-off properties apply, to the product of the partial transition probabilities

associated with each subgroup.

It can be shown [4] that the transition amplitude S({~k,T}) then satisfies a similar factorization property, which can be stated in terms of essential supports as : u

¢

E Sp

(S-

Q K

S K)

(7)

where

Q SK is the (tensorial) product of S-matrix kernels associated with each K subgroup of initial and final variables, and that (7) is itself equivalent, apart •

from exceptional

.

sltuatlons u

¢

(**)

", if they exist, not encountered later, to :

ESp

(S c) if there exists no connected D+(P,u)

.

(7')

(*)Vertices "at infinity" in some direction are allowed when all incoming and outgoing trajectories at that vertex are parallel. These trajectories then need not coincide, but are required, following [6], to satisfy "angular-momentum conser~ vation". (**)Even if there is no connected ~ P , u ) (7') does not necessarily follow from (7) if there exist different disconnedted diagrams D (P,u), D$(P,u) ..... composed of parts that relate together different subgroups o~ initial and final particles : see [4].

271

In (7) and (7'), u is defined again modulo {%kPk }. By the general results of essential support theory (or hyperfunction theory), (7') is in turn equivalent to : E Sp(f)

c

C+(P)

(8)

,

where C+(P) is the subset of Tp~ corresponding to all r~a;tiue configurations of external trajectories of connected diagrams D+(P)

with

external four-momenta Pk"

The set C+(P) is the union of the sets C+(P;G) corresponding to connected diagrams D+(P;G) whose topological structure is a given multiple scattering connected graph G. The +e-Landau surface L+(G) is the subset of the physical region ~{ composed of points P such that C+(P;G) is non empty. It is known [19] that these surfaces have at most codimension one, and are not dense, in ~4. Away from M ° p o i ~

(= points P

such that some initial, or some final, Pk are colinear) and from points P at which several surfaces are r~aJted (*), (which belong in both cases to subsets of codimension > I ) t h e surfaces L+(G) o f connected graphs are (if non empty) real analytic codimension one submanifolds, and C+(P;G) is correspondingly composed, at each P of L+(G), of a unique direction u+(P;G). The set associated with a class of related surfaces at a non M ° point P is still a closed convex salient cone. (See example at the end of Sect.3.2). If we leave aside the ~

o

points and the points P that lie on several non related

+e- Landau surfaces, the set C+(P) is thus known to be either empty (at non +e-Landau points), or to be a closed convex salient cone, which reduces in general to a unique direction u+(P;G). The points that have been left out lie in subsets of ~ of codimension strictly larger than one. In view of the general results of essential support theory, (8) therefore ensures that f is analytic outside the +e-Landau surfaces of connected graphs and is at almost all +G-Landau points the boundary value of a well defined function f , analytic in a domain of the complexified manifold ~ of M, from the pl~6 ig directions dual at each point P of a surface L+(G) to u+(P;G), or

to the cone C+(P) if P lies on several related surfaces. At non M ° points P that lie on several non re£a£ed surfaces, C+(P) is no longer contained in general in a closed convex salient cone : see Sect.4 ; f can then be decomposed locally, as a Sum of plus iE boundary values of analytic functions fB , each of which being associated with one of the surfaces, or classes of related surfaces involved at P. If the cones associated at P with each class of related surfaces

(*) Surfaces L+(G'), L+(G"), ... are related at P if P lies on these surfaces or their closures, and also on the surface L+(G), or its closure, of a graphG such that G',G",... are contractions of G. A contraction of G is either G or any graph obtained by removing certain internal lines and identifying the vertices where they were incoming and outgoing. Note that graphs that differ only by trivial changes which do not change the surface (such as the inclusion of vertices on certain lines or sets of lines) are identified.

272

are ~ j 6 % ~

(apart from the origin), each fB is well defined locally modulo an

analytic function. There are ~ priori ambiguities otherwise (see Examples2 and 3 in Sect.4). If P is on the other hand an M ° point, C+(P) is not contained in a closed convex salient cone. There is no natural decomposition of C+(P) into a union of closed convex salient cones, and hence of f into a sum of boundary values of analytic functions. Finally, the general decomposition theorems of essential support theory, or hyperfunction theory, also yield various decompositions of f, in bounded parts of the physical region, as a sum of contributions associated with the various + ~ Landau surfaces encountered. There are ~ priori a number of ambiguities that we shall not discuss here in general. See the example of Sect.4, where they are eliminated, in the region considered, by the "no sprout" assumption. Recent studies about macrocausality have been made in connection with the problems encountered in Sect.5. On the one hand, the situation at #~ points has o been analyzed [6'|2] : see App.3. On the other hand, a refined version of macrocausality has been proposed [12]. It gives information on the way rates of exponential fall-off tend to zero, in certain situations, when causal directions (= directions in the essential support) are approached. It follows, in the cases that have been considered, essentially from the same physical arguments as the previous version of macrocausality, and provides the regularity property R (see mathematical lecture) for the connected S matrix. For conciseness, a further discussion is omitted here, t h e reader being referred to [12].

3.2. Macrocausal factorization We now consider again wave functions of the form (5), but points (P,u) such that there may exist diagrams ~+(P,u). For simplicity, we shall restrict our attention to cases when these diagrams include not set of two or more internal trajectories outgoing and incoming at common vertices, and we shall consider here cases (which are the usual ones in a theory with only one type of particle - see Sect.7 for an extension) in which there is only one ~+(P,u), or such that all possible ~(P,u)

can be obtained from a unique ~+(P,u) by replacing certain interactions at

certain vertices of 9+ by no interaction, or by several subinteractions

: see

example in Fig.|. In this example, D+ is such that (i) the external trajectories 9, 10 coincide with the internal trajectories incoming at c and (ii) P4+P5=P7+P8 . !

The trajectories 9,10 in ~+ are issued respectively from b and a : although they meet at the space-time position of c, there is no vertex there. On the other hand, 4,5,7,8 have in ~$ a separate interaction at a vertex b' which is different from b, although b and b' have the same space-time position.

273 9

2

1

10

2

3

time

D+

0

//9

~

6 --10

3

~

Fig. I Apart from exceptional cases (see App.2), m a c ~ o c a ~

f~c~0~z~0n

is then the

assertion that the transition amplitude S({~Pk,T}) is asymptotically equal, modulo a remainder that falls off exponentially in the T ~ ~ limit (in a sense similar to that of Sect.3.1), to the term D({~k,T}), where D is the integral, over all possible internal on-mass-shell four-momenta associated with the internal lines, of the product of S-matrix kernels associated with each vertex of the graDh G that represents

the t o p o l o g i c a l s t r u c t u r e of D+(P,u). For i n s t a n c e , i f G = 2 6~ - -~ - ~ then

D(Pl,P2,P3;P4,P5,P6) = f S2,2(Pl,P2,kl,k 2) S2,2(k2,P3;k3,P 6) S2,2(kl,k3,P4,P 5)

]~

~(k~-~ 2) @(k%) o) d4k£

(9)

%=1,2,3 or in a diagrammatical notation :

D

=

3

D({~k,~}) is defined in the same way as S({tgk, }) in Eq.(l). The terms D are always well defined as kernels of bounded operators (see[12]), hence as tempered distributions, and contain an overall energy-momentum conservation ~-function. In terms of essential supports, macrocausal factorization is stated as : u

~

ESp

(S - D)

(I0)

where u is, as in (6) (7), defined modulo {%kPk } . In view of (7),(10) can be restricted to the case of connected diagrams D+(P,u). It can then be checked in usual (not always trivial : see example in Ch.lll of [I])

274

cases, such as all those encountered in Sect.4, that (10)

then entails also a cor-

responding property for the connected s matrix S c :

u E ESp (s c -

c

~ DB)

(ll)

where there is now one term D cB for each possible connected diagram P+(P,u) or ~$(P,u), and where D cB is defined in the same way as D, except that connected S-matrix kernels are now associated with each vertex. For instance, if we consider the simplified version of Fig.l, in which 4,5,7,8 are left out and 9,10 are relabelled 4,5, and

if there is no other connected D$(P,u)

-

3,3 where

~

:

-

3

2 ~--g-f~--~-

3 ~ ~

stands for connected S-matrices

5

(12)

"~-,,~ 6

c

$2, 2 •

Each term D6C can be written, as D, (away from points D. such that all Pk are colinear) in the form : D~(P I, • ..,Pm;Pm+1,'..,Pm+n ) = d cB × ~4(Epi_EPj )

(13)

where d 8c is defined on Mm,n ; (11) is then equivalent to : u

E

E Sp

(f-

~

e ds)

(14)

B where u is defined modulo %kPk + a for each k. In order to study the implications of macrocausal factorization, we first introduce

gen~ag&ed Fey~man i~tegraf~ associated

with graphs G B .

Let us consider a codimension one surface L+(G$)

of a "simple" graph GB(=graph

with no set of more than one line between two vertices), which, in a ziven domain R~ of the physical region, is related to no other +m-surface. In usual cases, such as all those encountered in Sect.4, the structure theorem on bubble diagram functions (see App.l) allows one to show that the essential support of d cB in R~ is (contained in) the union of the closures of the sets C+(G~) and C(GB) which correspond respectively to relative configurations of external trajectories of diagrams ~+(G B) and of other diagrams D , and which are disjoint (apart from the origin) at any P in R B . The decomposition theorems of essential support theory (or hyperfunction theory) c then provide a decomposition of d~ in RBof the form : =

+

(15)

where the only possible directions in the essential supportsAof (d~)+ and dBC are c those of C+(G B) and C(G~) respectively, and where (d$)+ and dcB are well defined modulo analytic backgrounds in R B . The essential support of (d~)+ is empty (apart ^ + from the origin) outside L (GB) and has the unique direction u+(P;GB) at any

275

+ c + point P of L (G B) in R B ; (dB) + is correspondingly analytic outside L (G B) and is + along L (GB) the boundary value of an analytic function from the plus ig directions dual at each P to u+(P;GB). The distribution generalization,

c (dB)+ thus defined can be considered as a physical-region

modulo an analytic background,

of the Feynman integral associated

with G B in R B : see discussion in [I|] and references therein. If L+(G B) is not + related, outside R , to other surfaces L (GB,) encountered in R B , then (d~)+ is, roughly speaking, the analogue of the Feynman integral associated with G B , the constants at each vertex being replaced by off-shell analytic continuations scattering functions associated with these vertices.

of the

(Appropriate cut-off factors

have also to be introduced in the integrand). This will be the case for instance in c Sect.4. Then (D~)+ = (d~)+X64(Ep~ - Ep~) will be written diagrammatically in the same J

way as D~c , but with + signs above each internal line (see example in Sect.4). If c L +(GB) is related to some other surface L + (Gs,) outside R~ , then (D~)+ as defined above corresponds

to the replacement of the constants at some vertices by off-shell

continuation of the scattering functions,

from which relevant singularities have + (D~)+_ would be singular also along L (GB,) in R B One generalized Feynman integral can alternatively be used for a given class of first been removed.(Otherwise

related surfaces.

m

In this connection,

see the example given at the end of this

section). R~k

: in usual cases, again such as those of Sect.4, the structure theorem

(App.l) shows that ESp (d),

at any point P belonging to L (G$) in R~ , is Composed

of u+(P,GB) and of the opposite direction u_(P;G~)

(i.e. C reduces,

if P C L+(G~)

,

_ (p;GB) . It is then easily checked that d cB is in the neighborhood of P the dise + e continuity of (dB) + around L (GB), i.e. the difference of (dB) + and of the "minus ig" + boundary value d~c of its minus ie analytic continuation around L (GB) from the to

"non physical" side of L+(G$) in R B (where d~ ~ O) : see details e.E. in Ch.lll of c

[I]. In the above mentioned case, (dB)+

can as a matter of fact be defined equiva-

lently modulo an analytic background in R B as the distribution which is analytic + c _ outside L (G B) and whose discontinuity around L+(G~) is dB

In the neighborhood of a point P of L+(G~) that lies on no other +c~-Landau surface or its closure, macrocausal f

--

factorization entails that : c (dB)+

,

(16)

where ~ means equality modulo an analytic background. In bounded domains R ofthephvsical + region in which all surfaces L (G B) encountered have the properties listed above (15), (14) entails more generally the following decomposition to which it is in fact equivalent : f ~

E

c (d~)+

(see example in Sect.4) inR (17)

276

Pr00~ : We prove below that (14) entails is

sufficient

to prove

that

(17).(The converse is proved similarly).It

f - Z (d~)+ is analytic

at

any "ooint P in R , i.e.

equivalently that ES~(f-Z (d c ) )~ is empty (apart from the origin). Being ~iven P, v o I~ + + ' ' let us divide the set of psurfaces L (G B)" that contain P (if there are) into subsets for w h i c h t h e d i r e c t i o n s thus obtained.

; ((d)+)

u.(P;G~)

coincide.

L e t ~(1),+ 0 _ ~ 2 ) , . . .

be the

If fi # a(1), ++ a!2~...+ then fi ~ ESp(f) by macrocausality

directions

and fi ~ ESp

c (see above). Hence fi ff ESp (f- Z (d~)+). If fi is for instance fi(1)+, then one

B

may write : f - Z (dCB) + B

where the sums E , ~ .... (I) (2) ~(1)

In view of (14) '

/N ( f - ~ d~) + Z d c (1) (1) ~

=

~ ESp (f - ~ d~). On the other hand, in view of the

+

fi(1) ff ESp( Z

(1)

'

+

...

refer to each subset and where (15) has been used.

(1)

essential support properties previously described of ( ~

c

- T (d13)+ (2)

c

(dB)+) , .

(2)

d/~'Band

fi(1)~ + ~

c (dB) + ,

ESp

Hence a (I) ~ ESp(f - ~ (d~)+) Q.E.D, ""

+

B

The extension of the definitions and results to regions that contain points P at which several surfaces are related on the one hand, and in which surfaces of graphs with sets of several lines between some vertices are encountered on the other hand, is possible in principle, although there are difficulties.

There is, roughly

speaking, one generalized Feynman integral, in a given domain R, for each class of related surfaces. We only give here a simple example.

Example

:

Let P be a point of the surfaces L+(~) ,

•

G=

L+ (G1),

L + (G 2)

2

6

2

where :

7

'

G1

8

' G2=

6

3

7 8

8

The su rf ac ~L + (GI), L + (G2) are codimension one surfaces in ~4,4 defined by the 2 2 2 equations k I = D 2, (kl) ° > 0 and k 2 = D , (k2) ° > 0 respectively, where kl=Dl+P2-P5 , +

k2=P7+P8-P4.

The surface L+(G) is here the intersection of L+(GI), L (G2). Bein~

given a point P E L+(G) that lies for simplicity on no surface (or its closure) other than L + (GI), L + (G2) , the vertices a,b,c of any diagram ~+(P;G) %IKI , c-b = %2K2,%1>0,%2 > 0 %2/%1

satisfy b-a =

and there is one direction @+(P;G) for each ratio of

. In the limim when %2/%1 tends to zero and to infinity, the directions

Q+(P;G I) and Q+(P;G2) are obtained. C+(P) is the closed convex salient cone composed of all directions @+(P;G) and of @+(P;GI) , G+(P;G2).

277

Macrocausal

factorization ensures that all directions ~+(P;G) are outside

ESp(f-d c) and that O+(P;Gi) ~ ESp (f-d~), i=1,2...

On the other hand @

ESp(f)

if Q ~ C+(P). These essential support properties determine f in a unique way in the neighborhood of P, modulo an analytic background, Feynman integral

as the following generalized

: f

a(P I ..... P8 ) ~

(18)

o

( k ; - ~ 2 + i g ) ( k ~ - D 2 + iE) where a is any locally analytic function such that a

2 2 2 k1=k2= ~

=

f2 2(PI'P2;P5'k1)f2,2(kI'P3;k2'P6)f2,2(k2'P4;P7'P8

(See e.g. App.C in Ch. III of [4]). In a diagrammatical notation

) "

:

2

6

43

4. THE THREE-BODY S-MATRIX BELOW THE FOUR-PARTICLE

(18')

THRESHOLD

In this section we consider a 3 + 3 process in the region R = {p;p E M3, 3 , (3~) 2 < s < (4~) 2 , 3 6 2 where s = (Zp.) 2 = (Zpj) I 1

p ~ ~A } o

(19)

4

+

In R, one encounters

18 surfaces L (G B) of connected graphs GB, including 9 I

surfaces of ~raphs with one internal line, such as G I = 2 1,.,==_~4 • 2 9 surfaces of trzangle graphs, such as G 2 = -- ~X X ~ ~'~" 5

4

~

5 3 .,,'~

and -"6 . These surfaces

are codimension one real analytic submanifolds of R and are not related at any point P in R or outside R.

Macroca~ality

yields decompositions

f

=

E

in R of the form :

fB

(20)

B

where each fB is analytic outside L+(GB) and is along L + (G 8) the boundary value of an analytic function ~B from the plus is directions dual, at each point P of L+(GB), to Q+(P;GB). We give below some examples of situations described in Sect.3.]

278

Example

I : A simple example (see Ch.ll of [I]) of points P where f is n0£ locally

the boundary value of a P2=P4 , P3=P5

unique analytic function are all Doints P such that PI=P6' + . These points lie on six surfaces L (GB) of graphs G B with one internal

line, which can be grouped in pairs in a way such that the directions @+(P;G~) are

opposZte in and

3

diagrams

Example

2

each pair. This is the case for instance for the graphs

I"

-

9+

46

,

(The t r a j e c t o r i e s

2

1 , 6 , as a l s o 2 , 4 and 3,5 c o i n c i d e i n t h e two

the internal trajectory coinciding with 1,6).

2 : An example of points P such that the directions @+(P;GB) now coincide,

and where the contributions

f~ to f in (20) are not ~ priori well defined, locally,

modulo analytic backgrounds,

is obtained []2] by considering the two graphs G],G 2

mentioned at the beginning of this section, and any point P of the (non empty) + subset ~ of L (G 2) defined as follows. Being given any P'=(PI .... ,P') 6 of L + (G 2) in R, the internal on-mass-shell

four-momenta of 9+(P';G 2) are well determined.

Then all

v

points P obtained from P' by replacing P¼, P5 by the two internal four-momenta of the

lines incoming at the vertex where 4,5 are outgoing belong to ~ . The diagrams

9+(P;GI)

and ~+(P;G2)

are similar to the diagrams 9'+ and 9+ of Fi~.l in Sect.3.3,

except that the external trajectories clearly @+(P;GI) = @+(P;G2).

4,5,7,8 of Fig. l are left out, and one has

(These points play an important role in Sect.5 in the

course of the derivation of the contributions associated with triangle graphs since th~yalways occur in integration domains

Example

: see Ill]).

3 : Examples of points P in which both phenomena occur are all ~oints P

such that PI=P6,P2=P4,P3=P5

and such that PI,P2,P 3 lie moreover in a common plane

(Z ~iPi=O, X1,%2,% 3 # 0). These points lie on eight surfaces

L+(GB) tangent at P,

four of which give rise to the same direction G+(P;G~), the four others ~ivin~ rise to the opposite direction. Each group contains three surfaces of graphs with one internal line and one surface of a triangle graph. ponding in each group to the triangle graph

The diagrams 9+(P;GB)

corres-

are shown in Fi~.2. Those correspondin~

to the graphs with one internal line are obtained by removinR one of the vertices. 5

t ~me Fig. 2

279 MacJl0ca~al f a c t 0 ~ z ~ 0 n

is equivalent to the decomposition

(17) in R in terms +

of generalized Feynman integrals associated with the 18 surfaces L (G~) encountered, which can be written in a diagrammatical

I

i

form (see Sect.3.2) as :

÷

i=],2,3 j=4,5,6 where, by definition,

Z

j

i

(21)

i=1,2,3 j=4,5,6

each term involved in the right-hand side is equal to one of

the terms (D~)+~ = (d~p~×~4 (pl+P2+PB_P4_P5_P6)

means

and where

previously

as

equality modulo an analytic background in R (after factorization of the 64-functions).

5. S-MATRIX THEORY Theorem [I|] "Refined macrocausality, the decomposition

unitarity and the no sprout assumption

(see below) imply

(21) of the three-body S-matrix in R in terms of generalized

Feynman integrals". We first give some explanations on the assumptions. exploitation of the following equations, which S-Is = l, and from the definition

÷

directly follow from SS -I= U, or

(3) of the connected parts of the S-matrix

~_

÷ ~

i=1,2, 3

The method is based on the

=0

j=4,5,6

(22)

i=l ,2,3 j=4,5,6

in the respective regions s < (3~) 2, and s < (4~) 2. The bubbles and

stand

their

kernels.

for

connected

operatoRS

diagram

functions"

The "bubble

:

c m,n

and

'

(S

FB associated

-lc )

m,n with

~

,

respectively, each

bubble

or dia-

gram B are defined, in a way similar to the terms D e in, Sect.3.3, as integrals, over all possible on-mass-shell

four-momenta associated with the internal lines,

of the product of kernels associated with each bubble. They are always well defined as kernels of bounded operators c

-Ic

(e.g.

~ ~ ' ~ ' ~. .

is the kernel of the product .

of S 3 3 and (S )3 3 ), hence as tempered dzstrzbutzons, ' 4 '. . an overall 6 -functlon (if all Pk are not collnear) : FB

=

fB

× 64 (EPi- Zpj)

.

and contain, as in Sect.3.3.

(24)

280

Refined ma~oca~gbty

(see the end of Sect.3.1) is the essential support (7), c • . < 2 f c c c • . used here for $2, 2 In the reglon s (3~) and _or S 2 3' $3 2' $3 3 in the reglon 2< < 2 . ', ' ~ c (3~) s :(4N) , plus the regularity property R whlch will be applled to $3, 3 at #~o points in the region (3p) 2 < s < (4N) 2 (see below).

Uvlitarity (S-| = S t) is used here to ensure that the essential support of the minus bubbles is opposite to that of the plus bubbles at any physical-region point (and to ensure the regularity property R for (S-I)~,3). The essential support properties (7) and the corresponding essential support properties of the minus bubbles, together with the regularity property R, will yield information on the essential support of the terms F B (see App. 1). Although we are c (7) of $3, 3 are

interested only in the region R , the essential support properties needed in the full region (3~)2< s < (4~) 2 since

~o points are always encountered

in integration domains for the individual bubbles, for instance for the t e r m ~ The regularity property R is needed because all points P=(PI,...,P6) are u=0 points for this term ; hence, the standard results of essential support, or hyperfunction, theory, which apply only at u#0 points, give no information on its essential support, or singular spectrum. This absence of information would completely disrupt the proofs. The u=0 problem arises, in the region (3p) 2 < s < (4~) 2, from the occurrence, just mentioned, of M ° points in integration domains.

No sprout assumption Let us consider the decompositions

(20) of f provided in R

by macrocausality~

Being given a system of real analytic local coordinates ql,...,q]4 of #~ at a pointP + + of L (G B) in R , chosen such that L (G~) is represented locally by q1=0 and Q+(P;GB) is the direction of the positive u]-axis, and being given any open cone F with apex at the origin whose closure is contained (apart from the origin) in the half-space ql >0'

the analyticity properties of f~ stated below (20) ensure the existence of

a complex neighborhood ~ of P such that ~B is analytic in ~ 0 {IImq|E F}, where denote the complexified variables of q. By definition, fB satisfies the no s~rout

property

at P if there exists ~ such that ~B is analytic in ~ 0 {Imql > 0} .

A slight extension of the decomposition (20) in the

re~ion (3p) 2 < s < (4N) 2

will be useful. Namely, macrocausality ensures decompositions in that re~ion of the form : f

=

~

fB

+

a

(25)

B Where a is analytic in R and where each fB has the same properties as those stated below (20) in a region R B slightly larger than R . For the ~raDhs G],C 2 of Fi~.1, R B is obtained by removing only from the reEion (3D) 2 < s < (4~) 2 the points D of + L+(GI) , resp. L (G2), such that pl=p2 or p5=P6 , resp. P|=P2 or p4=P5 .

281

No sprout assumption : "There exist decompositions of the form (25) such that +

each fB satisfies the no sprout property at each point P of L (G B) in R B

Lcymma (see [Ii] : "the terms fB in the decompositions provided by the no sprout assumption are uniquely defined in R B modulo addition of analytic functions". This lemma is an easy consequence of Bremerman's continuity theorem.

Proof of the theorem We now give some hints on the proof of the theorem. quoted,

one h a s t o show t h a t

associated

fB - ~ ( d ~ )

i n R . We o u t l i n e

w i t h t h e g r a p h G1 o f F i g . l ,

Eq. (23)

c a n be w r i t t e n

the proof

for

the term fl

[11].

in the form :

+

where FI = fl × ~4(Xpi_XPj ) ,

following

In view of the lemma just

~

3

4

+ H1

=

0

(26)

is defined as previously, as an integral

over internal on-mass-shell four-momenta (*) ,

3 ~Z~4

is defined from3Z~--~--4

in the Lame way as (DC)+ is defined from D c in Sect.3.3, and H 1 is the sum of all terms in the l.h.s, of (23), except that

~

,

are replaced by B#I E F B + A ' B ~ I ~ 4 + ~ 4

~

4

and 3 ~ 4 = 3 ~

and

3 ~ 4 4

3

~

4

(A = a x 64(Epi - Epj) where a is introduced in Eq.(25)). The two terms singled out with F 1 in the bracket of (26) are those which (after •.

.

.

4

factorlzatlon of thelr e.m.c. 6 -function) have at most, like fl' u+(P;GI) in their essential support along L+(GI ) and are analytic in R l outside L+(GI ). Moreover, their singularities are those that correspond to diagrams D B with a related topological structure (see App. 1). According to the ~eneral idea of "separation of singularities in unitarity equations" (see App .I), their singularities are expected to cancel among themselves. We shall prove below that this is the case, i.e. more precisely that h I is analytic in RI,

where H 1 = h I × ~4(~pi-~Pj). C

Once this is achieved, the desired result (fl e (dl)+) is obtained by replacing each term

~

i n (26) by

...~

,..~

internal on-mass-shell four-momenta), where

(again

~

in the sense of integrals

over

is as in Sect.3 the non con-

nected S matrix $2, 2 . In view of (22), the sum of the first two terms in (26) is

(*) F 1 is no longer k n o ~ to be the kernel of a bounded operator. However, this term, as also similar ones encountered below, are well defined by the standard u#0 results on products and integrals of distributions.

282

transformed into F I and the third term is transformed (see [11] into - 43 ~~ Hence :

FI =

3

~

4

~ E ~

4

(27)

The analyticity of h I in R I yields analyticity of the last term (after factorization of its e.m.c.

~4-function), and the result is therefore proved.

Finally, the analyticity of h| in R| is proved as follows : (i)

A detailed analysis[|0'|1]of each term contained in h| allows once to show

that ~+(P;G|) ~ ESp(h l) if P C L+(GI ) lies outside the union ~+ U N of an open subset ~+ of L+(G|) and of other lower-dimen§ional submanifolds . The equality (26) and the essential support properties of the three terms in the first bracket of (23) (see above) then imply that ESp(h I) is empty, i.e. that h 1 is analytic along L + (G|) outside ~+ U N

(ii)

It can be checked that the three terms in the bracket of (26) (after

faetorization of their e.m.c. ~4-functions), and hence h I by (26), satisfy, like f1' +

the no sprout property at any point P of L (G|) in R|. Hence by an application of Bremerman's continuity theorem (see[10,|1])h|,being analytic at some points of L+(GI ) is necessarily analytic all along L+(G|) in R I . O.E.D.

R~aYgks : I) the elimination in step (ii) of the possible ~+-singularities remaining in step (i) is crucial. Otherwise these @+-singularities would propagate all along L+(G|) for the term

~

4

and hence no result would be obtained anywhere.

We emphasize also that step (ii) applies to the Sum HI, but not to the individual terms in H! : some of them, such as

~

'3

'

'

are certainly expected to have Q+(P;G|) in their essential support along ~+ andoth~r submanif61ds (after factorization of their e.m.c.~4-functions). 2)( d ci)+ and hence fl

_

(d~)+

satisfies the no sprout property, as easily

checked, at all points P of L+(GI) in R|. However, unless "separation of singularities in unitarity equations" is

assumed, this fact cannot be used to prove ~ e o ~ y

from Eq.(27) that f1-(d )+ is analytic along L (G I) : the argument that would be similar to step (ii) above is not possible because several terms in such as at ~

or

3

--~--

4

have Q+(P;GI) in their essential support

points of L+(G|), (after factorization of their e.m.c. ~4-functions). 3) the proof would be completely discupted if macrocausality was used in

its form (6) rather than (7) : for instance it would not be possible to prove that

283

Q+(P;GI) is absent from the essential s u p p o r t o f ~ 4 . . + of its e.m.c.~ -functlon) at any polnt P of L (GI).

(after factorization

c 4) Previous proofs of the above result (f] ~ (dl)+) all start essentially modulo unessential variations, from the following equations, which replace (26)(27):

÷

14=0

(26')

(27')

where H is here simply the sum of all terms in the l.h.s, of (23) except that and

~

(i)

4

have been rem°ved and

A' ~ 4

3is replaced bY

3~

The "proof" of [5] relies on the assertion that all terms in H are analy-

tic, o r l i k e

,

zation

of their

The

proof

(ii)

e.m.e. of

[5]

i~ boundary values,

minuS

64-functions. is then

all

along L (~t) , after

This is however not correct

completely

factori-

( s e e Remark 1) .

iuined.

The proofs using the assumption of separation of singularities start

from Eq.(27'). The only terms whose singularities correspond to diagrams with a structure related to D+(P;G]) =

I-~-~=!~2-"_.5 are F I and 3 ~ 4 3 ~ 6 Hence the result is automatic if this assumption is used. Otherwise no result is obtained for the same reason as in Remark 2) above.

(iii)

The proof of [10] is close to that presented above, but there are some

complications and a supplementary step is needed, in particular because h, in contrast to hl, cannot be expected to be analytic along ~+ . (H-H| contains the term ~

4

which has ~+(P,G|) in its ess.support along ~+) -

Once the above result on f; (f; ~ (d~)+ in R]) is obtained, a similar analysis allows one to determine the contributions fB associated with triangle graphs, such as f2" One starts again from (23) and one now singles out the contribution F 2 to ~

, together with the terms

~

6

and

3

~

6

whose

singularities along L+(G2 ) correspond to diagrams with a related topological structure. The arguments are similar to above, although somewhat more subtle : see details in [11].

284

6. AXIOMATIC FIELD THEORY The

basic quantities of axiomatic field theory for the purposes of this sec-

tion are the connected "chronological functions" ~c which are, for each n, well defined distributions i n the space R 4n of n space-time vector variables x],...,x n and are in fact (possibly regularized) connected, "amputated" vacuum expectation values of the dhronological product T(x,j,...,Xn ) of n field operat6rs A(x~)i .... 'A(Xn)" . . . . 4n Their Fourier transforms T are correspondlngly deflned in the space R of enerF~yc momentum variables pl,...,pn o In view of the translation invariance of Tc under translation of all x k by a common space-time vector a, ~c contains an energy-momentum 4 4 n conservation B -function B ( ~ pk ). It can be shown (see below), as first established k=l in [20], that, being given any process m ÷ m', with m+m' =n, the distributions c can be restricted (as distributions) to the mass-shell Mm,m,={p=(pl,...,pn) ; 2 2 Pk = U , Vk, (Pk)o ~ O , k=],...,m , (Pk)o > 0, k=m +l,...,n}, and the following relation holds : c m, (Pl , •. .,pm;Pm+1 , . Sm,

= ~c(-p], .,pn.) . . .

,-Pm;Pm+], • ..,pn) IMm,m,

(28)

We shall also consider functions (Tc) I , where I is a subset of (|,..°,n),which are defined in a way similar to Tc, except that T(Xl,...,x n) is replaced by the product of T(x(J)) and T(x(I)), where J is the complement of I in (I .... ,n) and x(I), resp. x(J), denote the sets of points x k , k E I, resp. k C J.

Microcausality It asserts that the commutator [A(x),A(y)] of two field operators vanishes if (x-y) 2 < 0, where x2=x 2 - ~ 2 , O that :

i.e. if x-y is space-like, and entafls(see

Tc(Xl,...,Xn) = (Tc) I (Xl,...,Xn) where Z I = {(x I ..... Xn) , xj-x i ~ 0 The condition x.-x. > 0

if (x| ..... Xn) ~ Z I

[15]) (29)

for any pair of points x i in ~(I) and xj in x(J)}. •

-

2

means either that x.-x. is sDaceilke((x.-x.) < 0 )

or belongs

i ~ of polnts x in space-t ~melsueh-that x2 > ~ ' ~ o > 0) '. to the cone --+ V j(= set

The

spectral condition

is the assertion that the spectrum of the energy-momentum

operator of the theory is contained (for a theory with only one type of particle of mass N > 0) in the union of the origin (corresponding to the "vacuum"), of the positive-energy mass-shell hyperbolo~d V + (~) (P2 =~ 2 'Po > 0 ), corresponding to oneparticle states, and of the continuum V (2~) (p 2 _> (2U ) 2 , Po > 0). It entails[15]that (Tc) I has its support in the region Z jeJ

pj (= - Z pi ) E ~ ( ~ ) iEl E V+(2~)

U V+(2U), if ! < IjI < n-] (30) , if IJl = | or n-] .

285

Being given a point P in %,m''

let Sp be the set of all I such that (Tc)l

vanishes in the neighborhood of P. It is then easily seen that [15] • ESp (Tc)

=

N

2I

(31)

16 Sp

Proof

: By (29), TC-(TC) I = 0 if (x I.... 'Xn) ~ ZI" Hence, for any P, ESp(Tc-(Tc) I)

c ZI (see lemma below). If 1 6 Sp, (~c)l = 0 locally. Hence ESp(($c) I) is empty (apart from the origin) and ESp(Tc) ~ ZI ' since Tc=(TC-(TC) I) + (Tc) I. We have used the following elementary lemma (see e.g.[21])

: if the Fourier

transform f (here T -(TC) I) of a distribution f (here $c-(Tc)l) vanishes outside a cone C with apex at the origin, then the generalized Fourier transform of f at ~ny point P falls off exponentially in the appropriate sense outside C, i.e. ESp(f)c C. (Note that the analogues of the variables x of the mathematical lecture are here the energy-momentum variables pl,...,pn , and not the space-time variables). By standard results of essential support theory, (31) entails that the restriction of ~c to M,m,

does exist (away from M ° points), and it ~ives information on

the essential supports of Sm,m,C,hence of the saattering function fm,m' at any point P of Mm, m, : they are contained respectively in the sets associated with all possible configurations of trajectories (Pk,Xk) corresponding to all points (x|,...,x n) in ESp(Tc) , resp. relative configurations of such trajectories. Four-point function (two-body processes) For a two-body process (m=m'=2,n=4),

the result (31) reduces, at any physical

region point P, to : ESp(~c) c where V+ is

{x=(x I .... ,x4) , Xl=X2, x3=x4, x3-x I 6 ~+)

(32)

the set of points x in space-time such that x 2 ~ 0,x ° ~ 0 . By Lorentz

invariance, this result can moreover be improved, namely the Condition~x3-x I 6 ~+ can be replaced by x3-x1=%(P1+P2), % ~ 0 . The possible points in ESp(Tc) and ~Sp(S~, 2) are represented in Fig.3 time

Fig. 3

286

Let us consider a point P in the region (2H) 2 < s < (3H) 2. There, a better result is expected according to the macrocausality energy-momentum

ideas, namely x]=x2=x3=x4 , since

cannot be transferred from x1=x 2 to x3=x 4 by stable particles if

x 3 # x]. This result can be obtained (see [|~and references) by a regularity assumption which eliminates N priorl possible ~ la Martin pathologies,

corresponding physically

to the possibility of having sequences of unstable particles, with arbitrarily rates of exponential decay, by which energy-momentum might be transferred. exclusion of such pathologies macroeausality

is also part of the physical arguments upon which

is based : see Sect.3.|).

We admit below correspondingly ESp(~ )

small

(The

c

Xl=X2=X3=X4 }, V P s.t.(2~)2< s<(3~) 2

{(Xl,X2,X3,X4),

Equivalently,

that :

T , after factorization of its e.m.c, ~ 4 ' f u n c t i ° n ' c

(33) and f 2 , 2

are

analytic at P. Five-point function (2+3 or 3+2 processes) 2 2 We next consider a 2÷3 process in the region s<(4H) 2 (s=(p1+P2) =(p3+P4+P5 ) ). In this case,

(31) entails that : ESp($c)

c

C 3 U C 4 U C5

,

(34)

where 3,4,5 denote as previously the final particles and C k = (x=(x I ..... x5);Xl=X2,Xl.=., xj k=3,4,5,

(i,j,k) = (3,4,5). The cone

xi-x k C g +

Xk-X | E V+}

(35)

C 3 is represented in Fig.4a)

~3 4 = X5

Xl=X

time

/

/

a) Fig. 4

This result, which cannot be improved in a simple way by Lorentz invariance, is very far from the essential support expected from the macrocausality corresponds

ideas, which

in the region s < 4H 2, and away from M ° points, to x|=x2=x3=x4=x5

. We

explain below how the result (34) can be improved by using the further axiom of

a~ympto~c completeness,

together with the previous result (33) on the four-point

function. Let

( T c ) i , ~ = ~c -

I , where

I = (1,2,k),

(i,j,k)=(3,4,5)

and let

287

k

and

~

i

represent diagrammatically

T c and (Tc)i, j respectively.

Micro-

causality and the spectral condition yield the following information, which is the analogue of (34) :

ESp

(

2

~

b

) c C i U C.3 U

C-

(36)

k

__+ where C k = {x=(x I ..... x5) , Xl=X2, xi=xj, Xk-X | E V , Xk-X i E V+}. Asymptotic completeness yields on the other hand the followinF~ relations in the region s < (4~) 2, obtained by introducing in the appropriate place a complete set of intermediate states and by standard analysis of axiomatic field theory (see e.g. [22])

:

where the last term in the right-hand side is defined in the same way as bubble diagram functions in Sect.5, as aconvolution four-momenta

integral over internal on-m~s-sh6g£

(with the measure ~(k~-N2)@(k£)o)d4k£^ for each internal llne). The

only difference is that the external four-momenta are not restricted to the massshell. In

view

of

(33) (36), the standard

theory (or hyperfunction

u#0

results of essential support

theory) on products and integrals

(see App.l) show that

this integral is well defined and give information on its essential support. Moreover, a systematic use of the relations

(37) for the various values of k=3,4,5,

yields :

Theorem[13] "ESp ( ~

)

c

C = {x;x|=x2,x3=x4=x5,x3-xl

E V +}

(38)

at any point P of I~2,3 s.t. s < (4U) 2, (Pi+Pj-Pk)2 # U 2 where i,j,k is any permutation of (3,4,5)" The exclusion of the codimension one submanifolds the M ° points)

(Pi+Pj-Pk)2=~ 2 (which contaln

in the results obtained so far in [13], is due to reasons that we

shall not discuss here in detail. Lorentz invariance implies now, as for the fourpoint function, an improvement of (38) in which the condition x3-x I E ~+ is replaced by x3-xI=%(P]+P2 ), % ~ 0 . The situation is then the same as that shown in Fig.3 for the 2+2 processes,

except that x3=x 4 is replaced by x3=x4=x 5. To reobtain macrecau-

sality (see above), a regularity assumption for 2+3 processes would again be needed to avoid the ~ priori possible g la Martin pathologies.

Six-point function (3+3 processes) We shall now consider, as in Sect.5, a 3+3 process in the re~ion s<(4~) 2.

288

Generalized reynman integrals,

~ J i ~ "

such as

i

~

j

are

defined in a natural way in axiomatic field theory~4,13]where the bubbles are not defined ~ priori on-mass-shell, by associating Feynman-type propagators to each internal line. It can be checked that they coincide,

after restriction to the c mass-shell, and modulo analytic backgrounds, with the (D6)~ introduced previously.

The analogue of (34), which follows again from microcausality and the spectral condition, is here :

I

-

=

U

i=1,2,3 j=4,5,6

s=1,2,3 t=4,5,6 --+

where e.g.

--+

C 41 = {x,x2=x3,x5=x6,xl_x2 E V ,x5-x 4 E V , x4-x I C V +} (Fig.4b)).

The asymptotic completeness equations are the three equations (37) (with three lines instead of two on the left of the 3+3 bubbles) and three analogous equations ~volving terms

~

(i,j,k) = (1,2,3), k=1,2,3. As previously, micro-

causality and the spectral condition yield information also on

~ , or ~ r The full use of all previous informations, together with the standard

u#0 results, then yields : [13]

" ESp( ~

Theorem

c

- [

i ~

j

~- i ~ ~ ' ) (40)

{x; x1=x2=x3 , x4=x5=x6 , x4-x | E V+}

at any point P of M3, 3 s.t. s < (4~) 2, (Pi+Pj-Pk)2

¢ 2,

where (i,j,k) is any

permutation of (1,2,3) or of (4,5,6)" The condition x4-xl E V ~ 0 by Lorentz inv~riance.

can, as previously be replaced by x4-xI=%(PI+P2+P3 ), A supplementary regularity assumption (for 3+3 processes)

would be needed to avoid ~ la Martin pathologies, i.e. to replace this latter condition by Xl= ... = x6, and thus to reobtain the decomposition (21), equivalent to macrocausality and macrocausal factorization, on the mass-shell. As in the case of the five-point function, the submanifolds

(Pi+Pj-Pk)2=~ 2 are excluded so far in the

results of [13].

Remark : Although this has not yet been established so far, it is believed that the terms

~

, or

~

, after ~estriction to the mass-shell and facto-

rization of their e.m.c. 64-function, can be obtained from the scattering

289

function f3,3 by analytic continuation, on the complex mass-shell, around twoparticle thresholds. In S-matrix theory, these analytic continuations, and the mass-shell versions of Eq.(37) or of its analogues for 3+3 processes are not known at the outset, although one would like to reobtain them at a later

stage. Hence only physical

on-mass-shell nnitarity has been used in Sect.5.

7. TWO-DIMENSIONAL SPACE-TIME [16'|7] We first consider, as in [16], a theory with only one type of particle and a 3+3 process. The physical-region can be divided into sectors, each of which corresponding to a given relative position of the three initial two-momenta and of the three final two-momenta (in two-dimensional energy-momentum space). M ° points lie on the boundary of these sectors. In each sector, one encounters eight +s-Landau surfaces of connected graphs associated with graphs with one internal line and triangle graphs, which all coincide with the real analytic codimension one surface of the physical region defined by requiring that the sets of initial and of final two-momenta should coincide, e.g. p1=P6 , p2=P4 , p3=P5 in one of these sectors.(These conditions reduce to only one condition in ~3,3 in view of the mass-shell constraints and of energy-momentum conservation : pl+P2+P3 = p4+P5+P6 i . The situation is then the same as that described in Example 3 of Sect.4, except that these +s-Landau points P belong now to a codimension one surface and not to a low-dimensional surface of the physical region as in four-dimensional space-time. The directions @+(P;G B) and hence the plus i~ directions associated with two different subsets of these surfaces are opposite all along ~ and the scattering function f3,3 is thus not along ~ the boundary value of an analytic function. In contrast to the four-dimensional case, the structure of the connected S matric is not simpler in two-dimensional space-time than that of the non connected S matrix.

This is related to the fact that(non colinear)space-time trajectories ne-

cessarily meet in this latter case . In the neighborhood of a point P of a given sector that may lie on E but lies on no other +s-Landau surface, the structure of $3, 3 implied by macrocausality and macrocausal factorization, can be determined as follows. The terms D], D 2 associated with the two diagrams of Fig.2 of Sect.4 coincide (and vanish outside ~) : D]=D2=D with D

= R3,B(Pl ..... p6 ) x

~

s2,2(Pi,Pj;Pi+3,Pj+3)

(41)

i,j=],2,3 where two-body kinematics in two-dimensional space-time (~2(pl+P2-P3-P4) reduces to the kernel ~2,2 (PI'''''P4) of the identity) has been used. Macrocausality (Eq.(6) for 2+2 processes) ensures that each s2, 2 is locally analytic. Macrocausality (Eq.(6)) and macrocausal factorization (Eq.(10)) then imply

290

in a straightforward

way [16] that ESp(s3,3-d ) is empty (apart from the origin),i.e.

equivalently :

$3,3(P1 .... ,p6) = n3, 3 (pl,...,p6) x

]~ i<j

s2 ,2 (Pl 'Pj'Pi+3'Pj+3)+R(Pl ..... P6 )

i,j=] ,2,3 2

(42)

where R = r × ~ (p1+P2+P3-P4-P5-P6) and r is locally

an~y~7~c in

the neighborhood

of P : i.e. $3, 3 "factorizes into a product of two-body S matrices" modulo an analytic background. Let us now consider the two following properties that are established, or believed to hold, in a class of models : (i) absence of particle production (i,e. S

= 0 if m # n) and (ii) coincidence of the sets of initial and of final m,n two-momenta (i.e. for the three-body case $3, 3= 0 outside Z). These properties cannot be expected in four-dimensional space-time, but can be expected in twodimensional space-time. They are also related to each other by (heuristic) argument~ Property (i) entails that the only relevant +m-Landau surfaces are those mentioned above in each sector. Property (ii) then implies that r is zero outside Z

and is

thus identically zero since it is analytic in the neighborhood of P. Hence one obtains exact factorization. This result can first be extended by induction to the multiparticle S matrix Sm,m~ m > 3, in models in which properties (i) and (ii) are satisfied : see [16] . It can on the other hand be extended to models with several types of equal-mass particles as we now explain for the three-body S matrix. Being given the external particles, there can exist several diagrams ~+ for each one of the diagrams of Fig.2, depending on the types of particles associated with the internal lines. The term DI,D 2 involved in each case in the statement of macrocausal factorization are then sums of the corresponding contributions and are no longer g priori equal. However, it is easily seen that macrocausality and macrocausal factorization, together with properties (i) and (ii), imply that D I = D 2 and $3, 3 = D I = D 2 in each sector : see [17] . The equality D 1 = D 2 entails

co~istency r ~ £ ~ o ~ ,

also called

factorization equations, between sums of products of (non connected) scattering functions, such as those used in the lectures by Berg and references therein.

APPENDIX ] - ESSENTIAL SUPPORT, STRUCTURE THEOREM AND RELATED TOPICS The essential support [21] of a distribution f defined i n ~ n

(resp. in a real

analytic manifold M) is, at each point X, a cone with apex at the origin in the dual space (resp. in T X M) composed of "singular directions" along which a generalized Fourier transform Fx(f ) 6~ f at X does not decrease exponentially in a well specified sense ; f is analytic at X, resp. is at X the boundary value of an

291

analytic function from the directions of an open cone F (in the space of imaginary parts of the variables),

if and only if ESx(f) is empty (apart from the origin),

resp. is contained in the closed convex salient dual cone C of F . The notion of essential support characterimes more ~enerally possible decompositions

(in ~eneral non unique)

of f, in the neighborhood of a point, or over

real regions,

sums of boundary values of analytic functions from specified directions depend on the real point considered): This characterization

into

(which

see [2]] and references therein.

coincides with that obtained independently

in hyper-

function theory [23] in terms of the notion of singular spectrum, except that boundary values occurring in general in decompositions in [23] even when f itself is a distribution.

of f may be hyperfunctions

It is, however, proved in [24] that

the two notions do coincide. Following the language of [23], we say by definition that f is

~cT~oav~ly~¢

at the point X and in a given direction @ (of T ~ )

if

Q ¢ ESx(f ) E SSx(f ). It follows from the definition of the essential support that ESx(fI+f2) c ESx(f|) +ESx(f2) , and that ESx(f ) c ESx(f/) N ESx(f 2) if f=fl=f2 in the neiqhborhood of X. This latter result is a useful generalization of the edge-of-the-wedge theorem. These two results are used extensively in Sect.3 to 7.

Structure theorem on bubble diagram functions

(S-matrix theory)[~ '12]

Being given bubble diagram function F B such as those encountered in Sect.3 to 5 (= integrals, over internal 6n-mass-shell ' ~

, or

on-the-mass-shell),

~

four-momenta,

of products of distributions

, or FI,F 2 .... associated with each bubble and defined the general u#O results of essential support {or hyperfunction)

theory on products and integrals of distributions

entail that their possible sin-

gularities are associated with space-time diagrams,

as explained below. Being given

the bubble diagram B, let us first define diagrams ~B : they are collections of subdiagrams g b that are associated with each bubble b of B and "fit together". More precisely,

each gb is a configuration of incoming and outgoing trajectories asso-

ciated with the incoming and outgoing lines of b and representing

(see the beBinning

of Sect.3) a possible point in the essential support of F b. The various ~b fit together if, being given any internal line ~ of B, which is outgoing and incoming at a bubble b2, the two trajectories

from a bubble b 1

associated with % in eb] and gb2

respectively coincide. They are then identified as an internal line of gB " The various E b encountered in applications

are, by macrocausality,

rations of external trajectories of diagrams D+ if b is a bubble a bubble FI,F2,... , or (by unitarity a minus bubble. The configurations

configu-

----~÷~---

or

: S -I = S t ) of opposite diagrams 9_ if b is

of external trajectories of diaerams gB will thus

be configurations of external trajectories of diagrams D B whose internal lines are either the original internal lines of ~B or are internal lines of subdiagrams D b .

292

A point P = {Pk } (which is a set of external four-momenta)

is a u=O point of B

if there exist "u=O" diagrams EB , or ~B' whose all external trajectories pass through a common point, e.g. the origin, (and have the respective four-momenta Pk) , whereas some internal trajectories of gB do not pass through this point. Away from u=O points, the essential support of F B is (contained in) the set determined by all possible configurations of e x t ~ 7 ~

,~7~ajec£o~

of diagrams EB, or ~B" At u=0 points,

all directions may ~ priori be singular and F B is not necessarily ~ priori well defined. On the other hand F B is well defined (see [12]) if the bubbles are kernels of bounded operators, i.e. for instance if B =

~ ÷ ~ - ~

. If the regularity

property R holds for each Fb, then the essential support is contained in the set determined by considering all possible limit configurations of external trajectories of sequences of modified diagrams in which each internal trajectory is replaced by a pair of trajectories (one for each bubble) which must tend to each other in the limit. (The internal trajectories that

are considered here are g priori those of

gB " They may also be the internal trajectories of subdiagrams D b if the approach of [12] tO macrocausality is adopted, but their doubling does not seem to be required apart from exceptional cases,: see App.2). Configurations of external trajectories of different diagrams D B may coincide, for a common or for different bubble diagrams B. At u=0 points of a given B, all directions should on the other hand be associated with the above "u=0 '' diagrams DB, if u=O results are not known. " S e p ~ 0 n

of s i n g ~ ~ "

in unitarity equations

is the assertion in all cases that singular contributions to the various bubble diagram functions that correspond to diagrams ~B with a related structure (see details in Ch. III of [1]) should cancel among themselves, independently of the other possible singularities corresponding to diagrams with a non related structure.

Structure theorem (field theory) The bubble diagram functions encountered in field theory are again integrals over internal

on-m~s-shd.1 four-momenta

(see Sect.6), but the external four-momenta

are not g priori restricted to the mass-shell. A diagram c B is then a collection of subdiagrams gb' each of which is now a configuration (= set) of oo£~P~ in space-time associated with the incoming and outgoing lines of b, this configuration representing a possible point in the essential support of F b for given values of the external four-momenta and given values of the internal(on-mass-shell) four-momenta K% in the integration domain. The Eb must moreover fit together in the sense that, being given any internal line ~ of B, outgoing from b] and incoming at b2, the points (x%) I and (x~) 2 that represent it in gb] and Eb2

must satisfy : (x%) 2

where %~ is an arbitrary real scalar.

(x%) ]

=

%%~

(43)

293

A u=0 point P = {Pk } of B is now a point such that there exists a diagram ~B whose all "external" points have a common space-time position

(e.g. the origin),

whereas some "internal~! points are not there. Away from u=0 points, F B is a well defined distribution and its essential support is (contained in) the set corresponding to all possible configurations

of exter-

nal points of diagrams gB "

APPENDIX 2 - COMPLEMENTS ON MACROCAUSALITY AND MACROCAUSAL FACTORIZATION Macrocausality Exponential decay cannot be expected if there exists a diagram ~+(P,u), the trajectories

0h if

(Pk,Uk) can be obtained as limits of the external trajectories

a sequence of modified diagrams,

of

in which some of the constraints of the diagramsg+

are relaxed, if the violations tend to zero in the limit. These considerations

are

at the origin of the introduction of vertices at infinity, satisfying angularmomentum conservation,

in [6] , where certain violations to ordinary diagrams D + (with no vertex at infinity) are considered by analogy with results obtained there

on phase-space conditions).

integrals

(= violations of the loop equations and of the mass-shell

The consideration of limiting procedures

is then no longer necessary

(each sequence of modified diagrams giving rise to a limit diagram ~+(P,u)), from the special class of M

away

points P, if they exist, for which there exist "u=0"

o diagrams ~+(P) whose all external trajectories pass through the origin (and have the four-momenta Pk) , but such that the non parallel vertices do not all lie at the origin.

(A non parallel vertex is a vertex at which the incoming and outgoing

trajectories are not all parallel). A further physical discussion of macrocausality,

with similar conclusions,

is

given in [12], where the possibility of vertices at infinity is accepted from the outset, and in which, in addition to violations of strict locality at each vertex, a doubling of the internal trajectories

is considered.

This doublin~ accounts phy-

sically for the fact that, in the quantum case, internal particles cannot be strictly localized along classical trajectories trajectories

: the consideration

of all possible classical

(K~,W%) for each internal line ~ should thus be replaced by the consi-

deration of all possible wave functions of the form (5) corresponding trajectories

to all possible

(K~,W~), in agreement with the fact that these wave functions, with

X% = I and appropriate normalization

factors N(y,T),form a complete set of interme-

diate states : I

dK~ dW~

where W%=(0,W~).(K%,W%

~,T;K~,W~>

~,T;K~,W~

I

=

~

are for each ~P%,T the analogues of Pk ' Uk in (5)).

(44)

294

Macrocausal factorization We consider, as in Sect.3.2, cases in which there exists only one D+(P,u) and we shall denote here by (K~,W%) the well determined internal trajectories of this diagram. The variables K%,W~ in (44) will thus be replaced by K~,W~ . We denote below, on the other hand, by D(P,u)

(non causal) diagrams ~ (if they exist) with

the same external trajectories (Pk,Uk) ~B in App.1

,

obtained in the same way as the diagrams

by replacing here each vertex v of D+(P,u) by a subdiagram 9+ (not

necessarily reduced to a single vertex). The well determined internal trajectories of D(P,u) that replace the internal trajectories of V+(P,u) will be denoted (K%,W%). If there is no set of more than one internal trajectory between two common vertices ^

^

of D+(P,u), there is in general no D(P,u) whose trajectories (K%,w%) lie, for each ~, in a given neighborhood of (K%,W~) (~) Macrocausal factorization is then the physical requirement that S({¢.0k,T}) is asymptotically equal, modulo exponential fall-off in the T ÷ ~ limit, to the integral, over sufficiently small neighborhoods N% of (K~,W%) in (K~,W~)-space,

for

each ~ , of the product of transition amplitudes S({~0k,T;~p%,T,K.,W,} I % ~ ' ) associated v with each vertex v of ~+(P,u). In usual cases, such as all those encountered in Sect.4, there exists no D(P,u) at all. The methods of [12 ]

then allow one to show that the integral over the

complement of the product of the neighborhoods N~ falls off exponentially with T in the appropriate sense. The statement of macrocausal factorization given in Sect.3.2 follows,

in view of (44).

REFERENCES [1] - D .

IAGOLNITZER, The S Matrix, North-Holland, Amsterdam (1978)

(or for Chs.I and II, the earlier version of this book : I~oduc£~on to

S M~x

Theory~ ADT, Paris (1973)). [2] - G.F. CHEW, The Analy£~c S M ~ x , W.A. Benjamin, New-York (1966) [3] - D. IAGOLNITZER, H.P. STAPP, Comm. Math. Phys. 14, 15 (1969) This work is a development of several earlier works, including in particular : C. CHANDLER, H.P. STAPP, J. Math. Phys. 90, 826 (1969). [4] - D. IAGOLNITZER, in L e c t ~

i~ Theoretical PhysXc~, Vol.Xl D, ed. by

K.T. Mahanthappa and W.E. Brittin, Gordon and Breach, New-York (1969), p.229

:

(*) This is no longer true if there are sets of ~ > 1 lines between some vertices. Then, one is led to associate with these vertices modified S-matrices, corresponding to remove the interactions between the particles of these sets, and to add a new vertex on each set, to which an actual S-matrix is associated. In usual cases, this procedure leads alternatively to the statement (10) with D being defined as in Sect.3.2 with actual S-matrices associated to each vertex, except that a new vertex is introduced as above on each set, to which the inverse S -| of the restriction of S to'~ = Q ~'m is associated. m > ~

295

[5] - R.J. EDEN, P.V. LANDSHOFF, D.I. OLIVE, J.C. POLKINGHORNE, Cambridge Univ. Press.

(1966), ch. IV.

M. BLOXHAM, D.I.OLIVE,

J.C. POLKINGHORNE,

The Analytic S ~a~rix

J. Math. Phys. 10, 494, 545, 553

(|969) D.I. OLIVE, in Hyperfunctio~ and Theoretical Physics, Lecture Notes in Mathematics, Springer-Verlag, [6] - M. KASHIWARA,

Heidelberg

(1975).

T. KAWAI, H.P. STAPP, Comm. Math. Phys. 66, 95 (1979).

[7] - J. COSTER, H.P. STAPP, J. Math. Phys. 10, 371 (1969), 11,

1441 (1970), 11,

2753 (1970). [8] - D. IAGOLNITZER,

Comm. Math. Phys. 41, 39 (1975).

ST~tuctcota~.Analys%ya of Collision A m p ~ d e s ,

[9] - H.P. STAPP, in

and D. Iagolnitzer,

North-Holland,

Amsterdam

ed. by R. Balian

(1976), p.275.

[10] -D. IAGOLNITZER,

H.P. STAPP, Comm. Math. Phys. 57, I (1977).

[11] -D. IAGOLNITZER,

Macrocausality,

Multiparticle

[12] -D. IAGOLNITZER, [13]-H.

Unitarity and Physical-Region

S Matrix, to be published in Comm. Math. Phys.

Structure of the (1980).

Comm. Math. Phys. 63, 49 (1978).

EPSTEIN, V. GLASER, D. IAGOLNITZER,

[14] -J. BROS, Cours de Lausanne

in preparation.

(May 1979), and in preparation.

[15]-J. BROS, H. EPSTEIN, V. GLASER, Helv. Phys. Acta 45, 149 (1972). The direct proof given here of (31) is that of D. IAGOLNITZER, App. C in Part IV of Ref.1. [16] -D. IAGOLNITZER,

Phys. Rev. D, 18, 1275 (1978).

[17]-D.

IAGOLNITZER, Phys. Lett. 76B, 207 (1978).

[18]-F.

PHAM, in Hyperfunctio~ and Theoretical Physics, op. cit. in Ref.5 and

M. SATO. [19]-C.

CHANDLER, H.P. STAPP, Ref.3.

[20] - K. HEPP,

in

Lect~es in Particle Symmetries and Axiomatic Field Theory,

ed. by M. Chretien and S. Deser, Gordon and Breach, New-York [21] - D. IAGOLNITZER,

in

SOtu~alAnalys£~

vol.l,

(1966).

of Cogl~ion Ampli~de~,

op. cit. in

Ref.9, p.295. [22]- J. BROS, M. LASSALLE,

Comm. Math. Phys. 43, 279 (1975) and references therein.

[23] - M. SATO, T. KAWAI, M. KASHIWARA,

Equo.g6OVlTa,Lecture

in

~cTtofunetionS and Pseudo-c~ffgre~.Zal

Notes in Mathematics,

Springer-Verlag,

p.265. [24]- J.M. BONY, in Goulaouic-Schwartz

S@minaire, Paris (1976).

Heidelberg

(1973),

ANALYTIC 2-PARTICLE STRUCTURE AND CROSSING CONSTRAINTS

GUSTAV SOMM~R Fakult~t fHr Physik, UniversitHt Bielefeld, Germany

ABSTRACT : Some aspects of 2-particle structure and crossing symmetry in general local quantum field theory are reviewed.

I. TWO-PARTICLE-STRUCTURE IN GENERAL QUANTUM FIELD THEORY AND CROSSING SYMMETRY The n-particle structure of momentum space Green's functions in general quantum field theory has been studied extensively by Bros I) and by Bros and Lassalle 2) . The starting point was to define for euclidean momenta a Feynman convolution corresponding to a definite diagram with vertex functions having the analytic properties of general quantum field theory. This allows for an extension to the primitive holomorphy domain and for the definition of p-particle-irreduclble Green's functions through certain structure equations. For details see an article in these Proceedings 3) . Here we review what is known in the simplest case of many-particle structure, i.e. for the 2-particle structure (n=2) of 4-point functions 4). The aim then is to study the Green's functions far away from the euclidean region and from the primitive domain, and in fact near the intersection of two physical energy regions where extra constraints result from crossing symmetry and asymptotic completeness. The frame for the discussion of 2-particle structure given below is axiomatic Local Quantum Field TheoryS)(LQFT) together with Asymptotic Completeness

(AC). By LQFT we

understand here that there are given a) a unitary representation of the (proper) Poincar& group on a Hilbert space with an invariant vector

~

and such that the translation part of the representation

has the support of its measure in ({(pO)2_~2= o}U ~p°)2-~2=m2>o}u ~pO)2_~2= =4 m2+ ~+}) N{ p~° o}

(spectral condition), and

b) a Poincar~-covariant operator-valued tempered distribution (the "field") A(x), (x°,~) 6 ~4, acting cyclically on

~

and such that two fields

mute for space-llke separations (x-y)2< By

AC

A(x), A(y) com-

o (microcausality condition).

we denote the assumption that the underlying Hilbert space is equal to the

Hilbert spaces spanned by the incoming resp. outgoing states, these two spaces in fact being the same.

297

The assumptions of LQFT guarantee that functions H(q)(k)

defined in terms of the

Fourier transforms of 4-point generalized retarded functions of the type r(o)(XlX2XlX4):=(~j

E H ~=~(1,2,3,4)j=I

O~ 0 ( ~ ( x ° .... -x ° ))A(x_.)A(x ~)A(x ~)A(x ~)I ~) 3 3 ~3-l) ~j ~1 .z ~J H~

4

H(o)(klk2klk~)~(~j=ikj):=j~l(k~-m2) are holomorphic

•

(~r(o))(klk2klk~)

in certain tubes. In these definitions

trary sign factors between positions

o~3 = ±I

(o)= { (o~,..o~)}

fulfilling o j ~k~ = o~l V j#k

if

~k

are arbi-

is a transposition

(k-l) and (k) and ~ an arbitrary permutation of (l,2,3,4). Taking

into account the spectral condition all such functions

H(o)(k )

can be continued

through parts of the real faces of the tubes ("coincidence regions") and are in fact • domain is one slngle analytic function H(k) holomorph~c " ~n " the " pr~mzt~ve " " " " " whzch " the union of

~ : = { (k~ ,k2,k3,k~)IEk.:o,k.:p.+iq. ,p.E IR~, (q~l 'q~2'q~3 )EVO~U~ JJ J J ] ] ®3~ or

(q~1+q~2+q~3,q~2+q~,q~3+q~)£V+UV_}

with the coincidence regions in the boundaries of the tubes. From Poincar~ of the fields the function

H

on the manifold F = ( (k);Ik.=o} j 3

may as well be con-

sidered as a function !

,

H(s,ti~,~') = H(kl,k2,-k~,-k2) ,

,2

of the Lorentz invariants

~I

'

~I '

~i=ki,2 ~i=ki (i=],2) , ~=(~2),~=(~2,),

'

s--(kl-kl)2,t=(k1+k2) 2

and is an analytic function also in these variables ~(~) of a d o m a i n ~ c o r r e s p o n d i n g

to ~ .

From

AC

in the envelope of holomorphy

there exists a unitary S-operator

whose matrix elements between two-particle states can be expressed in terms of the function ties of

H . Also from H

partly reexpressable by

(])

, the discontinuities

H

properties,

A

across certain cut-singulari-

AC

Green's function

(~,R+R~),

itself, and have positivity relations of the type 6)

S A(K+I,K-I,-K+I',-K-I') wxw rr

for all test functions

ture,

AC

can be represented by syrgnetric Green's functions of the form

~(I)~(I')

~ £ ~ ( ~ r ), ~r C ~

d41 d~l ' ~ o

and

K

a fixed 4-vector. Using these

can help in various ways to elucidate the analytic structure of the H:(i) enlargement of the holomorphy domain,

(ii) monodromy struc-

(iii) crossing constraint equations on boundary values and the existence and

uniqueness of solutions.

Being interested primarily in (iii), further results of

the types (i) and (ii) are required for this purpose. Starting point for the considerations

is the fact, proven by Bros

allows one to define by the Bethe-Salpeter

type of equation

, that

AC

298

(2) H(k~,k2;k~

,

to

d;

2

.2

i= l

1

-i

u,

l,

,

,

~-k~')=Z~k~,k~,-kl,-k~)+ ½fZt(k~,k~, -k~k2)~(k. _~2) H(k~,k~,-k~rk2) "

,, ,, d4kld~k2

or, graphically:

'=

H

=

,

It

+

It Q~

H

the so-called t-channel 2-particle irreducible Green's function

It

which with

has the primitive analytic structure connected to the domain ~ a n d

H

moreover is

irreducible with respect to 2-particle intermediate states in the sense that it has vanishing discontinuity across singularity of

c~l:={(k)6F]4m2<(k1+k2)2~t<16m 2}

H . Inside the primitive analyticity domain ~

which is a cut-

resp. ~

, equation

(2) may as well be formulated in terms of the invariant variables:

(2') H(s,tl~,~') = It(s,t]~,!')+fds'ds"d2~"K(s,s',s",t

~,~",~')It(s',tl~,~')" • H (s",t I~",i' )

where

K

is a well-defined kernel function. Apart from the fact that equs.(2) or

(2') define I , in either case H can be represented as a Fredholm series of t iterated "Feynman convolutions" I t ~ t l ~ t . . ~ t l t . On the other hand, from the existence of fields commuting at space-like separations and from the resulting algebraic structure of retarded functions,

(3)

H(s,tl ~l~z'~ = H(t,s ~I~2 ~2~2''

H

has a property known as crossing symmetry

~ = H(u,tl ~I~2')

~2~1'

~1'~2''

'

and expressing the fact that the boundary values of

u=Z(~j+~j)-s-t, " 3 H

in the physical regions

corresponding to the 2-particle processes (I)+(I') ~ (2)+(2') , (I)+(2) ~ (T')+(2'),(l)+(2') ~

(2)+(T')

have to be equal (in the sense of distributions). Now this syrmnetry shows that can, e.g., also be represented by a Fredholm series of iterations of s-channel Feynman convolutions functions

~)s

IsQsI~s..~sI s

of s-channel 2-particle irreducible Green's

I . This fact strongly constraints the boundary values of S

H

H , and, in

particular, on those parts of the boundary where two physical energy regions intersect. To exploit such a constraint effectively, the simultaneous validity of the two Fredholm series not only in the primitive domain is needed, but rather near the interesting boundary point. While any term in the series is holomorphic (or at lea~t meromorphic) i~ ~ ( ~

, it is not at all trivial that any such term can be represen-

299

tea Dy an zterated convolutlon

~wzth integrations

in the analyticity domain) when

analytically continued to points which can be reached only by a complex completion of the primitive domain or by positivity arguments. Of particular connection is the discontinuity across

C := { (k)Eri4m2<(kl-kl')2zs <~} of equation (2) or (2') S

(A s )

is taken. Up to

,

some syrmmetrization (indicated by =) (4)

interest in this

equation which results if the discontinuity

* =

As H

As I t

+

=

it has the form

A sIt *t A s H

or graphically

+

with

* a convolution with compact domain of integration, so that the iteration t solution of this equation has only a finite number of terms at any value of s. In the s-channel elastic region

cel: = { (k)EFl4m2<(kl-k1')2~s<]6m 2} S

Asl t

--

of the 2-particle irreducible kernel

It

the discontinuity '

of the t-channel completeness

tion (2), i.e. kernel function of equation (4), can be expressed by the-mass-shell { k~E~o=m 2}

H

equa-

via the on-

s-channel completeness relation which is just the elastic

1

unitarity equation (with A indicating on-shell integration) (5)

AsH = Asl t = H ~

H

sE

C el S

or ~=~=graphically

Whenever f o r t h e v a l u e s

of ~ ,~ '

considered

and f o r

4%

plane analyticity in t, then the~) -convolution A H with another t

(6)

H

H ~

so that

A H s

type of composition

AtHOs

At H

~

s

sE Ce l t h e f u n c t i o n s

H

has cut-

in (5) can even be reexpressed by

with

sE cels

and the crossed channel discontinuity

A H t

are connected via the con-

straint

(7)

AsH

*

=

At H

~s

In this case the function rity at

sE C el s

At H H

can also be continued towards i-ts double cut singula-

sE C s , tE C tel leading in fact to two different constraints for the double A

discontinuity

At(AsH)(with * t ,

the restriction of

^

(Sa)

At(As H ) = Asl t *t AsH

(8b)

At(AsH) = AsH *t AsH

+ +

Aslt *t At (ASH) At(AsH) *t At(AsH)

*t

to the mass-shell)

300

They correspond to two different ways of continuing to the double cut, in case of eq.(8a) along { s~C s, t~C t} ~{SECs,

t¢C t} ~ { SECs, tEC~ I}

using the completeness equation (2), and in case of eq.(8b) along { s¢C s, tee t} ~ { s~C s, tec~ I} ~ { sec s, tEC~ I} using crossed channel elastic unitarity (mass shell completeness relation). Before explaining in Sections 3 and 4 to what extent the continuation to the different boundaries and the derivation of the different constraints which combine completeness and crossing symmetry has been achieved, we shall discuss briefly in Section 2 possible applications in connection with Banach space fixed point methods.

2. CROSSING SYMMETRIC BETHE-SALPETER-EQUATIONS AND BANACH SPACE METHODS In this Section we assume that the set

t E ~ ( [ o , 16me])O(f X{ 4m2+ IR+})

(9) v:= {(s,t,i,i')¢¢~I (i'~ ¢') ¢~r(~t x ~t) s E ~ \ ( {4me+ ~+} U{.E (¢j+¢j')-t-4m 2- I~+}) J=l with~denoting

(I0)

~t:={

complex neighbourhoods and with

1

¢++¢-

~ ¢ ~2 I ~ = ~(¢+-¢ )' ¢+e ~' ¢_ei ~, o~

belongs to the holomorphy domain of

H

and

_¢~
It, leaving open for a while the ques-

tion whether analyticity in this domain follows from LQFT and AC (what is expected, however). Then all the constraint relations of Section ! can be shown to hold 7) . From this, e.g., a system of nonlinear integral equations can be derived which in an abbreviated form (=) with *

(EU)

sE[4m2,~),

t£[o,16m2),(!,~')£~tx ~t

reads:

A

AsH = A t H ~ s A t H

on { sE[4,16m 2)}

rA H = A I +IX I * IX H - A I @ ~ (A H) I s st s t t s s t t t's (BS) LIX_(IXH) ~ A I ~_IX H + iX I * iX (A H) E s SEES sttt S (DD) [Aslt =, AsH ~ - Aslt*tAsH + A s I t tA sH ~ tAs H + A s I t~ tiXt ^(iX s "H)~ tiXt (As H) LASlt AsH-Aslt*tiXsH + iXslt~ tiXt (ASH) + At (IxsH) - AsH* EIXsH - A (A~ sH)*LA-(IXt E sH)

301

(IR)

AsH =* Asl t + A s l t * t A s H - Aslt~tAt(AsH)

+ Zs

with Z

s

=Al

+ Aslt*tAs H - Aslt*tAt(A~H)

s stu

Here

I is a linear combination stu ducible in 2 channels: (if)

I

:= I

(12)

Ist := I s + I t - H

stu

I

st

In all equations

tinuity

equations

structure

for

Z

equation

represent

(BS),

constraints

energy region

(DD),

of functions

nuous or analytic

~,~',

in

(IR)

< =o

,

of the Banach contraction

2-channel

2-particle

as given.

Then it results:

ii A

-

type

~'

is under

(EU), from

(BS), from the double discon-

In the last equation

irreducible

H

el

s

can be made a well-defined

other ex-

theorem and the Schauder

Green's

function

or rather

A I it s stu for A H s (b,b' suitable) and no

< b

< a and

II

,

A

s

I

where

s,t,

There exist also other results

yield different

~ H s

for other choices

of

and Z

s

theorem

w h e n the

is considered

stu

iS "IS

is the n o r m

-

A I s Stu

conti-

fixed point equations

A I exists w i t h s t solution w i t h o < Hi A

-

and different

map in a sui-

o < ~0 , P < )

of the system of integral

A I is given with a' < s stu , then one unique solution

in

and of

from elastic unitarity

(IR).

o < v < ] ,

thus enables us to study the solubility

b' <

of ~

F (s,t,~,~'), H S l d e r - c o n t i n u o u s in N and with growth properties such that e.g.

I sV(log s) l+°°(~') I-0 F(')I

If

to the components

of the Bethe-Salpeter

table Banach space ~

Theorem:

irre-

could have been used.

s

The set of equations (EU),

The application

functions w h i c h are 2-particle

ut

(DD) and from the inelastic

pressions

of Green's

symmetrization w i t h respect

stood. The different the general

+ I

su

- Asl t

H U < b', s

A I . s t or of the assumed growth

properties. We also remark that though invariant variables restrictions

to the dimension

d

the type described necessarily use

of functions with continuity

the constraints tions to measure

effectively O

sets.

s,t,~

of the underlying d >

4

must hold.

properties

restrict

have b e e n used there are

space-time.

For Banach

That this is so follows

and the fact that in

some of the integrations

d < 4

spaces of from the

dimensions

in the integral

equa-

302

3. TWO-PARTICLE STRUCTURE AND STABLE DOMAINS As pointed out before, each term in the iterative solution of eq.(2) with respect to

H

or

It

is analytic at least in the envelope of holomorphy of the primitive

domain of H. It is, however, not clear from the beginning that outside the primitive domain the analytic continuation of each term

H ~

H ~ t . . "~)t H is again of the

same type, i.e. can again be represented as in iterated convolution. hand, such a property is necessary for applications all variables

(s,t,~:,~2,~l',~2')

On the other

as indicated in Section 2, where

are allowed to stay simultaneously at or near real

positive values. Therefore the question arises to what extent at such points I

or their discontinuities

t nulng back to the primitive domain.

of

(Fl*tF) (s,t,~,~')

at the point

one has to know the numerical values or analytic structure, respectively,

Fl(sl',t,~,~" )

{ (s: ,~")' ~

and

Fe(s~,t,~",~r'')

at all points of the integration regions

~ [2m,~-2m] , ~~" E O~'t~ Is,~,~,}

parameter-dependent some

and

In fact, in order to determine either the nume-

rical values or analytic structure of the convolution (s,t,~,~')

H

can still be determined from each other without conti-

t E ~+

domains in

t with ~Is,~,~, well-defined compact, but

Cz. T~e~efore,

to determine

from each other via the Bethe-Salpeter

cessary (and sufficient)

that a domain

I t C Cz

{ASH} and

{AsI t}

for

type of equation (4) it is ne-

exists which is stable under the

map

*t : (F:'F2) ~ FI *t F~ in the following sense: We call domain

Tt

*t-stable (stable under the convolution

*t ) a

such that

(~,~') E 7 t x 7 t ~

U sE[4m~,~)

7t

s,~,~'

To find such a stable domain is a purely geometrical problem. Now for the purpose of applications as in Section 2 it turns out that it is sufficient to find and to consider such stable domains only on the "diagonal" manifold

(13)

I (~++~_) ZD : = { N~ E C 2 I ~ = 2~+-~_ N

i.e. to consider only ~2,~1',~2').

ZD = Z D x ZD

In fact, for

E ~D x ~ D implies

. ~+ .~ m .

~. E - ~

}

N

in the space

s E{4m 2 +~+}

Z = ~

of mass variables

, t E [4m2,16m 2]

~ts,~,$, C i .D To find then a stable domain

diagonal manifold still involves

IO real variables.

it holds that : t } : D ~. :t

Nevertheless

(~I, (~,~') on~Nthe

it is possible to

prove the following. Lemma: For any

t E (0, 16m 2)

fom the mass-shell

the smallest

~i z = ~2 2 = m z

~ 2 = ~22 <_ m 2 , is given by

*t-stable domain

~t

on

ZD, different

and containing at least one real point

:t :={ (~:,~2)

[ ~J. = ~2" , ~: 6 lit} with

303

(14)

K7 :={ ~ 6 ¢ I o < (~

The stable domain

~t

~m~)2 _< ~T - Re ~ }

agrees with eq.(|O). The knowledg~ of these domains~ inva-

riant under the 2-particle structure map

*t' enables us to determine the analyticity

properties of *

t

A I if those of A H are given in ~t x ~t , and vice versa. Because s t s involves only compact integration regions this immediately yields: A sH(s ,t,~,~') ~

Theorem: (a) If

for all

s 6 [4mZ, oo)

is holomorphic in all points of

V T := { (t,~, i') 6 ¢5 I t 6 { l!Zm - T+cI<8} n (¢ ~ { 4 m 2 + ~ + } )

(]5)

where o < T < 16, then Aslt(s,t,~,~') is also hol0morphic in VT e' U { t,~,i'[ t 6 { 4m 2, Tm z} , (i,i') 6 ~t x ~~t } for some ~ ' > o.

(h) If

A s It

s 6 [4m2,~)

for all

holomorphic in

VT E'

for some

is holomorphic in

VT g'

then

A H s

is

e.

This theorem supplies the information desired in applications of the type discussed in Section 2 provided the hypothesis of the theorem can be proved. The present state is discussed in the following Section.

4. ANALYTIC PROPERTIES OF TWO-PARTICLE IRREDUCIBLE GREEN'S FUNCTIONS In this Section we briefly review the present situation concerning the holomorphy properties of the 2-particle irreducible Green's function

It

as they are needed or

desired in applications of the type described in Section 2. First we quote for the full Green's function the following (partial) result: Theorem I:

H(s,t,~,~')

is holomorphic in a quasi-topological product of the cut

s-plane and a neighbourhood of (]6)

A 4 :={ t 6 [0,4m z) , (!,~') 6 H ~ x ~

}

The proof uses (i)

old results leading to the well-known on-the-mass-shell analyticity properties~

(ii)

the fact that ellipse type domains of holomorphy in

t

of A H

for

s

(~2~') £ ~

x ~

(corresponding to completions of the primitive domain along

Im (kl-kl')=o,Im kl=o)

are such that their real sections for

= (~++~-) , ~' = (~++~-) ~+-~_ ' , ~+-~_ contain

,

~+,~+ 6 ~

, ~_ ~' £ i ~

the corresponding real sections for

,

-

~ = ~

= o

,

304

(iii)

an analytic completion along IsI < e , t 6 [o,4m 2)

(iv)

Property (iv)

AsH

for

and

yielding holomorphy of

the fact that positivity of tivity of

Im(k1+k2)=o

AsH

for

Im k1= o or H

in

(~, ~') 6 ~

~ x ~4

Im k1'= o

x ~

for

, and

is implied by posi-

~_ = ~_' = o.

is an application of the Glaser-Positivity-Theorem6):

If (a)

F(k,l)

(b)

is analytic in

an open

~

r

c~ N 0 ~

~ x ~

exists with

S F(k,I) ~(k) ~(1)dkdl > o O~ x 6 0

r

V ~ 6~(~r)

r

Then: (A) for all open ~0 C

(17)

( CN x CN , and

S

F(k,I) ~(k) ~(I)

with d~Adk ~

~ C d[^dl (2i-~)> o

V ~ 6 e2(~), and

~ X ~

(B) a sequence of functions F(k,l) = Elf (k)=

fv(k)

f (I)

analytic in

uniformly in

~

exists with

~ x ~ ,

and (A) and (B) a r e e q u i v a l e n t s t a t e m e n t s . Because of the Theorem of Section 3 the results of Theorem | are sufficient to prove some of the desired properties of

I . In fact it holds: t

Theorem 2 "

the discontinuity

For all

s 6 [4m~,~)

ticle irreducible Green's function

Asl t

of the t-channel 2-par-

is holomorphic in a neighbour-

It

hood of (18a)

A:={ t 6 [0,4m2),(~,~ ' ) 6 ~ t

x ~~'t }

N

and therefore the dispersion relation for Knowing that the primitive domain for tion along Theorem 3:

It

continues to

is bigger than that of t ~m kl-k2')=o Theorem 2 can be supplemented by For all

s 6 [4m2,~)

I

H

the 2-particle irreducible function

A.

by a comple-

It

is

holomorphic at all points of

(18b)

A':= and

{t 6 AsH

~+, ~ - <

min [ 4 m , ~

is holomorphic in

- ~s] , (~,~') 6 ~ t x ~ t }

A'n (C s ~ { t

= 4m 2 + ~ + } ) .

Again the Theorem of Section 3 has been used, here, however, in order to improve the information on

AsH

from the knowledge of the bigger domain of

it was applied to improve the information on of

A H s

which possibly do not hold for

Asl t

Asl t , while before

from the positivity properties

A I . s t

One has good reasons to believe that also holomorphy in{

t 6 [4m2,16m2),

305

(~,~') E ~

x ~

} ~ A'

can be shown using positivity from completeness of the sub-

Hilbert space of ( n > 2)-particle states. In this case the structure considerations as indicated-in Section 2 could be put on a completely firm basis. In particular, if Glaser positivity, eq.(17), could be established for

Asl t , then the desired pro-

perties would result rather easily. This is because the *

t

-convolutiom has the

following Glaser-positivity preserving property: Theorem 4:

If

F1

and

F2

for

s E [4m2,~) , (t,~,~') E A , eq.(18a), have Glaser

positivity, eq.(17), then also

FI *t F2

has Glaser positivity.

REFERENCES I)

J. Bros, in: Analytic Methods in Mathematical Physics, New York 1970

2)

M. Lassalle, Commun. math. Phys. 36, 184 (1974) J. Bros, M. Lassalle, Commun. math. Phys. 43, 279 (1975), 54, 33 (1977) J. Bros, M. Manolessou-Grammaticou,

Saclay preprint DPh-T/78-109

3)

J. Bros, Proceedings Les Houches S u m e r

4)

G. Sonm~er, in preparation

5)

Institute 1979

R.F. Streater, A.S. Wightman, PCT, Spin & Statistics, and All That, NY. 1964 Res Jost, The General Theory of Quantized Fields, Providence 1965 H. Epstein, in: Axiomatic Field Theory, ed. M. Chretien and S. Deser, NY. 1966

6)

V. Glaser, R.C.P. 25, Vol. 16, Strasbourg 1973

7)

G. Sommer, in: Proceed. Steklov Institute of Math. --I (1978), 139.

ON THE SCATTERING MATRIX WITH INDEFINITE METRIC Xia Daoxing Research Institute of Mathematics, Shanghai,

Fudan University,

China

I. Quantum theory with indefinite metric. Some physicists

[|] developed a theory by using negative metric to get over the

difficulty of divergence ring problem concerning

in the theory of relativistic

Let K be a linear space with indefinite

inner product

pose that there are two linear subspaces K+ and K subspace K+ and K_ are mutually perpendicular product and become Hilbert spaces, x+,y+ E K and -[x_,y_],

quantum fields. This scatte-

indefinite metric can be formulated as follows.

if they

[x,y] for x,y E K. We sup-

in K, such that K = K+ + K_, the

with respect to this indefinite

are endowed with inner products

inner

[x+,y+],

x_,y_ E K_ respectively.

The subspace K+ is the space of all states of free particles with positive metric and the subspace K

According

is the space of states in which the fictitious

to this decomposition

of K, there is a natural

<x+,x_ly+,y_> for all x+,y+ E K+ and x ,y

The Hamiltonian

inner product

from K onto K+ is denoted by P+

[x,y] is <xlJly~.

of a system is an operator H self-adjoint with respect to the

indefinite metric. Let H duct. Then H

inner product defined in K,

= [x+,y+] - [X ,y ]

E K . If the projection

and J = P+ - P_, then the indefinite

particles appear.

be the operator adjoint of H with respect to the inner pro-

= JHJ. If we write the vector x

@ x+, x+ E K+ in a column as

fx_1

~xj then the operator H can be written in the matrix form

Lr* where H+ are the selfadjoint

K+

operators

H+ )

'

in K+ respectively,

and F is an operator from

to K_. Now we shall consider the space of all physical

with real local spectrum,

states. A vector x C K is called

if there is a vector-valued

analytic function x(.) defined

on the whole complex plane except the real axis, such that (H-~I) x(~) = x any a E K, there is a function f+ (C0,a) E L ] such that



]

=2-i-JJ

~f± (co,a) dCo

~:f

.

,

v;~o.

and for

307 Let K' be the space of vectors in K with real local spectrum. In the following, we only consider the case when K' is a closed subspace of K. And we shall adopt K' as the space of physical states. We denote the physical Hamiltonian HIK, by H'. 2. Formulation of scattering problem. The free Hamiltonian is a selfadjoint operator H in K. The wave operator o W+ = W+ (H',Ho), if it exists, is an operator from K+ to K, defined by

lim t~+ ~

I]

-itH e-itH' W+(H',Ho)X -e

and the scattering operator is

o xl I = 0,

for x C K+

S = W_(H',Ho)-I W(H',Ho). Since the operators H+ and

H ° are selfadjoint in Hilbert space

K+, the existence, unitarity and the expression

of the wave operators W+(H+,Ho) do not relate to negative metric and are considered in the ordinary quantum mechanics. On the otherhand, W+(H',H o) = W+(H',H+) W+(H+,Ho).

Thus S = W_(H,Ho)-I S W+(H+,Ho) , where S = W_(H',H+) -] W+(H',H+) Thus we may suppose that H+ = H ° and we shall only consider the existence, unitarity and expression of this scattering operator S. The main tool in our consideration is the following characteristic function. Let o be the spectrum of H+. For % E ~, we construct the operator h(%) = %I-H

+ F(XI-H) -I F ~

in K_. This operator-valued analytic function h(.) is called characteristic function and it depends upon the Hamiltonian H and the decomposition K = K+ + K_. 3. Some assumptions. Suppose that K+ is the space of strongly measurable and quadratic integrable vector-valued functions with values in a separable Hilbert space ~ and for ~(.) CK+,

II ~tI2=

f II ~(~)II 2d~, O

(H+~)(~) = ~ ( ~ )

,

where the measure d~ is the Lebesgue measure. Suppose further that there is an operator-valued function ~(~) such that ~(~) is the operator from [ to K , for ~ E o and

F~ = I J

the operator H

~(~) ~(~) d~ O

is a bounded selfadjoint operator.

308

4. The main theorem. Let h(~0+io) = s-lim

h(~0tio) be the boundary value in O of the analytic func-

~-> O +

tion h(1).

Theorem. If h(w+io) and h(w+io) -I exist for almost all real co and

~ll

h(~+i°) -I ~(~°)II2 de0 < ~.

Then the [x,x] > 0 for all x 6 K' and x # 0. The space K' endowed with [x,y] becomes Hilbert space. The wave operators W+ = W+(H',H+)

exist, are unitary from K+ onto K'

and

I

oh(~io)-I

W+~0 =

~(~) ~(~) de ,

~(.

+ i~(.)* I h(t±i°)-I

for cp ff K+.

6(t) ~(t) dt

t - (.rio) The scattering S = W

-I

W+ is a unitary operator in K+ and

(S~)(~) =

(I

- 2~i[(~)* h ~ + i o ) -I ~(~I ~(W)

,

~ 6 K+.

The method in the proof of this theorem is the complex analysis of operatorvalued analytic function defined in the upper or lower half-plane. 5. The complex Lee and Wick's model. As an example, we consider Lee and Wick's model. The Hamiltonian of the system is H = m ° V*V +

~(k) a*(k) a(k) ~ d3k +

Mo(k)(V*Na(~)

- VN a*(k)) d3k

v

where ~o(~) is a real function, m ° is the rest mass of V-particle and w(k) = /

2+~2

V

(~ > 0 ) . The o p e r a t o r s V, N, a, V*, N* and a* a r e t h e a n n i h i l a t i o n and p r o d u c t i o n operators of V-particle,

N-particle and G-particle respectively and they satisfy the

commutation r e l a t i o n

[a(k), a*([')] and anti-commutation

= 6(k-k')

relation {N,N*} = I, {V,V*} = -I

respectively.

Let the vacuum be denoted by ~o' [~o'#o ]

<%oI~o > = I. Then the state

in which the V-particle occurs is of negative metric, and [V*~o, V*% o] = -I. The space K (n) of all states [V+(n-l)e] @ [N+ne] is a reduced subspace of theHamiltonian and K (n) = K +(n) @ K (n) where K +(n) is the space of all states ÷ ÷ n) a * (kl)... ÷ ÷ 3+kl...d3[n~o ~q) = N* I k0(kl,...,k a * (kn)d

30~

with ;I~0[2 dkl'''dkn < co and K (n)_ is the space of all states + + , ÷ ÷ 3÷ 3-+ = V* ~(kl, .... kn_l) a (kl)...a (kn_l)d kl...d kn_10 o

~

[ 2 ÷ ÷ w i t h Jnl~l dki. . . . dk,n < ~° We identifYn t h e s t a t e s

~p and u@ w i t h ~p and ~ r e s p e c t i v e -

ly. The space K (n) is the N O.~..O sector. The indefinite metric is positive in K (n) (n) + i n K_ . I n t h i s c a s e ,

and n e g a t i v e

n H+~0 = ~ e(k ) ~0(k|. . . . ,~n) 1

H ~ = (m +

n-1 ~ e(

÷

)) @(kl,...,kn) ÷

÷

F%0 = n ]~0(kl,... ,kn_ 1 ,kn) ~0o(kn) d3~ n 6. The structure of the S-matrix in N@e0-sector. Lee and Wick have already obtained the explicit form of elements of scattering matrix in K (I) (the case n = I, NO sector) by solving a function equation and in K (2) (the case n = 2, N@0 sector) by solving a singular integral equation with one variable. But for n > 3, the problem comes to solving singular integral equation with several variables, and certainly we'll meet with some essentially new difficulty. The method used in [2] may be extended to the case n = 4

and will be omitted

here owing toits complexity. We'll only consider the ease n = 3 and find the concrete form of elements of scattering matrix in N~00 sector. In this case we solve

sin-

gular integral equation of two variables by the method of complex analysis to obtain the concrete form of the operator h(~+io) -!. This singular integral equation is B(~l+e2)Y(e1'~2) -

j

~ y(~l,~')~(e2)~(~') + y(~',~2)~(~')~(~l) ~ w' + e I + ~2 " e - io de'

= 6(~l-a|) 6(~2-a2), .where ~ and B are given functions determined by mv'° D and 4o(.) and y(~i,~2) is

unknown function

of two variables.

We denote the vector k in R 3 by k = ~%,÷ where e = Ikl, T is a unit vector. Then the S-matrix element is ÷

÷

÷

÷

÷!

÷!

.

!

,

,

÷

÷

÷

÷

÷,

÷V

S(kl,k2,k3;k{,k2,k3) = S(~1,e2,~3,ml,~2,e3) 6(TI,T2,T3;T{,T2,T3 ) and .

~

;

,

S(el,~2,e3,ml,~2,~ 3) = e

3 -i~(n (~J)+n (eJ)) { 1

3 ~(~

~2 ~3 ~] ~2 ~3 ) ~• ~(ej)~

. I V V +

6(~l,~2,e3,~l,e2,~3) ~6 ( w . - e ! ) f<e;)

+

' j');

where N is real and q9 ~, L, f, M are functions which can be expressed by ~ and B.

310

References I.

Lee, T.D. & Wick, G.D.

: Nucl. Phys. B., 9, 2 (1969), 209-243

1-10 ; Phys. Rev. D., 6(1970), 2.

Xia, D.X., Scienta Sinica,

3.

Xia, D.X., Acta Math. Sinica,

1033-1048.

18 (1975),

165-183.

19 (1976), 39-51.

; 10, | (1969),

ON THE REPRESENTATION OF THE LOCAL CURRENT ALGEBRA Xia Daoxing Research Institute of Mathematics, Fudan University Shanghai, China.

In the course of the past few years, several physicists have become attracted to the possibility that one can write down a complete physical theory in which the fundamental dynamical variables are local observables, such as current densities

([I],

[2], [3]) instead of the canonical fields. For simplicity, at present time, we only consider the non-relativistic field. In mathematics, it reduc~ to the representation of the group of diffeomorphisms. Let Diff(R n) be the group of all C~-diffeomorphic bijuctioms~p in R n, which is identical mapping beyond a compact set K , and with composition o

:

(~0,~)~+ ~(~(.))

as group operation. The group Diff(R n) is naturally equipped with Schwartz's topology to be a topological group. Let Do(Rn ) be the connected component containing identity of Diff(Rn). It is an infinite-dimensional Lie group. Let

K n be the Schwartz's n space of all C~-mappings g : R n + R m with compact supports. Then K n becomes infiniten

dimensional algebra with the following non-associative multiplication :

~x)~

[g,h](x) = X j ( h j(x) ~$(~) - gJ(x) 8x J ~xJ--J For every g C K n n' the solution ~p of the differential e~uation v - -

=

dt

gong

=

o

i

is denoted by exp(tg). Then exp : g~--+ exp(tg) is the exponential mapping from the Lie algebra K n to the Lie group. n

Let o(x) and JJ(x), j = 1,2,...n be the particle density and flux density. These are operator-valu~

distributions. We take f E K I and g E K n and construct the n

n

smeared currents P(f) = I P(x)f(x)dx

,

J(g) = j ! 1 1 JJ(x)gj(x)dx.

These are self-adjoint operators in the Hilbert space ~

of all physical states.

312

The commutational relation of the current algebra {J(x),p(x)} can be written as [p(fl),P(f2)] = O, [P(f),J(g)]

= io

gj

Sf

,

j-I [J(gl)J(g2) ] iJ(~gl,g2])Since p(f) and J(g) are unbounded operators, we shall use the unitary operators e itp(f) and e itJ(g) instead of them. If we denote U(tf) = e itp(f)

,

V(exp(tg)) = e itJ(g) ,

Then the commutational relation of the local current algebra is eouivalent to the following

where fl' f2' f E K ~

U(fl)U(f 2) = U(f| + f2 ) ,

(I)

V(
(2)

V(~)U(f) = U(f o~)V(<0) ,

(3)

and tpI, <92, [06Do(Rn). If we consider K In as an additive group,

then (1) means that U :

f l--+ U(f)

i s a r e p r e s e n t a t i o n o f K1 i n ~ . A l s o , (2) means t h a t V :~t'-'-~ V(~) i s a r e p r e s e n t a n tion of group D (Rn) i n ~ . o The problem o f f i n d i n g t h e r e p r e s e n t a t i o n s o f t h e l o c a l c u r r e n t a l g e b r a i s '

equivalent to determine those pairs of representations U of group K 1 and V of group n the commutational r e l a t i o n

Do(Rn), which s a t i s f y

V(<9)U(f) = U(fo[9)V([9).

(3)

In the following, we shall consider a kind of such representations which are closely related to the quasi-invariant measure and is seemly important in the quantum p h y s i c s . Let K be the dual of K 1 (i.e. Schwartz's space of distributions), ~ be the n n Borel field of sets in Kn, generated by the class of weakly closed sets~ in K . For , n any q06Diff(Rn), the distribution F(~0-l(x))D(x,~0-I) is denoted by ~9 F, where D(x,t0) is the Jacobian of ~ at x. The mapping [P ÷ ~0 is a representation of Diff(R n). Let be a measure on the space (Kn, ~ ) for every Cp6D (Rn) o ~ on (Kn,

*~):

Vt.0(E) = U(t0 E),

we consider the measure

313

where ~ * E = {~*FIF6E}

for E 6

~

. If ~

~ P for all ~0 6 Do (Rn)' then we say that

is a quasi-invariant measure with respect to the group Do(Rn ) . We consider a measure N on ( K n , ~ ) the Hilbert space

~

= L 2 ( K*n , ~

quasi-invariant with respect to D (Rn), and o

,N). We define

(U(f)~)(F) = ei(F'f)P(F).,

F6Kn,~(.)

6

(in this case O(x) =F(x)) ,,,. *_./dN(~*F)~ (V(~)~(F) = T<(9 F ) < ~ ) for f 6 K n! and ~ 6 D (Rn), where o

d~(~p*F) aN(F)

1/2

* ,

F 6Kn,~(.) 6

,

is the Radon-Ni~odym derivative of ~(~9"(.))

with respect to N(.). Evidently, U : f~-+ U(f) and V : ~-+ V(f) are group representations of Kln and Do(Rn) in ~

respectively,

and satisfy the commutational

relation

(3). The main problem is to determine the measure N (which is related the vacuum state) such that in this representation we have the prescribed'well'defined Hamiltonian. Following

[I-3], we considered the case, ~hen H K = I Ej I dx K J ( x ) ~

where K j(x) = ~

K j(x) ,

p(x) + 2iJJ(x)

~x j H

p

= V

o

+

~

m=l

Vm(X 1 ..... Xm)O(xl)...P(Xm)dXl...dXm

and H = K k + N . P Firstly, we consider the general form of Radon-Nikodym derivative. Under certain conditions, we called that S(g,.)=dN(~9

(.))/dN(.)

is an analytic functional [5].

Theorem[5]if S ( g , . ) i s an a n a l y t i c f u n c t i o n a l , then d~(F)

= exp C(

Fdx)(F,InD(x,~0)) +P(tp F)-P(F)

.

(4)

where C(.) is an analytic function, P

P(F) = mlI=

P(x I ..... Xm)F(xl)...F(Xm)dXl...dx m.

Example. We consider the Poisson measure [6]. Let f(x) be a fixed smooth function on R n, 0 < f(x) < oo and m be the measure on R n, : m(U) =]uf(X)dx.

If {x n} is a se-

quence of points in R n without limitingoopoint and x m # Xm ' for m # m', then we construct the corresponding dis~:ribution [ 6(X-Xm). Let ~/~ be the set of all such m= 1

314

distributions. tes on ~

There is a unique probability measure Do on (Kn, ~

) which concentra-

such that ~o {~m 6(X-Xm) 6 ~ I = e-lm(U)

there

are just n points of {xm} in U}

(%m(U)) n n!

for any open set U in R n, where % is a fixed positive parameter. This measure

~o is called Poisson measure. In this case, d~o(

I~ we take a measurable functional P(~)~@ueh that 0< exp(P(F)-(F,Inf) all F 6 K

n

< ~, for almost

and define a measure Z as ]~(E) = I

exp(P(F)-(F'Inf)d~°(F)' E

Then dB(k0 F) = exp{(F,InD(x,¢p)) + P(F )-P(F)) dD(F) and formally j=l'\~x J

Sx j ~F(x) 'gtx)

.

In [I], [2], [3] Menikoff and Sharp investigated the problem to find representation space of a current algebra. Now we have to find out

quasi-invariant mea-

sures ~ such that we can give a definite interpretation of the Hamiltonian H such that H becomes a non-negative self-adjoint operator and H~ = 0, where ~ is the 2 * ~ ,~). vacuum state I in L (Kn, Suppose that ~ satisfies the conditions in our Theorem. Let e. be the unit 3 vector whose j-th component is I and the other components are zero. Let

R.(x,F) = S(ej~(.-x),F) + $. F(x). J ~x j Then the problem of determining ~ is reduced to solving the equation n ,

F(x)

i

]

\

+ ~ Rj(x,F)Sjfjdx+Hp(F))d~(F)

= O, f 6 Kin

which is simply H~ = 0. Now we consider the case that C(.) ~ I in (4). Hence Rj(x,F) = F(x)

$ 6P(F) ~x j ~F (x)

(5)

315

We can prove that in this case (5) becomes i

n

~

1 ~P(F) ~ ~ F]dx 2 6F(x) j=1 ~xj

6p(F)~2

References [I] Goldin, G. Grodnik, J., Power, R. and Sharp, D.H., J. Math. Phys. 15 (1974), 88 [2] Menikoff, R., J. Math. Phys. 15 (1974), 1138, 1394 ; 16 (1976) 234, 2353. [3] Menikoff, R., Sharp, D.H., J. Math. Phys. 19 (1978), 135. [4] Xia Dao-xing, Measure of Integration Theory on Infinite-Dimensional spaces ~translated by E.J. Brody), Acad. Press. N.Y. § London (1972). [5] Xia Dao-xing, On the Representation of LOcal Current Algebra and the Group of Diffeomorphism (I) to appear in Scientia Sinica (1979). [6]Viechik, A.M., Gelford, I.M., Graev, I.M,

(1975), 6, I.

PART IV

TWO-DIMENSIONAL MODELS AND RELATED DEVELOPMENTS

S MATRIX THEORY OF THE MASSIVE THIRRING }©DEL

B. BERG

If. Institut fur Theoretische Physik, Universit~t Hamburg.

CONTENTS

I.)

Introduction

317

II.)

The factorization equations

318

III.)

The exact S matrix

323

IV.)

Boundstates and Levinson's theorem

325

V.)

The complete S matrix

33!

VI.)

Anomalous and virtual thresholds

335

VII.)

Sun~ary and outlook

338

References

339

Figures

341

Abstract: T h e S m a t r i x theory of the m a s s i v e T h i r r i n g model, d e s c r i b i n g the exact q u a n t u m s c a t t e r i n g of solitons and their boundstates,

is reviewed

T r e a t e d are: F a c t o r i z a t i o n equations and their solution, boundstates, g e n e r a l i z e d Jost functions and L e v i n s o n ' s theorem, states,

"virtual"

and anomalous thresholds.

scattering of bound-

317

I. I n t r o d u c t i o n

For a v a r i e t y of two d i m e n s i o n a l r e l a t i v i s t i c k n o w ncw the exact q u a n t u m S matrix. tum field t h e o r i e s w i t h Gordon theory q - m o d e l /3/,

field t h e o r i e s

physicists

I like to call these m o d e l s

soliton behaviour".

Examples

(alias m a s s i v e T h i r r i n g m o d e l ) / 1 , 2 / ,

the Gross N e v e u m o d e l /4/,

the last two m o d e l s /5/ and the c h i r a l

"quan-

are the Sine-

the O(N)

supersymmetric

non-linear

e x t e n s i o n s of

i n v a r i a n t SU(N)

Thirring model

16,71. I w i l l up to some side r e m a r k s c o n c e n t r a t e on the m a s s i v e T h i r r i n g model

(MTM), b e c a u s e

it is the p r o t o t y p e o f

a field t h e o r y w i t h

soliton

b e h a v i o u r and it's S m a t r i x t h e o r y is w i t h r e s p e c t to a n u m b e r of properties best understood.

Someone who understands

no d i f f i c u l t i e s w i t h the o r i g i n a l

this m o d e l

s h o u l d have

l i t e r a t u r e c o n c e r n i n g the o t h e r m o d e l s

I also have tried to be c o m p l e m e n t a r y to the r e v i e w of Z a m o l o d c h i k o v and Z a m o l o d c h i k o v /8/, w h i c h r e l i e s on the r e s u l t s of r e f . / 1 - 4 / previous

literature.

Following

the o r i g i n a l d e v e l o p m e n t ,

equations

to d e r i v e the S m a t r i x and the s p e c t r u m of the MTM.

"old fashioned", the m o r e This not

Our

because

we use the m e t h o d of f a c t o r i z a t i o n

it is n o w p o s s i b l e

fundamental Hamiltonian

is in

a p p r o a c h by m e a n s of the B e t h e ansatz.

s u c c e e d e d to solve the B e t h e ansatz

for

So far one has,

all m o d e l s w h e r e

however, the exact

is known.

lecture

is c o n c e r n e d w i t h d e t a i l s of the M T M S m a t r i x .

is c a l c u l a b l e , classical

This

to get the same r e s u l t s

is t r e a t e d in the l e c t u r e s of H o n e r k a m p .

S matrix

and

because

currents

of the S m a t r i x

in the ~ M

infinitely many higher

conserved

s u r v i v e q u a n t i z a t i o n /9/ and imply /lo/

(section II). A l t h o u g h there

The S matrix

factorization

is o n l y e l a s t i c

scattering,

nevertheless the S m a t r i x e x h i b i t s a s u p r i s i n g l y r i c h s t r u c t u r e . s o l v i n g the f a c t o r i z a t i o n e q u a t i o n s (section III)

one d i s c o v e r s

was p r e v i o u s l y known

in the p h y s i c a l

In the n o n r e l a t i v i s t i c

the two d i m e n s i o n a l can be shown /12/,

the b o u n d s t a t e ( b r e a t h e r )

from the q u a s i c l a s s i c a l

c r i b e d t h r o u g h simple poles t e r i n g amplitude.

for s o l i t o n - a n t i s o l i t o n

v e r s i o n of L e v i n s o n ' s

spectrum,

treatment/11~

After

scattering which

to be d e s -

sheet of the f o r w a r d

scat-

limit this is c o n s i s t e n t w i t h theorem.

that for the r e l a t i v i s t i c

More generally

it

S matrix Jost functions

318

exist in close analogy to p o t e n t i a l scattering

(section IV) .The d i s t i n c -

tion b e t w e e n b o u n d s t a t e and CDD poles is also d i s c u s s e d in section IV.

The complete S m a t r i x of the MTM /13/ includes the amplitudes t o n - b r e a t h e r and b r e a t h e r - b r e a t h e r

scattering

for soli-

(section V). In these

channels of the complete S m a t r i x one finds double poles, w h i c h populate the p h y s i c a l natural Another

sheets of the forward scattering amplitudes.

i n t e r p r e t a t i o n as anomalous thresholds /14/

They have a

(section VI) .

interesting feature arises from the fact, that b r e a t h e r - b r e a t h e r

b o u n d s t a t e s have to leave the s p e c t r u m of physical particles at a value of the coupling, where their mass is two times the soliton mass and still less than the sum of the masses This gives rise to a " v i r t u a l

of their b r e a t h e r constituents.

threshold"

(section VI).

In summary the S m a t r i x theory of the MTM is well u n d e r s t o o d and provides a lot of material,

w h i c h could be used for i l l u s t r a t i o n in an

introductory course about S matrix theory. An outlook is ~iven in the final section VII.

II. The f a c t o r i z a t i o n eouations The MTM is d e s c r i b e d by the L a g r a n g i a n ~

~Ay

~

-- ~V% ~

-- t % ~

~

with

~ , ~

of a s e l f i n t e r a c t i n g fermi field in I + I dimensions.

(1.a) B~ C o l e m a n ' s /15/

e q u i v a l e n c e the MTM is known to be e ~ u i v a l e n t to the Sine Gordon

(SG)

theory d e s c r i b e d by the L a n g r a n g i ~ n

of a s e l f i n t e r a c t i n g bose field. The c o r r e s p o n d e n c e

is such thae the

e l e m e n t a r y fermions f,{ of the MTM correspond to the solitons s,s of the SG theory.

The e l e m e n t a r y SG boson

c o r r e s p o n d s in the

MTM to the lowest ff boson b o u n d s t a t e b I.

The coupling constants have to be defined in terms of physicai quantities. Using the p a r a m e t r i z a t i o n of /15/ the c o u p l i n g constants lated by

are re-

319

tt'T-- 4) has dimension

(.mass) 2 and is related

The SG theory is classically verse

to the M T ~ fermion mass.

completely

integrable

by means

scattering method /16/. This implies the existence

set of higher conservation model.

(2)

of an infinite

laws /16/ which govern the dynamics

Their physical meaning

and outgoing momenta,

of the in-

is conservation

of the

of higher powers of in-

i.e.

in-m in m+. in,m , out.m . out, m+. +( out)m Pl ) +(P2 ) "" +(Pn ) = [Pl ) +(P2 ) "" Pn'

(m=I,3,5.

)

!

for

n-- 9 n

particle

exist in the classical these conservation

scatteringrsimilar

(Grassmann)

MTM /17/.In the quantized

severe consequences

is no production

for particle

in ~ ' out ' Pn = LPl "'"

Relying on axiomatic Iagolnitzer S matrix

theory theory

scattering:

or annihilation.

b) The sets of incoming and outgoing momenta [ in Pi '""

currents

laws survive to all orders of perturbation

/9/ and have the following a) There

higher conserved

are identical

out~ 'Pn "

S matrix theory it has been proven r i g o r o u s l y

/1o/ that properties

a) and b) imply factorization

in terms of two particle

scattering

by

of the

amplitudes.

Roughly: Sn

(P1"" Pn ) =

This factorization perturbation

theory

for the MTM fermion

elastic

scattering.

$2

(Pi' Pj)

has first been recognized ( 4/~ %

small)

fermion-fermion

(g small)

in SG

/19/.

Let us consider

and fermion-antifermion

(for distinguishable

is possible.

theory

conservation

shell condition p~ = p~2= m 2 (i = 1,2) scattering

for the SG boson

(3) is not well defined.

Energy m o m e n t u m

P2 = P½) and

(3)

/18/ and then also been verified

in M T M p e r t u r b a t i o n

As it stands equation detail

~

in

2- particle

p1+P2 = p~ + p½ and the mass

imply that only forward

particles)

backward

(p1=p~, - ,) (Pl = P½' P2-Pl

The S matrix acts on 2-particle

states by

320

IPl,P2> in = S I P l , P 2 ~ t

(4.a)

= u ( ~ ) I p l , P 2 > Out

and (4.b)

I p I ,p2>in--- Slp I ,p2>Out = t(~)Ip i ,p2>out + r(~)Ip I ,p2>°ut

Here ~ = (Pl + P2 )2; particles with momentum Pi are denoted Pi and antiparticles with m o m e n t u m Pi are denoted Pi" The transmission

amplitudes

plitude r = r(s)

are supposed to be real meromorphic

(f = u,t,s) an

u = u(s),

t = t(s)

of the complex M a n d e l s t a m variables,

(possibly)

infinitely many sheeted Riemann

points at o and 4 m 2. By standard convention - ~

and the reflection

to o and from 4m 2 to ~

functions

which are defined on

surface with branching the cuts are choosen

.Because of the conservation The assumption

am-

f(s)=f(s))

from

laws no other

singularities

are required.

is that only the minimal

singularities

are realized.

By putting the considerations

son's theorem

(section IV)

and anomalous t h r e s h o l d s ( s e c t i o n

about LevinV~)

at the

beginning one would have good arguments of s are real and above the elastic Following

the standard convention,

for this. The physical values threshold: 2= (p1+p2)2~ 4m 2 .

the physical values of the amplitudes

are obtained as limit

on a certain Riemannian ~he further discussion

lim f(~ +i~ ) t~ 0 + sheet, which is called

(f=u,t, s) the physical

is much simplified by introducing

sheet.

the complex

rapidity variable O defined by ch

The variable amplitudes.

~ = s-2m2 2m 2 O plays the role of a global uniformizing The conformal mapping

finitely many sheeted complex physical of the

(5)

sheet is required to be mapped 0

-plane. All other

or below the physical as functions [(s)=f(s)

of

e

strip

by f ( e )

is one-to-one

s-plane and the complex into the strip

sheets are mapped (cf.Fig.l.a,b)). = f (s(~)).

carries over into f (0)=f

between the in~-plane. o (72~

into similar

The ( ~

strips

The amplitudes

The Hermitian

(- ~).

(5) variable of the

above

are defined

analyticity

321

In the s-plane

the crossing

are

u(s+i~) =

lira

lim

r(s+i~) =

lim

of the amplitudes

t(s-4m2-i~ )

lira

~..~ o +

and

relations

t-~©÷

F.~ o ÷

lira

and unitarity

r(s-4m2-iE)

~.~o + u(s+i~ ).u(s-i~ ) = I

lim [ t(s+i~)ot(s-il) ~,,-~ 0 4.

+ r(s+i~ ).r(s-i~)]

lim ~ L-~ Q

+ r(s+i~)'t(s-i~)]

t(s+i~l.r(s-i~l

Using the Schwarz

reflection

that the following Crossing:

principle

equations

=

I

=

O

J

we obtain

from these relations,

hold for all complex @.

u(Q)

=

t(i,- e)

(6.a)

r(Q)

=

r(iW-O)

(6.b)

Unitarity: u(O).u(- e)

=

I

(7.a)

t(O)'t(-O)

+ r(O).r(-@)

=

I

(7.b)

t(e).r(-O)

+ r(O).t(-Q)

=

O

(7.c)

We are now prepared The 2-particle

for stating precisely

S matrix

is defined

the factorization

equations.

to act on a multiparticle

state by

means of S2(Pi'Pj) ~Pl"" Pi'" Pj'" Pn>°ut = u(%j)IPl..

Pi'" Pj'" P n > °ut

and S2(Pi'Pj) IPl "" Pi'" P j " P n > Out = t(s*ij)IPl "" Pi'" Pj'" P n > Out

where ~ij = (pi-Pj)

+ r(s~ij) IPl "" Pi'" Pj'" P n > Out

2

In lowest order MTM perturbation easily, S matrix

that the reflection is really

a matrix.

ments do not commute

amplitude

this task it is useful

(cf.e.~./19/)

the factorization elastic

~nd £he 2-particle

S matrices~f

the order of the factors to consider

one may check

does not vanish

The 2-particle

and therefore

only sense if we specify wave packets

theory

different

e~uation

on the r.h.s.

scattering

of well

argu-

(3) makes For localized

such that all the ½n(n-1) 2-particle interaction ~oints are I I far apart from another. Let us assume Pl ~ P2~..-~Pn ~' then an orclen which corresponds to a physiCally~possiblescattering process is

322

n-1 ~F i=i

Sn(Pl.. Pn ) =

n

(

~F s2(pi,pj) j=i+1

)

(8)

The product factors have to be written down from le~t to right. For 3-particle

S3(P1,P2,P3)

scattering

=

(cf.fig.2a)

this reads

S2(Pl,p2) S2(Pl,p3) S2(P2,p3)

By parallel shifting of particle to (8) can be obtained,

(9)

lines ordering procedures eouivalent

which have to give the same result for the n-

particle S matrix. Already the restriction scattering enables factoriz~tion

obtained from 3-particle

of n-particle

scattering.

one line the equivalent equation to (9)(cf.fig.2b))

S2(P1'P2) S2(PI'P3) S2(P2'P3)

=

By shifting

implies

S2(P2'P3) S2(PI'P3) S2(P1'P2)

(1o)

Equations of this type are nnown for a long time /20/ as consistency equations for factorization tivistic

~--potential

of multiparticle

scattering.

amplitudes

It has, however,

in non-rela-

only been recog-

nized much later by Karowski et al /2/, that one may go the other way round and use these equations as input for the calculation (even relativistic)

of exact

S matrices.

- --in p>OUt Applying (1o) to~Pl,P2,P3 ~ and looking for the final statelP1,~2, the following severe restriction on the scattering amplitudes is obtained: h(031+012 ) =

where h ( O ) @ij

t(~) r( • )

h(i~'+O31).h(Q12) + h(031)'h(i11-~12)

and ~31 = - e13. For i(j the rapidity variables s - 4m 2 are defined by c h e i j 132 m 2 (cf.(5)) .

323

III.The exact S m a t r i x We are now able to d e r i v e the main result. equation

Let us first note,

that

(11) has the unique solution

sh~O h(~)

= ~ A

.,

(12)

here ~ is an arbitrary real parameter, w h i c h will become related to the coupling constant.

The proof of

(12) is similar to that of the a d d i t i o n

t h e o r e m for the exponential function a l t h o u g h s l i g h t l y more c o m p l i c a t e d is real because h(8) = h(-~).

Let us assume the crossing and u n i t a r i t y r e l a t i o n s

(6),(7)

and e q u a t i o n

(12). Under the following a d d i t i o n a l two a s s u m p t i o n s the s c a t t e r i n g amplitudes become u n i q u e l y

determined:

(a) The phase ~(~) defined by

u(@)=~u(~)le i 2 ~ ( 0 )

is bounded in the p h y s i c a l strip

(b)

.

The~e exists a repulsive region in the p a r a m e t e r ~, where

no poles and zeros in the p h y s i c a l strip

and for a r b i t r a r y ~

t(@) has t( ~ ; k)

can be obtained by analytic c o n t i n u a t i o n from the r e p u l s i v e region.

The e x p l i c i t results for the a m p l i t u d e s t(O) where

are:

F(O) F(2~i-O)

(13.a)

T (2l+-~+~'-T)(21+-~+i~-'~T)

F(,)=T k l

i=0

(211+

+ ~%

One could write F(@)

21+1+ U

m__

+

) 11[

as infinite p r o d u c t of ~ - f u n c t i o n s ,

not really a simplification.

t(O) sin~ r(~) = ~ = s i n f u l % t(e) and

but this is

The other a m p l i t u d e s are

(13.b)

-- le

s ~ T ~(1- ..0..) t(0)

u(~) = t ( i l - e ) =

s~T% 0 iT

(13.c)

Remark: For the MTM the region w i t h g ( o

is known to be r e p u l s i v e

(cf.e.g./15/)

The assumptions of a bounded phase and the absence of poles and zeros of t(8)

in the p h y s i c a l stripe for the r e p u l s i v e region are r e q u i r e d

by L e v i n s o n ' s t h e o r e m

(cf.next section).

324 Proof of

(13.a) :

U n i t a r i t y and crossing relations yield h(~) h(i~ -~))

-

t(0) t ( i T + O )

(14)

Let C denote a contour, w h i c h is identical with the b o u n d a r y of the p h y s i c a l strip

, except that possible poles or zeros of

t(6)) on the

b o u n d a r y of the p h y s i c a l stripe are c i r c u m v e n t e d by small half-circle~ (lying inside the physical sical strip

strip ) and let 0 be an element of the phy-

. Under the done a s s u m p t i o n s we have in the repulsive re-

gion the integral r e p r e s e n t a t i o n in t(0)

_

1 r dz 21Fi 4 sh(z_O )

C

1 -

In t(z)

~w r dx

2~i

~ sh~x in%sh)~(iT-x) 1

sh{x-e)

The last equality is obtained by using

(12),

(14).

A f t e r some c a l c u l a t i o n one obtains

t(Q)

=

exp i

dx sh •

x

x

sin

x

(iT - 6))

sh ~ ch y m

Using Malmst~n's formula /21/, we finally arrive at (13.a). D i s c u s s i o n of the result: From

(13a) we can read off the poles and zeros of t(0)

the zeros of the factors. One i m m e d i a t e l y recognizes, and zeros are lying on the imaginary between fig.3.

1/k

and Re ~

with

~

In the p h y s i c a l strip

~k = i,(

1

= ~

O-axis.

that all poles

Simple linear relations

are obtained.

They are drawn in

there are only simple poles given by

~

(k:1,2 . . . . .

)

[~] is the largest integer smaller then ~ boundstates

by looking for

u

[~])

(15.a)

The poles describe

ff

(bosons) . By t r a n s l a t i n g the 6)-dependence back in the

s-plane we obtain a spectrum of q u a s i c l a s s i c a l type /11/

mk = 2 m s i n ( ~ )

(k=1,2 . . . . .

IX])

(15.b)

W i t h the r e l a t i o n s h i p

-~between ~

and the MTM

1

coupling g

(16) (SG coupling ~,

cf. (2)) the quasi.

classical s p e c t r u m becomes exact as has been c o n j e c t u r e d in ref./11/. Before the exact S m a t r i x was known,

the s p e c t r u m

(15) has also been

325

derived by lattice methods /22/ and by assuming f a c t o r i s a t i o n for the SG boson S m a t r i x /23/. In contrary to the spectrum the q u a s i c l a s s i c a l S m a t r i x /24/ is only exact for integer ~ . tion amplitude

These are p r e c i s e l y the points, where the reflec-

(13.b) v a n i s h e s and a new particle

the p h y s i c a l spectrum.

(cf.(15.a))

enters

Z a m o l o d c h i k o v /I/ first o b t a i n e d the exact ~ M

S m a t r i x by looking for the simplest analytic i n t e r p o l a t i o n b e t w e e n these points, w h i c h has a n o n v a n i s h i n g r e f l e c t i o n amplitude.

It has been

checked /25/ that t(O) agrees up to third order in g w i t h M T M p e r t u r bation theory.

IV B o u n d s t a t e s and L e v i n s o n s ' s t h e o r e m In the previous

section we have a s s o c i a t e d simple poles

always poles and zeros in the p h y s i c a l strip t (O) w i t h ff boundstates.

It is, however,

now)

(we consider

of the amplitude

known in p o t e n t i a l scattering,

that only those simple poles of the S m a t r i x d e s c r i b e boundstates, which c o r r e s p o n d to zeros in a related Jost function.

In the n o n r e l a -

tivistic limit of the M T M ff s c a t t e r i n g is known to be d e s c r i b e d by a smooth p o t e n t i a l /26/.One may t h e r e f o r e conjecture,

that results similar

to p o t e n t i a l scattering hold for the r e l a t i v i s t i c MTM S matrix.

This

question has been answered a f f i r m a t i v e by explicit i n s p e c t i o n of the S m a t r i x /12/.

Let us first review p o t e n t i a l s c a t t e r i n g in one space d i m e n s i o n /27/)

dx

(1+~x~)I V(x) j ( ~

The Jost solutions f(x,k)

and

V(x)

= V(-x),

of the S c h r ~ d i n g e r e q u a t i o n

d2 I ~(x) + V(x) rk~/(x) = E ~'(x} 2m dx 2 are d e f i n e d by the b o u n d a r y c o n d i t i o n f (x,k) ~ f(x,k)

(e.g.

for a well behaved symmetric p o t e n t i a l V(x) :

e ikx

as

k2 E=~-~

x --~

is analytic in the open and c o n t i n u o u s on the closed upper half

326

of the complex k-plane.

The scattering solution of the S c h r ~ d i n g e r -ikx from the right

equation c o r r e s p o n d i n g to an incoming plane wave e is given by

(x,k) = f(x,-k) + r(k).f(x,k) = t(k).f(x,-k) r where t and r are the t r a n s m i s s i o n and r e f l e c t i o n coefficients. parity even

(+) and odd

The

(-) channels of the S m a t r i x are

s+(k)

=

t(k)

t r(k) .

They

obey the u n i t a r i t y c o n d i t i o n ~s+(k) I :I for k real. The Jost funcdf tions of the parity ~ channels are defined to be (fx = ~-~) f (O,k) Xik

f+(k)

It can be proved

(e.g.

and

f_(k)

=

f(O,k)

/12,27/) , that the Jost functions are related

to the a m p l i t u d e s by f+(-k)

st(k)

-

f+(k)

(17)

and I t(k)

:

(18) f+(k)" f_(k)

F u r t h e r m o r e it can be proved, that the Jost functions f+(k) have in the upper half complex ~ p l a n e no poles and only a finite number of simple zeros.

The zeros are located on the imaginary k-axis.

They are diffe-

rent for f+ and f_ and c o r r e s p o n d to boundstates with even or odd wave functions respectively.

If a zero is located at k = i ~ ,

~)o

the energy

of the c o r r e s p o n d i n g b o u n d s t a t e is E= _ ~ 2 .

We r e c o g n i z e that all poles

in t(k)

whereas

c o r r e s p o n d to p h y s i c a l boundstates,

in the s+ channels

r e d u n d a n t poles are possible.

We now state L e v i n s o n ' s theorem. For real k the phase shifts d e f i n e d modulo

T

by s+(k)

=

e2i

6t(k)

are real functions. For the Jost functions f+(k)=f+(-k) phase shifts are d e t e r m i n e d m o d u l o 2 ~ f~(k) = The functions f+(k)

(cf. (17))

holds and the

from

lf~(k) I e - i ~ t(k)

are analytic

in the upper half plane and we can

327 a p p l y the a r g u m e n t

principle

to g e t the n u m b e r s

of p a r i t y +

boundstates

' k ) f+(

I 2~i

n+

dk

--

Here

- --

_

f+(k)

C

f(k)~k~%~for

Equations

(19)

half plane,

The i n t e g e r s ~ ± of s+_ v i a

+

I + ~ (s+(o)

are L e v i n s o n ' s

theorem

(19.a)

-

half plane,

behaviour =

I -- TI~I~+

+ (~)) -

the u p p e r

i n s i d e the u p p e r

k-~o is assumed.

f r o m the t h r e s h o l d

- ~~

-

the c o n t o u r C e n c l o s e s

small half c i r c l e

I (2 1T ,v + (o)

-

avoiding

where

k=o on a

the b e h a v i o u r

can be o b t a i n e d

- I )

(e.g./12/

(19.b)

for one d i m e n s i o n a l

potential

scattering.

In the n o n r e l a t i v i s t i c complex 0-plane

l i m i t of the M T M the p h y s i c a l

becomes

the u p p e r

turn to the r e l a t i v i s t i c

Because

these

1

states

S2

parity

± eigenstates

+ i#1,P2>

( [p1,~2>

the t w o - p a r t i c l e

=

S matrix

the

s+

channels

with

the

s+(e).s+(-e)

Explicit

formulas

in the M T M are

s+

=

diagonal

t

+ r

.

--

unitarity

=

(21)

)

becomes

S

In

Let us n o w

I eigenstates

.+_ I> On

k-plane.

of the

case.

of CPT and T i n v a r i a n c e

i d e n t i c a l w i t h C=±

half complex

strip

relations

s _ (e) . s _ (-e)

(7.b,c)

=

read

I

for s+ are

sh2~-s+(e)

= - sh2~---

s_(S)

= -

e+i~ ) S-iT

t(ilr -6))

(22.a)

t(il~-S)

(22.b)

)

and ch2~---

8+i T )

ch2~---- ( S - i T )

We n o t e strip

from t h e s e e q u a t i o n s only p o l e s

for c e r t a i n

t h a t the

s+ a m p l i t u d e s

h a v e in the p h y s i c a l

i m a g i n a r y • and no zeros.

Let us e x c l u d e

328

= i n t e g e r for the moment, function of ~ in fig.4a,b).

then the poles are given g r a p h i c a l l y as The boson b o u n d s t a t e s

simple poles in s_ for k=odd and in s+for k=even.

(cf.(15))

energe as

T h e r e f o r e the bound-

states have parity P=(-I) k. This is consistent w i t h the opinion that the lowest

(=lightest)

b o u n d s t a t e b I is a pseudoscalar,

which corres-

ponds to the e l e m e n t a r y SG field.

It is not d i f f i c u l t to compute the residua

of the poles with the

result (+i).Res s+ = R k ) o

for

P=+1

(i.e. k even)

(23.a)

for

P=-I

(i.e. k odd)

(23.b)

@=e k (+i) Res s

= Rk
@=e k Here

Res f(@) = +lim (O-Sk).f(8). This is in a g r e e m e n t with the e=e k 8~8 k

result+): a) Symmetric wave functions correspond to n e g a t i v e and a n t i s y m m e t r i c wave functions c o r r e s p o n d to positive residua. general S matrix theory

This is required by

(e.g./28/) ; poles with the wrong r e s i d u u m give

rise to ghosts with n e g a t i v e norm. b) F e r m i o n - a n t i f e r m i o n b o u n d s t a t e s of p a r i t y P=-I have symmetric wave functions and of parity P=+I have a n t i s y m m e t r i c w a v e functions P u t t i n g a) and b) together we p r e c i s e l y arrive at

(e.g./29/)

(23).

Besides the physical b o u n d s t a t e poles we notice in s+(fig.4)

further

poles w h i c h are redundant. F o l l o w i n g the s u g g e s t i o n s - o f p o t e n t i a l scattering

(17),(18)

one can introduce "generalized J o s t functions"

by the

ansatz g+ (-e) s+(e)

-

-

(24) g+(e)

and I

t(@)

=

(25) g+(e).g

(e)

+ ) F o r m u l a t e d in the s-plane. R e s i d u a in s- and e - p l a n e are related by (i~)

Res = Res 8k sk

with

~)

o.

329

and finds the solution 2 k e (21-I+--~--f) • (21+1

Im ~ g+(e)

= T• = o = k--"1 =

)

2k-1

8 (26)

- - 2k+I 8 . 12k~2 8 21+2+---~----~) • (21+2 -iT )

X

and g_(~)=g+(O-

21+I12k+I~ -i'll'8). (21+3+2k~2

). The generalised

i )~ 8

Jost functions have no poles

and for imaginary ~ a finite number of simple zeros in the physical strip

. Parity ± boundstates

The functions Levinson's

g+(g)

theorem.

correspond presisely

to these zeros of g+.

can be used to obtain the generalized The numbers of parity ~ boundstates

version of

are

!

g+(8)

_

n~

I

2~ i

~

-

j d8 - -

¢

=

g±(e) (27 .a)

1 (~+(o)

- ~ +(,~)

)

I (~ ( 0 ) -

+

I~'(~)

)

- ½1~+

Here the contour C is identical with the boundary

of the physical

except t h a t 0 = o

inside the physical

strip

is avoided on a small half circle

, where we have the behaviour

g(O)~ 011+ for O ~ o .

can be obtained from the threshold behaviour

4+

I

=

¥ y (s+(o)

strip

The i n t e g e r s ~ +

of s+ via

- I ) (27 .b)

The sign difference shold behaviour case),

compared with

of bosons

cf./12/. Equations

theorem.

(19.b)

(potential

scattering)

(27) constitute

By explicit calculation

relies on the different and fermions

thre-

(present

the generalized Levinson

of the phase shifts we may obtain the

already known results n_= The

term

[(k+1)/2]

,

~I (~(o) - ~ (~)), which reflects

n+ + n _ = crossingi

[~]

is defined by

,

330

u(e)

= lu(e)I, e2i ~ (e) and gives the c o n t r i b u t i o n

In the semiclassical

1-(~-I)

limit all results agree w i t h those computed pre-

v i o u s l y /30/. This shows again, that the semiclassical method is reliable in the SG theory.

Let us close this section w i t h some remarks on the so f~r excluded case =integer.

For this case the r e f l e c t i o n amplitude v a n i s h e s and we

have s+(e) = s_(e) = t(e) T h e r e f o r e poles with the wrong r e s i d u u m

(ghosts)

appear in the s+ chan-

nels. F o l l o w i n g K a r o w s k i /31/ a c o n s i s t e n t particle i n t e r p r e t a t i o n is g u a r a n t e e d if there are no transitions between genuine particles and ghosts. Let us denote the ghost with mass m n by b'. A n t i c i p a t i n g m e t h o d n and notation of the next section the f bn-gf b' t r a n s i t i o n amplitude is n °ut~/b'f~b f~in ~ n ~ n ~= u23"t13 - t23"u13 This v a n i s h e s for u=+t , as is indeed the case for ~ i n t e g e r

(13.c).

S i m i l a r l y all other transitions between genuine p a r t i c l e s and ghosts vanish. For S matrices w i t h higher symmetries

(e.g. the chiral SU(n)

Thirring

m o d e l /6/) analog c o n s i d e r a t i o n s are quite r e s t r i c t i v e and imply, that only a finite number of b o u n d s t a t e poles can be introduced consistently. Further there are poles in the p h y s i c a l strip

possible, which are not

interpreted as b o u n d s t a t e s or ghosts. They are called CDD /32/ poles. If such poles are allowed,

i . e . a s s u m p t i o n b) of section III is given

up, the solution of the f a c t o r i z a t i o n equations for the t r a n s m i s s i o n amplitude becomes

sh½(e+ei/- ch½(e-ei) T(e)

=

~

sh½(e-Si)'ch½(8+ei).t(e)

where t ~ ) is the previous If the r e l a t i v i s t i c model

(minimal)

solution and the 8i; are imaginary.

(like the SG theory)

behaved p o t e n t i a l in the n o n r e l a t i v i s t i c

is d e s c r i b e d by a well

limit, CDD poles are excluded

/33/, because they w o u l d be in c o n t r a d i c t i o n to the existence of Jost functions.

331

V The complete S m a t r i x The complete S m a t r i x includes scattering ref./13/,

of boundstates.

Relying on

this section is devoted to the d e r i v a t i o n of the b o u n d s t a t e

scattering amplitudes.

The ff boson b o u n d s t a t e of mass m k as given by

(15.b)

is d e n o t e d b k.

There is the p o s s i b i l i t y of b o s o n - b o s o n s c a t t e r i n g bkb I and of bosonfermion scattering bkf. For the d e s c r i p t i o n of scattering of p a r t i c l e s with d i f f e r e n t masses m k , m I we use the r a p i d i t y v a r i a b l e O k l , d e f i n e d analog to

Skl

(5) by 2 2 m k + m I + 2mkmlCh(Skl) .

=

(28)

Skl is the complex M a n d e l s t a m variable, A

2

w h i c h for p h y s i c a l v a l u e s

reduces to _ Skl=(Pk+Pl ) . The l.h. cut Skl=(mk-ml )z and the r.h. cut

in the Skl-plane starts at 2 at Skl=(mk+ml ) . We further use the

variable ~kl

=

--Skl iT.

(29)

If no m i s u n d e r s t a n d i n g is p o s s i b l e the indices k,l are dropped.

The f e r m i o n - f e r m i o n amplitude u(8)

(13.c)

is c o n t i n u o u s at the bound-

state poles 8 k as may be seen from fig.3. T h e r e f o r e and from the discussion of the s+ amplitudes -

fermion sectors the residua

(cf.

(22) ff.) we conclude,

of the S m a t r i x at s12 =

(P1 and P2 are at u n p h y s i c a l values) presisely,

that in the

(PI+P2)

2

2

=m k

act as p r o j e c t i o n operators. More

the o p e r a t o r s P12kdef_ Rkl

p r o j e c t for ~ i n t e g e r

~=2 ( I Pl 'P2> +

Res2

S(PI'P2)

s12=mk onto c h a r g e l e s s (-I

(30)

states

)kl 1,p2>)

with m o m e n t u m p=p1+P2 , mass m k and p a r i t y

(-1)

k

They are identified with the b o u n d s t a t e s b k. Let us first consider the S m a t r i x for b o u n d s t a t e - f e r m i o n It is defined by

scattering.

332

S(P1+P2,P3) l(p1+p2 )2=mk2 = 2 2 P3 = m

Using the factorization

Rkl

Res2

S12S13S23

s12=mk (31 .a)

equation

(Io) and the definition of the pro-

jector we find I Rk

Res

2 s12=mk

Formulas

k k k $12S13S23 = P12S13S23 = S23S13P12 = P12S23S13P~2

(31) have a graphical

interpretation

(31.b)

as given in fig.5a).One

can associate with b k a fermion and antifermion with "parallel" momenta Pl and P2" Since for real and parallel momenta the mass shell condition 2 2 Pi=m implies (p1%P2)2 = 4m 2 , the momenta have to be complex and only the real parts are parallel, p=p1+P2 is real, because it is the physical momentum of b k. A convenient choice is p1=(p °, Repl+iImp I) and p2=(p O, Repl-iImp I) with p°real.

(p1+~2 ~

can take all values between o

and 4m 2 for convenient choices of Imp . We now calculate the amplitudes explicitly. There is no reflection, k since the projector P12appears on both sides of equation (31b). It is convenient to use amplitudes defined as functions of ~ by f ( ~ ) ~ _ f (~(~)). The boson-fermion becomes now

scattering

out tbkf(~)

=

amplitude thkf(~)

out

I

--

where

t23 = t( T 2 3 )

and

~13 =

Using equations

(cf.(29))

(t23u13 + (-I) n r23r13 + u23t13)

etc.

~ + 1(1-~ ) '

~23 = ~

1 ( I - ~ )"

(13) for t,r and u one finds after some calculation

333

k I k sin~(~ +7~-+~)

tbkf (~.) =

(-I)

sin~C~

k I -2--~-+~) k I sin~( ~ +2--]~-~) k I sin~ ( ~ -~--~--~)

(32) sin~( ~ + ~

k-2j 2

sing( T

'=

+ ~)

Similarly boson-boson scattering can be treated. The S matrix for scattering bnb m is defined by

S(PI+P2,P3+P4

(p1+P2) 2 = ~ (p3+P4) 2=m2

I Rk

Res

2 s12=mk

I R1

Res

2 $12S13S14S23S24S34 s34=ml

cf.fig. 6b).Again there is only forward scattering. After some calculation one finds for k &l:

tbkbl(~)

tg~(~ +2-]C) k+l. tg~-(~ +2~) tg2( ~ -2--~ -)k+l" tg~(~ _~_~)l-k I~ ~j=1

2

k+l-2j

tg~( '~

'~

(33)

/ )2

A consistency check is to build up the t b b amplitude from bkff k 1 analog checks were scattering. The result is the same. This and carried out by Karowski and Thun /13/. All amplitudes are in agreement with previous calculations /24/ in the semiclassical limit.

334

Particularly

simple is the scattering

amplitude

for two elementary

SG

bosons bl:

tblbl ( ~ )

At

~=3

•

sin~ ~

+ sln~

sin,~

- sin~

or equivalently ~ 2 = 2 T ,

where m1=m and m 2 = m ~ ,

-u(O)

= tb1~(~)

one recognizes

/34/ = -t(e) = tblf(O)

= tblb1(e)

and tb2f(e)

= tb2~(e)

This can be interpreted are in an isotriplet

= tblb2(e)

as a hidden SU(2)

symmetry.

and b 2 is in an isosinglet.

this symmetry was first observed by Coleman

The particles

f,bl,f

Using different methods

/35/ in his investigation

of

the massive Schwinger model. Finally

let us consider

the poles

(there are no zeros of t b f and k . In the boson fermion amplitude tbk f there

t b k b l i n the physical strip are two simple poles at +k -

+

(34.a)

I

and double poles at

±9 (34.b)

+ I 2

2~

....

with j=o,2,.. ,k-2 if k is even and j=I,3.., Let us choose k&l for the boson-boson

k-2 if k is odd.

amplitude

tbkbl.

Then we find simple poles at k+l = 2--~ '

~

l-k = 2-]¢

(35.a)

and double poles at

= k+l-2 j

(9=I . . . . . k-l)

(35.b)

335

F u r t h e r we encounter the poles r e q u i r e d by crossing symmetrie at i - ~ .

For the special c a s e s t b~f

and tbo Ab the poles are drawn in fig. 6a)

and b).At the first i o o ~ it is surprising,

that besides simple poles

double poles are found. This and related q u e s t i o n s are

c o n s i d e r e d in

the next section.

The simple pole at ~ =

(k+l)/2 k

in the b o s o n - b o s o n amplitude tblblK

has

the i n t e r p r e t a t i o n as boson b o u n d s t a t e bk+ I. This is seen by c a l c u l a t i n g the associated mass and checking the residuum.

A l s o the simple pole in the b o s o n - f e r m i o n amplitude tblf interpretation.

has a natural

It's associated mass is p r e c i s e l y m. T~e m e a n i n g is,

that the fermion f can be interpreted as bk~ boundstate.

This is consi-

stent with the picture /36/ of the SG soliton being a c o h e r e n t boundstate of SG bosons. The soliton may "eat" a boson and remains a soliton. A g a i n the check of the r e s i d u u m is consistent. The sign of the r e s i d u u m formula

R b of a b o u n d s t a t e b is given by the compact

(b=boundstate)

Rb~ b / ( ~ ) w h e r e ~ b k = ( - 1 ) k, ~ f =

~=i

~ o and

*,~

(36) are the b o u n d s t a t e

constituents. M o r e r e c e n t l y /37/ the i n t e r p r e t a t i o n of the fermion as b o s o n - f e r m i o n b o u n d s t a t e has been used to r e c o n s t r u c t the M T M fermion S m a t r i x from the boundstate S matrices.

This m e t h o d can be used /38/, to obtain the

kink S m a t r i x of the Gross N e v e u model, where the b o u n d s t a t e S m a t r i x is known /4/.

VI A n o m a l o u s and virtual The double poles

(34.b),

thresholds (35.b)

of the p r e v i o u s

section turn out /14/ to

336

be anomalous what

threshold.

an anomalous

For physical

We follow

threshold

external

for those values

time

graph of a process

on mass

shell,

energy-momentum

/39/.

involving forward

conserving

The d i s c o n t i n u i t y

normal L a n d a u

of the momenta,

all m o v i n g

the

singularities

for w h i c h

classical in time,

interactions

of the s i n g u l a r i t y

One e v a l u a t e s

ref./14/. Let us first recall

is.

momenta

occur

closely

space-time

of the S m a t r i x

one can d r a w a space-

particles,

which

and i n t e r a c t i n g

localized

is o b t a i n e d

are all

only through

at a space-time by the C u t k o s k y

graph as if it were a Feynman

point. rules graph,

w i t h two exceptions:

a) The point b) F e y n m a n

Singularities this

interactions

propagators

are r e p l a c e d

are r e p l a c e d

below threshold

case one may choose

to be real and the

are called

The rules

S matrix

elements.

~-functions,

@(p~ ~ (p2_m2)

anomalous

the time c o m p o n e n t s

space c o m p o n e n t s

is called p s e u d o m o m e n t u m .

by actual

by m a s s - s h e l l

singularities.

of the external

to be imaginary. for finding

In

momenta

Such a two-vector

an a n o m a l o u s

singulari-

ty are /40/:

(I) The graph must be a g e o m e t r i c a l (2) W i t h every

internal

figure

line there must be a s s o c i a t e d

of lenght m, the mass of the p a r t i c l e (3) The p s e u d o m o m e n t u m (4) A l l p s e u d o m o m e n t a

Thus,

locating

plane

geometry.

The C u t k o s k y branch

thus we obtain

must be p a r a l l e l

singularity

As an example

show that

We have

a pseudomomentum w i t h this

line.

line.

sum to zero.

becomes

let us consider

a problem

the graph

in four d i m e n s i o n s

discontinuity

space.

to the a s s o c i a t e d

an e i g h t - d i m e n s i o n a l

a finite

In two d i m e n s i o n s

associated

at a given v e r t e x must

an anomalous

rules

point.

in E u c l i d e a n

the

integral

in E u c l i d e a n

of fig.7).

singularity over

is a

six ~ - f u n c t i o n s ;

over the cut.

we have only a f o u r - d i m e n s i o n a l

integral.

Thus we

337

have

two ~ - f u n c t i o n s

left after

is a double

pole.

the F e y n m a n

diagram associated

two virtual

is best

particles

the F e y n m a n

integral

as d e r i v a t i v e tor.

This

w i t h respect

larity of its d e r i v a t i v e

to show,

an a s s o c i a t e d

mA2 For fig.9

to be a p o s s i b l e

The angle ~ amplitude.

equations

dimensional have

are

~(bk,bl;bk+l) = ~

fermion

will

38)

the

of the s i n g u l a r i t y

it is d e t e r m i n e d

in the

to locate

tAB

(38) the double

poles

corresponding

for any two

to fig.7.

A l l we

masses.

the r e l e v a n t

angles

of the three p a r t i c l e

to be (39.b)

(39.c)

(39.d)

all double

scattering:

into either

By equation locations

poles

can be explained.

Let A be f and B be b

into a fermion

(37) and

i 2

poles

j +

2~

where

Therefore

(38) we obtain

of the d o u b l e

the

the initial initial

pair or two bosons.

the case,

pair.

For example

. While

and a boson,

a fermion-antifermion only consider

=

thus

to be

~(B,c;b)

into a f e r m i o n - a n t i f e r m i o n

in fig.7.

intersect,

k ~(bl'bk+l;bk) = |-21-

for i l l u s t r a t i o n

decays

a and b m u s t

l k ~(bk,f;f) = ~ (| + ~ )

can only d e c a y

m a y dec a y

A

of a,c:

(39.a)

these rules

fermion-boson

to the r a p i d i t y

exists

(37)

process

computed

there

(a,c;A)~

in the p a r t i c l e

(15.b)

readily

k ~(f,~;b k) = ] - ~

Using

-

scattering

For the SG s p e c t r u m

(35.b)

For the d e c a y of a p a r t i c l e

the lines

geometry

are all we need

forward

(34.b),

{ I

~(A,c;a)

to do is to plug

vertices

graph,

pole

is related

is the location

By e l e m e n t a r y = 2 -

T hese

in fig.8

~(b,c;B)

of fig.9

the

with only one such fac2 2 is (Pc-mc) . Thus the singu2 mc 2,) and the to I'" /tPcsingu-

is p r o p o r t i o n a l

and vice versa.

+

of

conservation,

is a double pole.

defined

~(c,a;A)

the s i n g u l a r i t y

By m o m e n t u m

to m c of an integral

2 + 2mamcCOS1~ ~ ma2 + m c

=

singularity

w h i c h we denote Pc" Thus 2 2 I/(Pc-mc) and can be w r i t t e n

two factors

that for each d o u b l e

diagram

the angle ~(a,c;A)

shown by c o n s i d e r i n g with fig.7.

of the n e w i n t e g r a l

of the n e w integral

It remains

The a s s o c i a t e d

carry equal momenta,

contains

The d i s c o n t i n u i t y

larity

integration.

the initial

We boson

a=b I , b=f and c=f

2k-l))~

are given

boson

by

A n d by e q u a t i o n

338

with

j=21-k,

1=I ....

~k-~]

range given

in

(34.b).

c onsid e r i n g

the decay

The d i s c u s s i o n %f

and

ffKand The

~b

this

is seen to d e s c r i b e

The other part of the range of the initial

can be c o m p l e t e d . Especially

boson

/14/

there

a part of the

is o b t a i n e d

into bosons.

and explains

all d o u b l e

poles

are no a n o m a l o u s

thresholds

in the

in

ff amplitudes,

because (37) cannot be f u l f i l l e d for this case. l-k simple pole a t ~ = ~ - (35.a) is e x p l a i n e d by the tree graph of fig.9

Let us finally

consider

fig.6.

The pole given by ~ = 6 k c o r r e s p o n d s

)~%6 to the mass m 6 of the b2b 4 b o u n d s t a t e to be present

in the M T M

spectrum

of ff b o u n d s t a t e s

Precisely

at this point

the pole at

~=6/2)%

and the c r o s s i n g

pole give

a double

in the p h y s i c a l of their

point

strip

residuum,

longer natural.

corresponding

virtual

For

to the a n o m a l o u s

because

threshold

theory

of the ~ M

b) Jimbo, Miwa,

soliton

a resonance.

laws.

This

The remnant

is

is a

considerations

amplitude.

lecture

is the c o m p l e t e

behaviour.

functions.

at this c o n f e r e n c e Sato and c)

functions

of the MTM means

approaches

program

Green's

closely has

con-

to this pro-

of a) Honerkamp, program

outlined

functions

related

in the

from

to the present

lead to a simple

of the Ising m o d e l

closed

solution

The b o o t s t r a p

to c o n s t r u c t

and fairly

solution

in the lectures

is t h e r e f o r e

So far the b o o t s t r a p

of the Green's

Complete

Different

Karowski.

of Karowskit~ies

exact S m a t r i c e s a n d

lecture.

is no

be a b r a n c h

b2b4--gf~ , the b o u n d s t a t e

pole. A n a l o g

~kbl

the sign

as b o u n d s t a t e

is a well u n d e r s t o o d

open p r o b l e m

with

of the Green's

blem are treated

in the

threshold

in the

changed

there w o u l d

sheet and become

in form of a double

A challenging

and other m o d e l s

known

however,

of the c o n s e r v a t i o n

symmetric

and o u t l o o k

The S m a t r i x

struction

have,

(15).

again two simple poles

such that an i n t e r p r e t a t i o n

apply to the b l+k b o u n d s t a t e

topic.

we have

. The poles

into the second

here,

VIISummary

4 ~ 6

In a theory w i t h production,

pole w o u l d move impossible

pole.

for

b 6. At )%=6 the b o u n d s t a t e

b 6 ceases

/41/

by

construction

scaling

limit.

Acknowledgement It is a p l e a s u r e during

to a c k n o w l e d g e

this conference.

and P.Weisz

for u s e f u l

the kind h o s p i t a l i t y

I also w o u l d discussions.

of P r o f . I a g o l n i t z e r

like to thank M.Karowski,

H.J.Thun

339

References: I.

A.B.Zamolodchikov,

2.

M.Karowski,

3.

A.B.Zamolodchikov

(1977)

Phys.5_~5 (1977)

T.T.Truong

and P.Weisz,

183 Phys.Lett.

67B

321

(1978) 4.

Commun.math.

H.J.Thun,

and Aj.B.Zamolgdchikov,

Nucl.Phys.

B133

and Aj.B.Zamolodchikov,

Phys.Lett.

72B

525

A.B.Zamolodchikov

(1978)

481 5.

R.Shankar

6.

B.Berg and P.Weisz,

and E.Witten,

and P.Weisz,

Nucl.Phys.

7.)

V.Kurak and J.A.Swieca,

8.)

A.B.Zamolodchikov (1979)

9.)

Phys.Rev.

Nucl.Phys. B157

(1978)

(1978)

(1979)

Phys.Lett.

2194

2o5; E . A b d a l l a , B . B e r g

387

82B

(1979)

and Aj.B.Zamolodchikov,

289

Ann. Phys.

(N.Y.)

12o

253

M.LOscher,

Nucl.Phys.

P.P.Kulish

and E.R.Nissimov,

B.Berg,

D17

B146

M.Karowski

B117

(1976)

475;

JETP Lett.

and H.J.Thun,

24

(1976)

Nuovo Cim.

38A

R.Flume and S.Meyer, Nuovo Cim. Lett. 18 (1977) J.Lowenstein

and E.Speer,

1o.) D.Iagolnizer, 11.) R.J.Dashen,

Phys.Rev. D18

B.Hasslacher

V.E.Korepin

Commun.math. (1978)

63

and H.J.Thun,

11;

(1978)

Phys.Rev.

Theor.Math.Phys.

W.R.Theis

(1977)

238;

1275; Phys.Lett.

and A.Neveu,

and L.D.Faddeev,

12.) B.Berg, M. Karowski,

Phys.

220;

76B

D11

25

97 (1978)2o7

(1975)

(1975)

Phys.Rev.

3424;

Io39

D17

(1978)

1172 13.) M.Karowski

and H.J.Thun,

A.B.Zamolodchikov, 14.) S.Coleman

and H.J.Thun,

15.) S.Coleman,

Nucl.Phys.

Preprint

Phys.Rev.

D11

(1975)

(1973)

D.J.Kaup,

61

(1978)

31

2088

Fortsch. Phys.

A.C.Newell

25

and H.Segur,

and L.D.Faddeev,

(1977)

3o3

Phys.Rev.Lett.31

Theor. Math.Phys.

Infinitely many higher conservation M.D.Kruskal Summer

295;

152;

L.A.Takhtadzhyan

and D.Wiley,

N.Y.,

Chu and D.W.

21

American Mathematical

Society,

ed, A.C.Newell,

July 1972. For a review see A.C.Scott,

McLaughlin,

(1975)

Io46

laws were first obtained by

seminar on nonlinear wave motion,

Potsdam,

(1977)

(1977) u n p u b l i s h e d

Commun.math.Phys.

For a review see: J.A.Swieca, 16.) M.J.Ablowitz,

B13o

ITEP-12

Proc. IEEE 61 1

(1973)

1443

F.Y.F.

340

17.) P.P. Kulish and E.R. Nissimov /9/ and references given therein. 18.) I. Ya. Araf'eva and V. E. Korepin, JETP Lett. 2o (1974) 312 19.) B. Berg, Nuovo Cimento 41A (1977) 58 and references given therein. 2o.) J.B. McGuire, J. Math. Phys. 5 (1964) 622; E. Brezin and J. Zinn-Justin, Ccrmpt. Rend. Acad. Sci. Paris B263 (1966) 67o; C.N. Yang, Phys. Rev. Lett. 19 (1967) 1312 and Phys. Rev 168 (1968) 192o 21.) I.S. Gradshteyn and I.M. Ryzhik, "Table of Integrals Series and Products", Acadestic Press (N.Y.1965), 8.341 (7), p.939 22.) A. Luther, Phys. Rev. B14 (1977) 2153 23.) S. Nussinov, Phys. Rev. D14 (1976) 647; B. Schroer, T.T. Truong and P.H. Weisz, Phys. Lett. 63B (1976) 422; cf. also M. Karowski /37/. 24.

V. E. Korepin and L.D. Faddeev /11/

25.

P.H. Weisz, Nucl. Phys. B122 (1977) I

26.

V.E. Korepin, JETP Lett. 23 (1976) 224; for a review see /8/.

27.

L.D. Faddeev, Am. Math. Soc. Trans. 65 (1967) 139

28.

W. Brenig and R. Haag, Fortschr. Phys. 7 (1959) 183

29.

J.D. Bjorken and S.D. Drell, "Relativistische Quantenfeldtheorie", B.I. Hochschultaschenb~cher 1oi / 1oi a+, S. 124

3o.) R. Jackiw and G. Woo, Phys. Rev. D12 (1975) 1649; S. Coleman, Phys. Rev. D12 (1975) 165o 31.) M. Karomski, Nucl. Phys. B153 (1979) 244 32.) L. Castillejo, R.H. Dalitz and F.J. Dyson, Phys. Rev. Ioi (1956) 45~ 33.) R. Haag, Nuovo Cimento 5 (1957) 2o3 34.) S. Meyer and P.H. Weisz, Phys. Lett. 68B (1977) 471 35.) S. Coleman, Ann. Phys. (N.Y.) 1oi (1976) 239 36. ) R. Jackiw, "Collective Phenmmena in Quantt~n Field Theory", Cracow lectures 1975, Acta Physica Polonica B6 (1975) 919 37.) M. Karowski, Kaiserslautern lectures 1979, to be published by Plenum Press (Londcn) 38.) M. Karowski and H.J. Thun, to be published 39.) R.J. Eden, P.V. Landshoff, P.I. Olive and J.C. Polkinghorne, "The analytic S matrix", Cambridge University Press (Cambridge 1966) , p.87 ff. and p.11o ff 40.) J.D. Bjorken and S.D. Drell /29/ S.232 ff. 41.) B. Berg M. Karowski and P.H. Weisz, Phys. Rev. D19 (1979) 2477

341

s-prone

0 -plane

11;:

l,m 2 s-plane

O-plane

fig. 1 a

Fig. lb

t

t,

51,2 S 1,3

52,~

P1

P3

52,3 $1,3 $1,2 /

I

Fig. 2a Re ~o

Fig.2 b

Equivalent possibilities for factorization of 3-particle scattering.

Re%O 0

= 11~, 1

0

Fig./,a a) Poles and zeros of the amplitude s+ = t+r.

Oo

141/3 1/2

b) Poles and zeros of the amplitude s =t-r.

Fig./, b

1

1/;~

342

Redo

ImO 11:

='--

¸

3

2

physicoL sheet votues

-I

-2

\

--3

o ~6 ~,,'~3 ~7 ~. =,,.~,

I

1 X= I

~=0 g-'re/2

~g~o --,,,~

~:~" poles ~

[3z= 8~

zeros

Fig. 3 Poles

(

) and zeros ( - - , - - )

of t(O) f o r a r b i t r a r y

coupling.

343

Y/

D

:}'

f P; P2

P3

bsk,,//'\bt

P

Pt÷ P2

Fig.Sa - Composition of a fermionantifermion pair to a boson b k.

P;+P2

P3"P4

Fig.5b Composition of two fT pairs to bosons bk,b I. -

Re %0

Re ~o 1

S 1/;~

o Fig.6a - Poles ( ) and double poles ( .) of tb4 f .

0

0

Fig.6b - Poles and double poles of tb4b2 .

B ~T~ lP~.~A (e,c+b)(~,-~~

Fig.7 - The graph that produces double poles as anomalous thresholds in two dimensions.

A,c,o )

bt

bk

,~~q~(o,c iA)]c

A Fig.8 - Graphical representation of the ~p(a,c;A).

bt

/\ bk

Fi~,.9 - Graph explaining a ~ e pole.

FIELD THEORIES IN ]+I-DIMENSIONS WITH SOLITON BEHAVIOUR : FORM FACTORS AND GREEN'S FUNCTIONS +) M. KAR~gSKI Institut f~r Theoretische Physik, Freie Universit~t Berlin D-tO00 Berlin 33, Arnimallee 3

Abstract : For a number of field theoretic models with soliton behaviour the on-shell solutions have been obtained in the last few years. In this paper it is shown how to calculate offshell quan'tities like form factors and Green's Functions by means of the known S-matrices, general properties of quantum field theory and minimality ~ssumptions. i. Introduction

Various i+i dimensional field theoretic models with soliton behaviour have been investigated recently, e.g. the sine-Gordon the O(N) nonlinear

al~as the massive Thirring model ~],

~ -model and the Gross-Neveu model[2]

[3] and a Z(N)-Ising model in the scaling limit~].

, the chiral SU(N) model

The later will be used in this

paper in order to demonstrate the bootstrap program for the construction of offshell quantities like form factors and Green's functions from the known exact S-matrix, general properties of quantum field theory and some minimality assumptions. A Z(N)Isingmodelin

the scaling limit is defined by taking the correlation functions of

a Z(N)-Ising mod~l on a 2-dimensional lattice in the limit lim T+T with

Ri ÷ ~

ll- T - n B T c

o

and

: S(r1"''rn) n

ri= RiII-T/Tcl

fixed. By analytic continuation one

obt&ins from the euclidian Schwinger functions the Green's functions in Minkowski space. For N ~ 2

there exist several Z(N)invariant nearest neighbour interactions.

Let us make the assumption that one of these interactions enforces soliton behaviour for the model in scaling limit.

For field theories with soliton behaviour the dynamics is govern~by an infinite number of higher conservation laws. In particular for scattering processes they imply the

absence of particle production and factorization of the n-particle S-matrix[5] n-i s(n)(pl...pn ) : i=l

where the two-particle S-matrix on the particles with momenta

S(2)(pip j ) Pl

and

n

j:i+l

S(2)(pi,pj )

(i)

applied to an n-particle state acts

pj

(2) S(2)(pi,Pj)

I...~i(Pi)...~j(pj)...>in=

I...~'.(pi)...~'j(pj)...> in

S

(Pi,~)

~,~

~.~.

13

13

The ~'s label the kinds of the particles. Together with unitarity, crossing and minimality assumptions this property allows the exact calculation of the S-matrix

345

[1,2,3,4] . Let us repeat br&efly the argument for the Z(N)-Ising model with soliton behaviour

[43

. Assume the existence of an elementary boson b I corresponding

to the order variable O(x) and the existence of a two-particle bound state b2: (blb I) . Then [6] we get the series of bound states b k with masses sin a k sina

mk : ml

(3)

Since the order variable 0 assumes the values - the N roots of one - we have o

+

= o

N-I

(4)

which suggests the assumption that the antiparticle bl is identical to the bound state bN_ I. Then from ml=mN_ I we get for the parameter a the value

a = ~ Tr

(S)

Minimality means that the transmission amplitude for the scattering of two particles b I has only the po&e corresponding to the bound state b 2. This implies uniquely[4] 2zi SII ( 8 ) =

(sh8 +~ ½~ i(@sh

(6)

where @ is the rapidity difference defined by plP2 : m2ch 8 .

(7)

From eq.(6) we obtain for the scattering of the boundstates b'. and b k

[ j -k I

j +k

Sjk(8) : exp 4 7 d__~x c h x (iN )- c h x ( l o x sh x th ~

8 N ) shx:-IZ

(8) •

N

This formula is consistent with crossing ,

more general even with

Sj~

bk = bN_ k

the

above assumption bl=bN_ I and

since

(8) : Sjk(iz-8) : S.3 N-k(8)

(9)

This model is simple because of absence of backward scattering which is a consequence of bl: bN-i [4] . For N=2 the model is much more simple since there is

only one kind of particles

b=b and the two-particle S-matrix is s (2): - I

(i0)

Because of this simplification the bootstrap program has been carried through for this model

[8]

S48

2. Watson's Theorem ~] For form factors,i.e, matrix elements of local operators,we derive a set of equations which follow from general principles of quantum field theory, maximal analytieity,and the S-matrix faetorization. For simplicity we first conSider the case where we have only one kind of bosons in the model and a hermitean operator O(x). If we define the function F(@) (for 8 c.f. eq.(7)) by follows from CPT-invarianee >out F(@)=
in : F(@)

it

°ut: F(-8)~ unitarity and factorization , and crossing = F(ix-@) .

equations:

F(8) : F(-8)S(8)

(i1)

F(iz-8) : F(iz+8)

The corresponding equations for the general case where the ~'s denote the different kinds of particles read °ut<~l(Pl)'''~m(Pm ) ]O(°)I'''~n(Pn) >in = F~I..~ n (8ij ,iZ_ers,ek~) :

S (m)

(8. .) ,

~1"''~m where

..@1

el "

i]

m

F

(-@ij,i~+Srs,-Sk£) ~,...~T

1

l(i<j(m , l(r~m<s~n , m
and

S (n-m) sT

n

m+l"

..~V

n

(@k~) ..d

m+l "

n

o~m~n .

Solutions of these equations are only known for simple cases.

3. Construction of Form Factors The construction of form factors has been performed for models with an arbitrary number of kinds of particles for the oase n=2.

Hence

one particle expectation values

of local operators, the usual form factors are known exactly for various models

[7].

Absence of backward scattering enablesus to make proposals for more general form factors with n>2 , e.g. for the sine-Gordon

~ ] and the Z(N)-Ising model. Further-

more arbitrary n-particle matrix elements are known for the Z(2)-Ising model in the scaling limit [8]. There are two distinct problems in this procedure

.

a) the "matrix problem": The set of matrix equations (12) has not been solved for the general case. For n=2 we diagonalize the two-particle S-matrix and get for each eigenvalue the simple equations (ii) .

847

Theorem ~] An analytic function F(@) fulfilling eqs.(ll) is (up to a normalization) uniquely determined by the positions of the poles (and zeros) at O< Im @ < z

8=i~ k in the physical strip

: •

°

F(@) = K(8) Fmmn(@) where K(8) = const

(13) -i

sh-~ (@-i~ k) sh-~ (8-i~ k

(13a)

is a solution of eqs.(ll) with S--I and

l?

Fmin(@) = exp 4--~m_~

l ch 7 @

dz i

sh -;-( 2 z-@ )

ch

1

7

£n S(z)

(13b)

z

( ] For vanishing refl&ction S "n" is diagonal, and a solution of the general equations

(12) is F i..C~n(Si2...)

= K

~i.-an

(0i2...)

II i<j

Fmin (@ij) ~'(~'i ]

(14)

where the function K is a solution of eqs.(12) with S=I, this means it has no branch cuts in terms of the variables

pip j .

b) The "pole problem": The poles in the physical region, of the function in F~i..~n (8i2"'') : which are contained in the function K, are determined by one-particle states in all subchannels

~ .... ~. . If we make the minimality assumption that there are no 11 lj further so-called redundant poles and no zeros in the physical region, we can make

proposals for exact form factors, which can be checked by perturbation theory.

4. Examples a) The Z(N)-Ising model in the scaling limit with soliton behaviour. The minimal form factor function corresponding to the external legs bj and b k for the Z(N)-Ising model follows from eqs.(8,13b): min

%k

(i 0):exp ?o -dx x

ch

) sh2x

th

chx(

j+k, ~ (i- c h x "

)

(15)

348 Since there is no backward scattering in this model the general form factor is given by eq.(14). The determination of K as a rational~function of piPj is in general a complicated problem. Let us consider in more detail the case N=3. Then we have two particles bl,b2=~1 with

mass m and the two-particle

Sll(e)=S12(i~-@) =

i

S-matrix

2~i]

(i~)

sh ~ (e- -~-.

The minimal form factor functions are

Fminll (iz-@) : exp ~

min(i~.e) : exp ~ Flz 8

dx x

dx x

sh

3 (l-chx~)

sh2x

2x sh-3 sh2x

(17) (l-chx~)

The matrix elements of the order variable o(x) must have poles corresponding to the bound state

b2:b I of two particles b I . Hence we make the proposals:

i in : const (Pl+p2)Z_ m z

i n l r

= const

min Fll (812)

(18a)

+ 2 (p1 1 ~ Fmin (8 ) + P22) m 2 )z_ (pl p2) (pl+pz+p3 m 2 i<j ~.~.- iji 3

(18b) •

_min

,8. ,

(pi+Pj) 2- m 2

(18c)

The numerators in eqs.(18b,c) prevent reflection in two-particle scattering and particle production 2+3, respectively. The proposals (18) fulfil the consistency conditions

Res

812~ i

in = const in

(19a)

349

in Res @ = 2~i iz 3

in = const (19b)

Furthermore from the four point function (18b) one gets back the two-particle Smatrix eq.(16):

$11(012) = 1 + 4mZish@12

pl+p2+p3)Z-m2)

(Pt+P2+P~)2:m2 (20)

We have up to now no other possibilities to check these results, because we do not know, as mentioned in the introduction, which interaction is that one which enforces the soliton behavioum ~6r the Z(3)-Ising model in the scaling limit.

b) The Z(2)-Ising model. For the Z(2)-Ising model in the scaling limit the problem of calculating arbitrary matrix elements of the order variable

< o [ a ( o ) l p l . . . p n > in = F(n)(812)..) has been s o l v e d c o m p l e t e l y

~].

(21)

The minimal s o l u t i o n of Watson's e q u a t i o n s (11) w i t h

S(2)=-I is obtained from eq.(15)

(22)

Fmin(8) : i sh ~

2

Since there is no reflection we can use formula (14). The function K (n) contains the poles from one-particle states in all subchannels. Because there is only one kind of bosons b with mass m, we obtain for n=3 K(3)

=

const (PI+P2+P3)z'm2 =

Z (3) , ch 7e12 - ch ~@13 - cn

8_%3 2

(23)

This proposal fulfills the consistency condition that F (3) with the leg corresponding to o(x) on-shell, as in eq.(20), has to reproduce the correct S-matrix 8 (2). From this condition we also obtain the constant to be Z(3)=2.

For n~5 the arguments are just slightly more involved. The function K (n) must have poles at (pi+Pj+Pk)2=m 2

where any set of particles i,j,k is

combin/d to give a

350

one -particle state. However, the residues of "higher poles" where more than three particles build up a one-particle state must vanish, since the presence of such poles would be in contradiction to the absence of particle production. Hence the f

%

minimality hypothesis implies that K in) has the form n-3 K(n)

const [i~j (Pi+PJ)~ -~-=

z(n) -

(24)

2 m] i<~
~ ch 28-5-j i<j

The numerator is necessary in order to modify the severe singularity of the denominator at pi+Pj=O

into~ ~ simplen,±poles. The constant

z(n)

can again be

determined to be Z ~n; : 2--2- by crossing and taking the off-shell leg o(x) on-shell. Finally we obtain


• .

"Pn

>in

n-i =

(2i)-7-

~ i<j

th

kJ

(25)

2

for odd n. The matrix-elements vanish for even n because of Z(2)-invariance.

5. Green's Functions Till now the problem of constructing Green's functions has been solved only for the simplest field theory with soliton behaviour, the Z(2)-Ising model in the scaling limit T÷T c [8J. Having deduced all matrix elements of the field u(x), we can immediately write down expressions for the Green's functions. For example the twopoint function is • (p) = /d2x

0(< 2 ) e ipx = i / d< 2 5T~_K-T~+i ~

(26)

with the spectral function 0 given by

O(p 2) :

~ n odd

i I ~!

~ 47

..I "

den 47

27

6(2)(p_Zpi)

~ l~i<j(n

th 2 ~

(27) 2

There are similar formulae for the n-point Green's functions which are all in agree ment with those derived in Ref. L ~

by different methods. This agreement supports

the validity of the minimality hypothesis and gives some confidence in the feasibib ity of the bootstrap program. The determination of the correlation functions for one of the more involved soliton open problem.

field theories remains, however, a challenging

351

Acknowledgement: This talk is based on collaborations at the Institut fur Theoretische Physik FU Berlin,with B.Berg, V.Kurak, B.Schroer, R. SeiiAr, H.J.Thun, T.T.Truong and P.Weisz. I would like to thank them for many discussions. References [i]

A.B.Zamolodchikov,

Comm.Math. Phys.55 (1977)183;

M.Karowski, H.J.Thun, Nucl.Phys.B 130(1977)295; M.Karowski, H.J.Thun, T.T.Truong, P.Weisz, Phys.Lett.67B(1977)321. ~]

A.B.and Ai.B.Zamolodchikov, Nu~l.Phys.B133(1978)525;

Phys.Lett.72B(1978)72B;

B.Berg, M.Karowski, V.Kurak, P.Weisz, Phys. Lett.76B(1978)502. ~]

B.Berg, M.Karowski, V.Kurak, P.Weisz, Nucl.Phys.B134(1978)125; B.Berg, P.Weisz, Nucl.Phys.B 146(1978)205; V.Kurak, J.A.Swieca, Phys. Lett.

~]

R.K~berle, J.A.Swieca, Factorizable Z(N)Models, S.Carlos preprint (1979).

~]

P.P.Kulish, Theor. Math. Phys.26(1976)198; M.Karowski, H.J.Thun, ref. I ; D.Jagelnitzer, Phys.Rev. D18(1978)1257; S.Parke,Harvard preprint HUTP-79/A037.

~]

B.Schroer, T.Truong, P.Weisz, Phys. Lett.63B(1976)422; M.Karowski, FU-Berlin preprint HEP 7/79

~]

M.Karowski, P.Weisz, Nucl.Phys.B139(1978)455.

~]

B.Berg, M.Karowski, P.Weisz, Phys.Rev. Dl9 (1979)2477.

~]

R.Z.Bariev, Phys.Lett.55A(1976)456; B.McCoy , C.A.Tracy, T.T.Wu, Phys.Rev. Lett.38(1979)783; M.Sato, T.Miwa, M.Jimbo, Proc.Japan Acad.53A(1977)6.

+) Supported by Deutsche Forschungsgemeinschaft,

contract Schr 160/5 and Schr 415.

ONE AND MULTIDIMENSIONAL COMPLETELY INTEGRABLE SYSTEMS ARISING FROM THE ISOSPECTRAL DEFORMATION D.V. Chudnovsky CEN-Saclay Bolte Postale No. 2 91190 Gif-sur-Yvette, France

Introduction.

In mathematical physics there is one subject that now attracts the

attention of mathematicians working in different fields.

This is the study of dif-

ferent "solvable" or "completely integrable" models, some of which are very important field theory models.

Perhaps for the first time since Jacobi and Welerstrass,

the whole flow of integrable systems of mechanical and physical impQrtance appeared in direct connectlonwlth the problems of functional analysis and algebraic and differential geometry.

Different mathematical methods such as the inverse scat-

tering method [59], the monodromy theory [18], the Riemann boundary value problem [43]p the Riemann-Roch theorem [46], classification of vector bundles, etc. are the natural tools for the modern study of completely integrable systems.

Recently

new developments have appeared, such as: explicit examples of complete integrability in 3 and 4 dimensions (cf. [5], [17], [19]); the quantum inverse scattering method [39], [40], [56]; the inverse spectral problem for operators of an arbitrary order; and relation with the monodromy properties for irregular singularities (Stokes phenomena).

The latter direction is deflnltely very promising; we devote a separate

paper to the discussion of this problem.

In the present paper you can find all the

information necessary to begin the corresponding discussion for the matrix llnear differential operators of an arbitrary order.

The present paper is devoted only to

the linear differential operators of orders one and two.

We cannot cover the sub-

~ect of completely Integrable systems even in a big treatlse, though there is none. For this reason we have decided to restrict this paper to the consistent representation of a very general class of completely integrable systems (containing almost all the practlcally interesting ones) together with the exposition of an entirely new description of their Hamiltonlan structure.

We consider here classical completely

integrable systems and simultaneously, their operator generalizations [13], where the solutions are operators on a Hilbert space H depending on tlme-space variables x,t,y.

The most interesting among these systems are operator generalizations of the

sin-Gordon equation (§§2,5), containing all the SO(N) nonlinear ~-models and prlnclpal chlral fields (§§1-2). main result.

This operator formalism allows us to establish our

We show how two-dlmenslonal completely integrable systems arising

from the isospectral deformation for linear differential operators of order one and two can be decomposed into the common action of two simple commuting one dimensional Hamiltonlans.

In general, these Hamiltonians arise as natural generalizations of a

very special and important completely integrable Hamiltonlan system

353 n

n

fijxx" lifi+ fi j=~l fjgj ; gi,xx" ligi+ g i ~ f j g j : i - l,...,n.

This system, studied for a century in [35], [41], [51], [23-25] is

examined in §6, where following [35], we call it the Russian Chain [14], [16], [17]. The infinite-component and operator generalizations of the Russian Chain play the crucial role in the decomposition of two and three dimensional completely integrable systems into commuting one dimensional Hamiltonians.

The grounds for this decomposi-

tion are connected with a very peculiar phenomenon: the coefficients of a linear differential operator L -

uj ~

S-I

can be expressed in terms of products of eigen-

j

functions of the direct L~ = kn~ and conjugate L*¢ - kn¢ eigenvalue problem for L. For n = 2 and the Schrodinger operator -y" + u(x)y = -12y this was observed for the first time in [15], [23,26]; or in [42], [52], [30].

In §ii we establish the repre-

v

sentation U = - 2 ~/ ~ _ dE

for an operator spectral measure dE

associated with the

eigenvalue problem ¢~,xx " U~V + ~ 2v~; $~,xx " ~ U + ~2~.w in the case of an operator U.

We develop the general form of such a representation in §§9-10 using the resol-

vent expansion [32], [37] and the matrix moment problem. The moment problem associated with the linear operator (cf. the Green formula)

imposes interesting constraints

v

on

elgenfunctions

¢=, _$,,,in

the form of algebraic

relations such as

%-

1.

These constraints (§§§7, 9, 10) lead to the conclusion that two and three dimensional completely integrable systems which arise from the isospectral deformation can be viewed as a free motion constraint to an infinite dimensional symplecticmanifold= explicltly defined by the intersections of quartics (cf. the discussion in §§7= 10). From the three dimensional systems we have chosen only the equations of the KadomtsevPetviashvili type [59]= §§1, 12: 3Uyy

to §12, u = u ( x , t , y )

m

(4ut+Uxxx-12UUx) x.

In this case, according

can be rap,scented as u . _f~ tr[$~dZ $=] for $~ - ~ = ( x , t , y )

satisfying an operator Russian chain in x and two commuting systems in y and

t.

In this presentation we restrict ourselves only to iaoepectral deformation equations.

The more general approach is that of the isomonodromy deformation treated

in [18], [19]. lar

It should be noted that the Russian chain equation is a very particu-

case of the general Schlesinger isomonodromy deformation equations [16], [35].

We also do not consider here

explicit solutions of completely integrable systems in

terms of %-functlons [4], [33], [49].

Nevertheless, such e-function objects as

Baker functions on Riemann surfaces [4], [I0], [18], [46] immediately lead to the Russian chain and its generalizations. This paper is organized as follows.

In §i we define the most general class of

isospectral deformation equations in dimensions from one to four. sln-Gordon and SO(N) and SU(N) G-model equations.

§2 is devoted to

In §3 and §4 we explicitly define

the spectral transformation associated with the operator Sch~odinger equation. §5 contains the general form of isospectral deformation equations together with numerous

354

examples of operator two-dlmenslonal equations.

We would like to attract attention

to the most promising example of the operator sin-Gordon (5.11) or (5.14), containing all SO(n) O-models. In §6 the Russian chain and its history is described, and a detailed study of its relationship with the resolvent expansion and the higher KdV equations is

presented.

In §7 we study the constraints associated with the Russian chain

and their appearance in the SU(N) ~-model [26]. tion of the Russian chain.

§8 deals with the formal quantlza-

In §§9-10 we describe isospectral deformation equations

~enersted by linear differential equations of the first order.

We decompose this

system into a co.non action of cowmutlng one dimensional Hamiltonians using the operator analog of the Russian chain (systems (10.7), (10.8)).

In §Ii, on the

basis of the investigation of the operator Russian chain [14] i we study the decomposition of operator equations of the KdV type.

Three dimensional operator systems

of the Kadomtsev-Petviashvili type [17] are introduced in §12~ where they are decomposed into a common action of three co-~.uting one dimensional operator Hamiltonlens.

This paper is concluded by a short discussion of the operators of order three. The author is deeply thankful to Professor D Ja~olnitzer

for the kind invitation to present this paper.

Division of ~N-Saclay for stimulating discussion and support. from ONE and NSF is gratefully acknowledged.

and to CEN-Saclay

I thank the members of the Theoretical The partial support

355

§i. %Sospectral deformation equation in two, three and four dimensions.

Chiral

field theories and different O-models. 1.0.

We present here the description of two-dimensional systems of nonlinear equa-

tions associated with commuting linear operators (isospectral deformation equations) Special attention is devoted to chiral and gauge two-dimensional field theories. 1.1.

The most general approach to the completely integrable systems is based on

the representation of the system of nonlinear equations as the condition of commutatively of linear differential operators in partial derivatives.

The first such

representation in one-dimensional case belongs to Burchuall and Chaundy [I0] (1922). The corresponding idea is presented here along the lines of the Zaharov-Shabat approach [59], [27].

For the (x,t)-space-time dimensional case, x I = x, x 2 = t, we

use light-cone coordinates 2~ = t - x, 2~ = t + x. The most general compatibility condition in (x,t)-dimensions has the following form

(1.1)

i~

= u~;

i~n = v~

for (complex)operators U, V and eigenfunction ~ depending on ~, q.

Then (i.i) is

equivalent to

(1.2)

U n - V~ + i[U,V] = 0.

The system (1.2) itself is trivially solvable if there are no restrictions placed on U, V.

The general solution has the form

u = i~

-i,

v = i~n~

-i.

The system (1.2) becomes, however, nontrivial if the spectral parameter I is introduced, in which case U and V must be rational (or in general meromorphic) functions in I. Now assume that according to the Zakharov-Mikhailov-Shabat

scheme [59], [27],

U and V, as functions of l, have poles of fixed orders ml,...,m n at fixed points ll,... , o l°n of a l-plane.

Then the system (1.2) becomes a "completely" integrable

system of nonlinear equations on the coefficients of the U and V: i.e., on the residues of U and V at i = ~ . The Lax representation [47]

(1.3)

dL d-~ = [A,L]

for two linear differential operators A and L in d w i t h

coefficients depending on

356

x and t, is imbedded into the scheme.

To get the Lax representation (1.3) from

(1.2) we demand U and V to be matrices having poles only at one fixed I

(say, O

1

= =) with orders of poles at I = i

o L and A.

o

for U and V equal to the orders of operators

The most interesting for the gauge theory is the case of U and V being rational functions with only one pole for U and one pole for V. In the simplest case one can set:

U=

A i+ I ,

B I- i

V

for A, B independent of I.

Then from (1.2) the main system of equations is obtained

[58], [591, [271, [20]: i A~ = ~ [A,B],

(1.4)

i B~ ~ - ~ [A,B].

This system of equations (1.4) is equivalent to the condition of consistency for the system A

(1.5) 1.2.

i~

k+ 1

'

-%

B

=

~- 1

The system (1.4)leads to field theories, e.g., to principle chiral fields and

o-models, connected with Lie groups. Let us suppose that we have at any point (~,n) some element g(~,n) of Lie group G.

(i,6)

We consider the following equations of motion

g~n

=

i,

Y~g~g

-i

gn+gn g

-i

.

g~)

and the corresponding action

S : fd~dq y T r (

g~g

The equation (1.6) can be reduced to the form (1.4).

(1.7)

A = ig~g

-i

,

B = igng

We define:

-I

Then in the notations (1.7), the system (1.6) is equivalent to the system (1.4):

(1.4')

A n - B~ - i[A,B] = 0,

A n + B~ : 0.

The system (1.6) is called principle chiral field theory associated with G.

357

The most interesting

from physical and mathematical

points of view are two cases

G = SO(N) and G " SU(N). 1.3.

Moreover,

it is often necessary

ciple field g in (1.6).

For example,

to consider also some reduction of the printhe most important systems,

o-models,

corres-

pond to restrictions:

(1.8)

g

2

= 1.

Under the constraints

(1.9)

(1.8) the system (1.6) is reduced to the system:

i g~q = Y(g~gg~+gqgg~ )"

The restriction

(1.8) is consistent with the equation

Let us see to what systems of equations duced for G = SO(N), SU(N).

(1.!0)

We present g in the form

g = 1 - 2P

where p2 = p, i.e., P is an operator of projecting,

(i. I0')

we can put

g = 1 - 2Pk,

where k is the dimension of the tion.

(1.9).

the systems (1.8) - (1.9) can be re-

image

of P, the number k is unchanged

In other words, the system (1.9) with the constraints

in the evolu-

(1.8) is indeed a sys-

tem of equations:

(i.ii)

[(Pk)$N,Pk]

with the a¢tion

= 0: k = i ..... [~]

[58]:

1 S k = ~ fd{dN Tr(Pk~PkN).

In other words one can consider each of the systems the space of projectors

of a given dimension

Grassman real variety ~ , k

k.

(i.ii) as the field defined on

This space in the real case is the

and in the complex case is the complex Grassman variety

N,k"

~N

SO(N) ,k = O(k)SO(N-k)

r¢ '

SU(N)

N,k = U(k)SU(N-k)

~

The most important case is k = i, when we have l~PN-I or ~pN-l. projector P1 can be written in the form

In this case the

358

N

(1.12)

(q)ij

and Y-niE i

: ni~j

Thus we have two nonlinear O-models A) Real

[54], [58], [20], [27].

p,pN-i or SO(N) ~-model

n~q +

(1.13)

: l.

i=l

(n~,nq)n = 0;

(n,n) = i;

cpN-i or SU(N) O-model

B) Complex

-~

(l.i4)

-+

-> -+

->

+nEq + ~((n~,n ) + (nn,n$) + (n~q,n) - (n,n~,n))n -> -> -

(n,n~)nq-

-~-~

-+

(n,nq)n~ + 2(n,nD)(n,n~)n = 0

and -> ->

(n,n) = i.

Here we put n = (nl,...,nN) and (a,b) = in the complex case. ÷ ÷ ÷ ~ - - ai~i i=l It is clear from (1.13), (1.14) that both SO(N) and SU(N) nonlinear O-models can be generalized to infinite-dimensional mensional Hilbert space H.

case, when ~ belongs to an infinite-di-

This generalization

is given below in §3-

In §2 we give a different form of equations of SO(N) nonlinear O-model with examples for the case N = 3,4.

The study of nonlinear o-models is continued later

in §§2, 5~ 7. Other two-dimensional

examples of systems (1.2), (1.3) will be presented in

§5, where we associate with the linear problem (I.I) the spectral and inverse scattering problems. mation equations 1.4.

Equations

(1.2), (1.3) are usually called isospectral defor-

(in (x,t)-dimensions).

General commutativity conditions can be used also to write three- and four-

dimensional equations suspicious for being "completely integrable". dimensional

(x,y,t)-systems are of Zakharov-Shabat

[59] type.

These three-

They can be written

as:

(1.15)

~--~ L n - ~ t Lm = [Lm,Ln] fl

for matrix linear differential operators Ln, L

of orders n and m in n=- with coeffim

cients being functions of x, y, t. for

Lm~=

~yy '

Ln~=

ax

Such a system is the condition of compatibility

"~K "

359

In the scalar case (n = 2, L 2 = -d2/dx 2 + u, L

is of third order) we get the som called Kadomtsev-Petvlashvill equation that will be treated later:

3Uyy " ~ x

(4ut+ Uxxx-12UUx)"

Four-dimenslonal systems that can be considered along the lines of compatibility conditions are those closely related to four-dimenslonal self-dual Yang-Mills ~elavln-Zakharov

[5] representations) and have the form * -i * Vl~ - ~ V2~

Vl~ = .%V2~;

* -i * [V1 + IV2,V I - ~ V2]

or

§2. Examples of principle chiral fields~ SO(N) nonllnear o-models and sin-Gordon equations. 2.0.

We study here in detail the equations of SU(N), S0(N) principle chiral fields

and SO(N) nonlinear o-models.

All these equations are considered as generalizations

of sin-Gordon equation a~n = sin ~[1],

[59].

We present different forms of the

equations defining these chiral (or gauge) theories and the Lax representation of an arbitrary SO(N) o-model. sin-Gordon equation.

Later, in §3 and §5 we define the most general operator

The SO(N), SU(N) principle chiral fields and SO(N) nonlinear

o-models are particular cases of the operator sin-Gordon equation of §5. 2.1.

Sin-Gordon equation [I], [59]:

~q

m

sin

a

or

(2. I) ~xx - ~tt " sin a. This is S0(2) principle chiral field. O-model; or the SU(2)(£P I) 2.2.

At the same time this is the SO(3) nonlinear

o-model.

The following system is known under different names: Complex sin-Gordon equatlona or Fohlmeyer-Lund-Eegge

[54], [58] system, or

Getmanov's Lorentz invariant system [58]. This is a system of equations for two scalar fields ~ and ~:

360

1

SEn + ~

(~nS£+~ESn) - o;

(2.2) ~En + sin e -

sin(~/2) 2 cos3(~/2)

For 8 = 0 one g e t s s i n - G o r d o n .

8E8 ~ = O.

The s y s t e m ( 2 . 2 ) i s s i m u l t a n e o u s l y SU(2) p r i n -

c i p l e chiral field and at the same time SO(4) nonlinear O-model. 2.3.

Now it is possible to write for any n ~ 3 the equation of SO(n) nonlinear

O-model as the system of relativistic SO(n-2) covarlant differential equations involving n-2 scalar fields. Pohlmeyer

men +

w~EI =

We follow here K. Pohlmeyer and K.H. Rehren [54].

[54] makes the following transformation.

For SO(n) o-model (i.13):

( ~ E , ~ r l ) ~ - O: |~1 ,, I ,

[~ i = I we consider an orthonormal basis n

- cos~ ~ b-I = n' bo " hE' b l =

q sin

, bk: k ffi l,,..,n-2

n . . . . . + in • if a = arc cos(n ,nn). Let S;k = (bjE,bk); S._ ffi (b. ,b_). Now S._ _ _ _ . _E JK J~ ~ lJn_2 (fiE,fj); Sij (~i~,f~) for fl,...,fn_2 forming an orthonormal basis I n ~ . Now defining $ by

we reduce the equation of S0(n) o-model to (n-2)-component equations without constraints.

This equation has the form

+ (~'~q)~ +

(2.3)

-E~

~l -

n$12 3 " 0 ,

i - i$i 2

Of course there exists a symmetric equation corresponding to another choice of the basis b_l, hoi b I"

i

(x,×~)x~

XE~ + I - I X I 2

- i~l 2 ~ - 0. +

2.__4. There exists another form representing SO(n) o-models

[54].

o-model we have the following 4-component field theory in 1%4. vector product in 1%4: [A,B ,C] i - EijkEAjBkC~. Then the equation takes the form:

In the SO(6)

Let [.,.,.] be the

381

(2.4)

~n +

(~'~)~n+ (~'~n)~- (~'~)~ z - i~! 2

which is already symmetric in ~ and n.

[ ~ ' ~ C ~ n ] / 1 " 1~12 + (1-1~12)~ " 0

This is a 4-component equation of the S0(6)

nonlinear o-model. 2.5.

We can call equations

tion (2.1).

(2.2) - (2.4) to be generalizations

These generalizations

of sln-Gordon equa-

can be written down for arbitrary compact Lie

group G and are naturally connected with the principle chiral fields. G = SO(N) these generalizations belong to Budagov and Tahtadzan

For G - SU(N),

[8] (or [58]).

Let A,B,R,W be diagonal matrices; A,B,R be hermitian and W anti-hermitian. Then the "sln-Gordon equation for the group SU(N)" takes the form:

Another equivalent representation

for the "sin-Gordon equation for the group SU(N)"

is:

(2.6)

(0{10~)n- (o~lw~TDn- [Q[1B0/,A]. In the case of SO(N) we have W = 0 and for SO(N) matrices there exist the

similar equation:

( T x)x -

(2.,)

All these equations

+

" o.

(2.5) - (2.7) are very special examples of operator (matrix)

equations of the order two (§3).

They are particular cases of the "Operator sin-

Gordon equation" proposed below in §5. We note that equations clple chiral field equations

(2.5) - (2.7) are equlvalent to SU(N) and S0(N) prin(see §1).

More precisely,

Mikhailov [58] have proved the equivalence

Budagov [8] and Zakharov-

(guage equivalence)

of the Lax representa-

tions for "sin-Gordon for the group G" with the Lax representation chiral field equation on G, both for G = SU(N) (equations

for the principle

(2.5), (2.6)) and G - SO(N):

(2.7) cases. 2.6.

There is one way to write down the Lax representation

corresponding to SO(N) nonlinear O-models.

because this was one obtains a very simple spectral problem. two spectral problems

and isospectral datae

This form is preferable to that of §1 The difference between

(1.4) and (2.8) is the same as between u~n = sin u and

Uxx - utt = sin u equations

(look at [I] and [59]).

We consider SO(n) nonllnear O-model as (n-2)-component

system with components

~l,...,~n_2 in the form (2.3) (without constraints):

The essential point is Pohlmeyer's method for the introduction of matrix Riccatl equations [54].

Let ri: i - 1,...,n-2 form the basis elements of the

362

Clifford algebra Cn_2: {Fi Fj} - 2~lJl. The lowest-dimensional representation of % - 2 matrices.

are in 2 [(n-2)/2]

Then the Lax pair of SO(n) O-model equation

~

- I

i~'i 2

× 2 [(n-2)/2]

has the following form [54]:

o

(2.8)

i

7

= (4x) -1

7.

Here n-2 (2.9)

~ = ~_

~jr j .

J-l

The conditions of consistency of two equations (2.8) is exactly the SO(n) system of equations (2.3) for {~l,...,~n_2 }. Equivalent representation of commuting linear differential operators arises from Clifford algebra Cn_l:

Fj

: J - l,...,n-2;

~

.

Then two linear systems have the form: a

Y-UY;

~y=VY

and

. V

n2

+ j-i

1

2

[

n2j%]

- (4),) -1 -'~/1 - i$~ 2 ~n-I + Z

j=l

§3. Operat0r ~enaralizatiogs ' of classicaI!woTdimensional 3.0.

equatigns.

One of the main purposes of this paper is to consider operator generalizations

of two and three dimensional completely integrahle systems and to solve them in terms of one dimensional operator Hamiltonian systems.

In Chapters 3 - 5 we de-

scribe operator generalizations of two dimensional equarions and the most general form of operator nonlinear evolutionary equations, associated with isospectral deformation for llnear differential equations of order two with operator function coefficients. 3.1.. We can put in correspondence with any equation of Zakharov-Mikhailov-Shahat type an operator equation [3]. The most general such equation is the condition of compatlbility of two linear problems [59], [27].

363

s~--~- ucx)¢, "~n" v(x)¢, where U(~), V(~) are rational functions in the ~-plane.

In the classical cases

U(A), V(A) are N x N matrices and all such systems are characterized by: i) the posltions A 0 of poles of U(A), ViA) that are fixed, and il) orders of poles of U(A),

J V(A)

at A = ~

that are constant.

In order to replace classical systems by an operator one changes U(~) D V(A) to be operators keeping the same poles A~ and order of poles fixed [13].

In

general in the quantum case there appears also the necessity to rearrange the order of terms in U(A), V(A) and the equations themself.

Hence, in each concrete case

this must be done separately. One should not think that the tranformation

from usual equations to operator

ones is immediate and that this operation merely involves the replacement in the equation of c-functlons by operators.

First, for certain equations it is simply

impossible, e.g., in the sln-Gordon ~xt = sin $ the function ~ cannot be substituted by an operator, if complete integrability is to be maintained~ of the equation must be changed (see §§2,5). ordering of operators.

instead the form

Second, there appears the problem of

The only way to get the true operator generalization

is to

return back to isospectral equations and then generalize them to the operator case. 3.2.

As an example of operator generallzation we can consider the equation (1.4)

or prlnciple chlral fleld equation

(1.6) in the case of Infinite-dimensional

group

G. For this we consider the most general u-model for an operator-functlon g(~,~)

being an operator on L 2 ( ~ , d ~ 1

g -

and satisfying

-1

together with the condition g

2

- i.

We consider only the case when g = 1 - 2PI, where P12 = P1 is one-dimensional projector corresponding t o two vectors n and m from L2(~,do):

(P1¢) (~) = ~(~)~(~l ) ~ (~l) d~l. Then equations for ~-model are equivalent to the equations

(3. l) ~'~

+ { ( ~ , ~ n ) + (~n,~',~) + (~,~n)}~ + (n,m~)n n + (n,~n)n ~

+ 2(n,m )(n,m~)n = 0;

=~n + { ( ~ ' ~ n ) + (nn'=~) + (~'~n)}~' - (n'm~)mn - (n'mn)=~ + 2(~.,~n)(~.,~)~ = o together with constralnts: (n,m)

- i.

364

In the case dim L2(~,do) - n the condition n = m leads us to SO(n) o-model (RP n-I case) and the condition m = n* is equivalent to SU(n) o-model (~pn-i O-model). In both cases we take n, m from £2 with the scalar product in the sense of £2. n

3.3.

n

For these lectures we decided to restrict ourselves only to those operator

nonlinear evolutionary equations that are associated with linear differential operators of order at most two with operator function coefficients.

In principle linear

differential operators of any order can be treated along the same lines.

This is

shown brlefly later for linear differential operators of order three. We call [13], [14] an isospectral deformation equation of order two (or at most two) if this equation admits the Lax representation (1.3): dL d--~ = [A,L] d where A is a differential (or integro-differentlal) operator in ~ x and L is a dlfd ferential operator of order at most two in ~ x . In other words, either

(1)

L = A~x

(ii)

d2 L =---~+ dx

+ V, where A is a diagonal operator, V = [A,U];

Or

U.

Both cases are considered here.

The case (i) is treated later and chapters

3-5 are devoted to (il). Of course, one can reformulate the definition of our class of equation in terms of the Zakharov-Shabat scheme.

If we consider two dimensional nonlinear operator

equations arising from the scheme of §i:

~xl(Xl,X2,X) - u(~)~; ~x 2(x l,x 2.1) - V(1)~ then we call the corresponding nonlinear equation u

- v

x2

+

[u,v]

=

0

x1

to be of order two if each of the operators U(1)~ V(1) have only one pole in X-plane, but orders of these poles can be arbitrary. In this class of equations one includes the majority of known two-dlmensional completely integrable systems: Kd V, m Kd ~ nonlinear Schrodinger, sin-Gordon, chiral fields, ~-models, Gross-Neveu model, Thirring model, etc. One of the interesting features of equations of order two is the existence of a large class of solutions (so called finite band potentials) expressed in terms of 8-functions of hyperelliptic curves.

The presence of hyperelliptlc curves is an-

other common feature for the class of equations, called "of order two". generate hyperelliptlc curve one gets just soliton solutions.

For a de-

365

The theory of matrix Lax systems of order two was developed by Gelfand and Dikij [36], who introduced Hamiltonian two dimensional structure on these systems and gave recurrent formulae for conservation laws.

However, we need some

other simple way to present equations of order two and to connect them directly with the evolution of scattering data.

This will be done in §5 and for this we

use in §5 certain single integro-differential operator [ associated with potential U(x).

Such Integro-differential operator L had been introduced by Ablowitz, Kaup,

Newell, Segur [i] and then by Calogero and Degasperls [ii], [31] in the matrix case. §4. 4. 2 .

Scatterin ~ data for operator SchrodinKer equation. In this chapter we describe briefly the inverse scattering data

Schrodinger equation with operator-function coefficients.

for the

In the appendix we il-

lustrate how isospectral deformation equations (of order two) are linearlzed via inverse scattering method.

The corresponding scheme is due to Kruscal-Miura-

Gardner-Green and Zakharov-Shabat. One can propose a scheme of inverse scattering for the operator Schrodinger equation on a Hilbert space H. case.

Not too much is different from an ordinary scalar

Classical operator approach in the arbitrary Banach space belongs to Kreln

[45], Berezansky [6], [7], Nizhnik [53], Levltan [48] and in the matrix case from the point of view of inverse scattering to Calogero, Degasperis

[11], [31], Wadati,

Kamljo.

4~I.

we assume that ~ x )

is an operator on H with the norm

[Iu(x)II vanlshinz

asymptotically exponentially or faster (as Ixl + ~). We consider Jost functions being operator solutlonsof (two) Schrodinger equations with the potential U(x), corresponding to the continuous spectrum: (4.1)

Sx~x(x,k) - U(x)~(x,k) - k2~(x,k)

and

(4.2)

~xxCX,k) - ~(x,k)U(x) - k2~(x,k):

k ~ 0

together with the boundary conditions: (4.3)

~(x,k) ÷ T(k)exp(-ikx):

x ÷ -~

(4.4)

~(x,k) ÷ exp(-Ikx) + R(k)exp(Ikx):

x ÷ +~

and ~(x,k) ÷ T(k)exp(-ikx) :

x + -~ ;

(4.5) x,k) ÷ exp(-ikx) H e r e R(k)

is

the reflection

course we have (4.6)

R(k) = R ( k ) .

operator

+ R(k)exp(ikx) and T(k)

is

i f now x ~ + ~ . the transmission

operator.

Of

366

The most important property of scattering data is the analytic continuation of the R(k), T(k) into k-plane.

Under the conditions of fast asymptotic vanishing of U(x),

both R(k), T(k) are meromorphic in the whole k-plane. In this case, R(k) may have N simple poles at the values X~ : J = I,...,N. J Then X~, J = 1,...,N are exactly discrete eigenvalues of the problem (4,1). Of course, X~: J = i, .... N are also eigenvalues of (4.2).

There are the corresponding

eiganfunctions

(4.7)

~(J)= ~(Jlu(x) - x~(~): S

l, ....N

=

xx

Scattering data corresponding to the eigenvalue xj2 have the followlng form:

lim

([k

k÷xj

-

xj]R(k))

=

P~:j

J

=

1, . . . . N .

The system

S = {a(k);xj,Pj:

J = 1 . . . . . N)

i s c a l l e d t h e system of s c a t t e r i n g

d a t a a s s o c i a t e d to the p o t e n t i a l

U(x).

I n the c a s e o f n o n d e g e n e r a t e e i g e n v a l u e s a l l Pj a r e p r o j e c t i o n o p e r a t o r s of rank one. 4.2.

Now one can generalize in a normal way inverse scattering method consisting

of reconstruction of the potential U(x) from the scattering data S. The best tool for this is Gelfand-Levitan

equation.

First of all we construct

initial data associated with S N

FCy)=f;R=(k)e(iky)dk+)i=l Pie(-iy). Then we write Gelfand-Levltan

equation as the Fredholm operator integral equation

K ( x , x 1) + F(x'I-x1) + Now t h e p o t e n t i a l

K(x,z)F(Z+Xl)dZ = O: x 1 ~ x °

U(x) i s r e c o n s t r u c t e d from K(X,Xl) in a v e r y simple way

U(x)

=

d -2 Tx~(x,x).

In the case of U(x) + = U(x) the situation is simplified considerably. We have in the case U(x) + ffi U(x) only finitely many eigenvalues discrete.

Also R+(k)

= R(-k*)

and

~ ( x , k ) - [ O ( x , - k * ) ] +.

that are

367

!ndependent of the condition U+ = U we have the classical formulae for scattering data T(-k)T(k) + R(-k)R(k) - I . To establish this identity we look at the Wronskien (which is constant): WCx,k) = ~(x,-k),xCX,k ) - ~x(X,-k)*(x,k)0 In the case of hermitian U (U+ = U) and real k we have the well-known unitarity equation:

T+(k)T(k) + R+(k)R(k) - 1. Appendix to §4.

Inverse scatterin~ method as a "nonlinear Fourier transform".

The essence of the inverse scattering method is very simple: one reduces a complicated nonlinear evolution equation on U(x,t) to a linear equation of the evolution of scattering data for U(x,t).

Of course, these equations (isospectral de-

formation equations) are rather special. The process of solving Cauchy problem can be explained as follows:

[

Initial data U(x;O)

direct spectral problem

,

i

at t = 0 I' nonlinear evolution of U(x;t)

Solution

,

,

.,

Spectral data S(k;O) = {R(k;O); P j ( k ; O ) }

inverse ~ scattering problem

of u(x;0) I

linear evolution of the scattering data in t

Spectral data S(k;t) - {R(k;t); Pj(k;t)}

U(x;t) at any t

k

Such a scheme has obvious anisotropy in x and t. Remark.

This simplicity is the consequence of the restrictive condition U(x) ÷ 0

aslx[÷~. §5. Two-dimensional o p e r a t o r systems o f . " o r d e r 5._~0.

two".

We d e s c r i b e h e r e t h e g e n e r a l form o f t w o - d i m e n s i o n a l o p e r a t o r e v o l u t i o n a r y

e q u a t i o n s a s s o c i a t e d w i t h the i n v e r s e s c a t t e r i n g

method f o r S c h r o d i n g e r o p e r a t o r .

These e q u a t i o n s (of o r d e r two in the t e r m i n o l o g y o f §3) a r e d e s c r i b e d in Theorem 5.1.

MOst o f t h i s c h a p t e r i s devoted to examples of e q u a t i o n s of o r d e r two [13],

[14]: o p e r a t o r Korteweg-de V r i e s , o p e r a t o r mKdV, o p e r a t o r n o n l i n e a r S c h r o d i n g e r , a u d o p e r a t o r sin-Cordon e q u a t i o n s .

We c o n s i d e r ,

n o n l i n e a r S c h r o d i n g e r e q u a t i o n [15] and i t s

as a p a r t i c u l a r stationary

c a s e , a multicomponent

version -

a Russian chain

368

[16], [25], [35]. This Russian chain and its operator version become the subject of our studies in §§6 - 9.

The most interesting in this chapter is, perhaps, the

introduction of the operator sln-Gordon equation. 5.1.

We consider now the most general two-dimenslonal operator evolutionary equa-

tions for U(x,t) associated with the operator Schrodinger equation: d2~ = U~ - k2~; dx2

~ - ~(x,t,k)

d2~ - ~U - k2~; dx2

~ = ~(x,t,k) .

(5. I)

We consider equations of order two (in the sense of §3) for which the evolution in t of the scattering coefficient R(kDt) of (5.1) is linear.

For this we consider

the integro-dlfferentlal operator L (see [i], [II], [31]): +~ 4LF(x) - Fxx(X ) - 2{U(x,t),F(x)) + C I dx'F(x')

+~

(5.2)

OF(x) = {Ux(X,t) ,F(x) ) + [U(x,t), I dx' [U(x' ,t) ,F(x') ]]. X

Now the class of the equations under consideration can be described by iteratlve applications of the operator L.

we remind that [A,B] = A B

- BA, {A,B} - AB + BA.

Theorem 5.1 (cf. [ii], [31], [14] and [I], [52]). For fixed entire functions a(z), ~(z) and fixed constant operators M, N the following nonllnear operator equation for U(x,t): (5.3)

U t ( x , t ) = a([)[N,U(x,t)]

+

8(/)'G'M

is equivalent to the linear differential eequatlon for the scattering coefficient R(k,t) of (5.1): Rt(k,t ) - ~(-k2)[N,R(k,t)] + 2ikB(-k2){M,R(k,t)}. Of course, this i s n o t t h e c o m p l e t e p i c t u r e o f e v o l u t i o n f o r s c a t ~ e r l n g d a t a , s i n c e we need to know t h e e v o l u t i o n f o r a d i s c r e t e p a r t o f t h e s p e c t r u m . The e v o l u t i o n o f a discrete part o f t h e s p e c t r u m can be o b t a i n e d from t h e e v o l u t i o n o f t h e c o n t i n u o u s spectrum, and we have Xj, t = 0: J = I,...,N PJ,t = ~(-X~) [N,Pj ] + 21Xjg(-X~) {M,Pj }. The i m p o r t a n c e o f o p e r a t o r L f o r our p u r p o s e s i s v e r y s i m p l e , i n t h e f o l l o w i n g way on t h e p r o d u c t ~ ( x , k ) of the operator Schrodinger equation:

of the left

The o p e r a t o r L a c t s

and r i g h t e i g e n f u n c t i o n s

369

(5.41

-

The action of L on the mOmentae

Remar k 5.2.

n

"

[ kn*(x'k1~(x'k)dk

Is very simple:

Let us now present examples of operator equations of order two, following the statement of Theorem 5.1.

5.2. Examples. 1) The 9perator K d V (5.41

equation [13]:

~t " ~xxx + 3(~x¢ + ~¢x )"

Here M = I ,

8(L)

= L;

a(L)

= 0.

2) The operator modified Kd V equation [13]:

(5.5)

~t = ~xxx - 3(~x~2+~2~x)"

Here

~x

~2

, M =

0

I

, 8(L1 = L; c¢(L1

O.

31 The operator coupled nonlinear Schrodinger equation [13], [14] : (5.61

i~t = Sxx + ~ ;

"itt " txx + ~

and nonlinear Schrodinger equation

(s.7)

i~t = Sxx + ~ + ~

(suggested by A. Neveu [60]).

Here =

~e

, ~-

, a(L) - L, 6(L) = O.

Operator equations 2) and 3) (but not 11) give us multi (infinite-) component equations if one considers operators $ of rank one [13], [14]. We consider H to be L2(~,dp) with the scalar product: (a,b) - f a (~)b(~)*dp.

Then, as a particular case

of the operator nonlinear Schrodinger equation we get a multicomponent nonlinear Schr~dinger equation

370 and a multicomponent modified Kd V equation:

(5.9)

St = Sx~. -

~x'($,$)

3

- 35"($x,$)

f o r ~ = $ ( x , t ) from B = L2(~,d~). As we see in §12, the combination of these two equations gives us KadomtsevPetviashvili equations in (x,y,t)-dimensions. In the case of finite-dimensional

H, dim(H) - n, we obtain the n-component

coupled nonlinear Schrodinger equation: n

I i*~,t = ~

*j*J°*~

- *~,XX '

n

The most

interesting case is that of the stationary equation, when ilgt

¢2 " ¢ ~ ( x ) s

-il~t

, ~ = ~(x)e

.

We obtain in this case the following completely integrable Hamiltonian system called Russian chain [14], [15], [16], [18], [19](first appearing in [35],

[22]-[26]): n (5.10)

n

~,xx

=~

with arbitrary ~ I " ' ' ' ~ "

Cj~j'~ + ~ :

g = 1 .....n

For the detailed description of the Russian chain see §6,

5..3. The most interesting example of equation of order two is the operator sinGordon (or nonlinear a-models). 4) Sin-Gordon:

~xt = sin ~,

~(xpt) ~ 0(mod 2~):

Ixl ~ ~ .

Here

H=I~ 2 , U=

;2x/4

The operator, even matrix complicated form.

I,

~(L) = O, ~(L) = L- I ,

M = I.

generalization of sln-Gordon equation is of rather

The natural operator analog of sin-Gordon equation corresponds

to the space H x H and potential of the form

371

U=

1¼P2 !p1 2 x

The corresponding operator equation of the order two arises in the situation ~(L) = O, 8 ( t ) "

M=

0

l

' Then "The Operator Sin-Gordon" equation

takes the form: Pxt " - YI {p ' I_~(P2)tdx, } + P'

(5.11)

In the scalar case for P - ~x we get the usual sin-Gordon equation Sxt = sin $. Here is another form for the "operator sin-Gordon equation".

We take, as in the

previous case,

for V - ~

. Here the potential U corresponds to the Schrodinger equation 2 (-

~x [+U)~= E¢

and the potential V corresponds to a Dirac equation d c3

~ ~ = v~ +

~

choice e(z) = z-I , 8(z) = 0 in Theorem 5.1 is presented as the following (x,t) equations on the operator P: 2Pxt = {P,T}; (5

2T x = _(P2)t ' This system is equivalent to the "operator sin-Gordon" proposed above in (5.11) : ext = - ~i {p, I~(p2) tdXl} "

(5.14)

This operator sin-Gordon equation (5.11), (5.14) contains "sin-Gordon equations for SU(N), SO(N)" from §2, (2.5)-(2.7). Let's show how SO(n) nonlinear o-model is obtained as a trivial reduction of this general sin-Gordon equation.

For this, one uses Clifford algebra approach

(of §2, 2.5) and imbeds SO(n) nonlinear o-model in the form (2.3) into systems (5.12), (5.13). Let FJ: J = l,...,n-2 be generators of the Clifford algebra Cn_ 2.

We assume

now that (5.15) If

P belongs n-2

(5.16)

P " ~-1 rjrj;

to

a vector space, generated by matrices FJ: J =l,...,n-2.

372

then by the equations

(5.13): n-2

2Tx . - ~J=l 2rjrjtI or we can put

(Xn_2

l

T- ~ ~

rjrj td~l I .

O t h e r equations a r e

r~xt - T: J - I ..... n-2 and then putting r j -

rj

get T = ! system

, which is equivalent to the system

of equations

(2.3) is equivalent

,jx~/l,

,,~,,2 , we

of equations (2.3).

Thls

to an SO(n) @-model by §2, 2.3.

Of course, isospectral problem (5.1) for this particular choice of U in (5.12) is equivalent to a Lax representation of SO(n) nonlinear o-model in §2, 2.5. §6. The Russian chain in the scalar case. 6.0.

In this chapter we discuss the Russian chain (5.10) (the stationary multl-

component nonlinear Schrodinger).

We decided to deduce this system from the iso-

monodromy deformation for the Fuchsian llnear differential equations

[18].

Indeed,

as it was shown in [16], [18], [19] the isomonodromy deformation equations are closely related to classical completely integrable systems.

In the one-dimensional

case isospectral deformation equations are equivalent to simplified Schlesinger equations

(see (A) below).

In multidimensional

cases we later see how isospectral

deformation equations in two- or three-dimensions commuting Hamiltonians of the Russianchain 6.1.

can be reduced to two or three

type.

We can start to consider the Riemann monodromy problem [18], [19], [28]

immediately from

Fuchsian linear differential equations.

For this we consider

the simplest linear differential equations: n+2 Ai

(6.1)

for ~ -

~ : 7dx

(yl,...,ym)

singularities

i= 1

x-t i

and A i being m

x

m constant matrices:

i - l,...,m and n+2

tl,...,tn+ 2 (where, e.g., tn+ 1 = 0, tn+ 2 - 1).

One of the solutions of the Riemann monodromy problem is based on isomonodromy deformation equations, an example of which is given by Palnlev~ transcendent Vl [18] (the case m = 2, n = 1). Schleslnger

The general Isomonodromy

equations were written by

[55] who used Riemann theorem [28] that the system of functions satis-

fying Fuchsian linear differential linearly dependent over ¢[z].

equations with the same monodromy group are

In other words~Schlesinger

used the "consistency

condition" as the form of representation of a system of nonllnear "completely integrable"equations.

373

Schlesin~er's The0rem 6.1. ([55], [18], [16]) matrix solution of (6.1); Y(Xo;Xo) = I.

Let Y

=

Y(Xo;X) be the fundamental

The necessary and sufficient condition

for Ai: i = l,...,n+2 as the function of the parameters tl,...,tn+2,Xo to have the fixed monodromy group for Y(x),is the following completely integrable system of total differential equations

4

~!.

dAj = - i ~

[Aj'Ai]d log Xo_t i • J = l,...,n+2.

This Schlesinger system can be written in a classical form [55] (1908-1912): ~Aj

=

~ti ~Ai (6.2)

~t i

( i _ [Aj,A i] tj_t i

1 ) Xo_t i , ~ # i

~ =_

I

j#.

~Ai .~-~xo

[Ai,A j ]

ti-t j '

[Ai,Aj] Xo_t j :

i = i .... ,n.

This system of equations is the source of "classical" completely integrable systems like stationary KdV, e.g. There is, of course, a natural temptation to identify the deformation equations of the type (6.2) with the Lax type of isospectral deformation. the case at all!

This is not

The deformation equations (6.2) are reduced in the most interesting

cases to the equations, differential with respect to a spectral parameter [18], while Lax's equations represent equations rational with respect to a spectral parameter. In the Schlesinger system the following transformation due to Painlev~ and Gamier

[35] is performed t i + a i + £t i A i ÷ E-IA i

: i = l,...,n+2

for constants ai, where, in particular, an+ 1 = 0, an+ 2 = I.

If we put e ÷ 0 then

the system (6.2) takes the followin 8 form [16], [18], [19], [35]:

~Aj ~t i

[Ai'A~] : J # i aj-a i

n+2 j=l ~ i

= 0

: i " l,...,n.

The system (A) is indeed completely integrable and, moreover, can be solved in Abelian integrals (of the first and second kind) corresponding to a certain algebraic curve [16], [18], [35]. Moreover, any completely integrable one-dimenslonal system connected with the algebra of commuting differential operators can be represented in a form (A) [16], [18], [19].

We call the system (A) a "simplified Schlesinger equation" [16].

374

6.2.

Let us consider the simplest example of simplified Schlesinger system (A)

which has a rather long history. The case n = 0 is trivial (no deformation), but the case m ~ 2; n = 1 is already extremely important.

In this case we obtain the following system of non-

linear differential equations equivalent to (A) for n = 1, m ~

2:

m.

f~'" fi = lifi - 2fi .

fjg~ ;

(R) or m

(6.3)

m.

gi = ~igi - 2gi .~0 fJgJ: i = 2

for m-2 arbitrary constants %2,...,lm. This system was derived from Schlesinger's

equations by R. G a m i e r

[35].

The system with different constraints was studied beginning in 1830, especially for m = 3 (cf. Jacobi [41]); C. Neumann [51] (m = 3,4). R. G a m i e r

[25] gave the solution of (R) in terms of hyperelliptlc

integrals ,

and he emphasized the achievements of the Russian school. Since S. Kovalevski relation

[44] (1880) the system (R) was studied by Russians in

the motion of a heavy rigid body. Some of the people who studied (R)

in the XIX century are: S. Tchapligin, G. Koloaoff, D. GorJachev, V. Steklov, G. Appelrot, N. Delone, P.Nekrasov

[3].

Because of their work we decided to call (R) the "Russian chain". R. G a m i e r

was the first who remarked (in 1919) that ( R ) w a s

possibly con-

nected with the system of equations written by J. Drach [32] (1919). know, J. Drach indeed

As we now

discovered a stationary KdV equation [15].

Without knowing about Garnier's remark, D.V and G.V Chudnovsky walked into (R) in 1976-1977 and indeed proved the equivalence of (R) to the stationary KdV equations in 1977 in a series of papers [22]-[25].

Our attention to this problem

was attracted by Professors V. Glaser and A. Martin to whom D.V. and G.V. Chudnovsky are deeply grateful. Remark.

Of course, the system (R) or (6.3) arises not only from isomonodromy de-

formation equations

(simplified Schleslnger system (A))but also from Isospectral

deformation equations

(of §5).

Indeed the system (R) is identlcal to (5.10): to

the stationary multicomponent nonllnear Schrodinger equation (5.8). Now we know that this system and its infinlte,component

and operator general-

izations contain all two-dimensional nonlinear systems associated with inverse scattering method.

In other words, these two-dimensional

systems (such as KdV, nonlinear Schrodinger,

completely integrable

sin-Gordon, O-models)

can be decom-

posed into common action of two commuting Hamiltonians on an inflnite-dlmenslonal symplectic manifold; one of these Hamiltonians is Just an operator Russian chain and another is one of its first Integrals

[14], [19].

375

6.3.

Infinite-c0mponent Russian chain. In order to prepare for operator generalizations of the Russian chain (6.3)

we consider a Russian chain on the Hilbert space L2(~,d~k) corresponding to a measurable function A:fl ÷ C.

Let us define the following Hamiltonian system for

f(k) = f(x,k); g(k) = g(x,k):

~

(6.4)

f(k)x x = -2fflf(~)g(~)d~'f(k) - %(k) f(k); g(k)xx =

-2f~f(~)g(~)d~%'g(k)

~(k)g(k).

This system has the Hamiltonian (6.5)

H I = f~fx(k)gx(k)d~ + f~%(k) f(k)g(k)dp k

(/~f(k)g(k)d~k)2 ,

+

where gx(k) is a variable conjugate to f(k) and fx(k) is a Variable conjugate to g(k).

The system (6.4) possesses a lot of internal symmetries.

Their existence

can be explained because (6.4) (or (6.5)) admits a large group of symmetry transformations of Darboux-Backlund type.

These transformations will be described

elsewhere, but they are simple consequences of the Darboux-B~cklund transformation for the second order linear differential operator [29], [62].

Here (6°4)Skow$

that f(k), g(k) are elgenfunctions of the Schrodinger operator with potential V =-2f~f(~)g(£)d~£, corresponding to the elgenvalue A(k). There are different ways to represent internal symmetries of (6.4) (cf. below); the simplest is to write them as quartics in canonical Theorem 6.1.

variables.

We have the following first integrals of the system (6.4) K[k] = fx(k)g(k) - f(k)gx(k);

(6.6)

du E C[k] = fx(k)gx(k) + ~(k)f(k)g(k) + f(k)g(k)fflf'gd~ - ~ ~(~)-X(k) x X(fx(E)z(k ) - f(£)gx(k)) × (gx(E)f(k) -g(E)fx(k)).

The first integrals K[k]; C[k] are in involution, and independent.

There are

additional series of first integrals:

C2[k] = fx(k) 2 + %(k)f(k) 2 + f(k)2~fgdV% d~ -~

%(E)'l(k) (fx (£)f(k) - f(E)fx(k))%x (£)f(k) - g(£)fx (k));

C3[k] = gx(k) 2 + A(k)g(k) 2 + g(k2)f~fgdBA

-

-

du E ~ l(£)-%(k) (fx (i)g(k) - f(Z)gx(k))(gx (£)g(k) - g(£)gx (k))'

There are certain relations between K[k], C[k], Ci[k] ,

e.g., we have

{K[k'], Cj[k]} = 0 if k' @ k; {K[k'], K[k]} = 0; {K[k], Cj[k]} = 2Cj[k]: J = 2,3. The proof of Theorem 6.1 follows by direct computation; however, time can be saved considerably if you notice that expressions like (fx(~)g(k) - f(~)gx(k)) are Wronskians of two solutions of linear differential equations of the second order.

376

New Hamiltonians, commuting with H I can be represented in the following form: Hh(k) = lh(k)C[k]d~. In particular, H I = IC[k]d~k.

Momentae Hamiltonians Hi(k) n

are the most important

and are connected with rasolvent expansion for the Sturm-Liouville problem. 6.4. Resolvent expansion and the Russian chain. The Russian chain (6.4) represents equations for elgenfunctlons f(k), g(k) of the Schrodinger operator

(6.7)

-y" + [u(x) + ~]y - 0

with the potential (6.8)

u(x) = -21 f(k)g(k)d~k

and the aigenvalue ~ = -X(k).

The resolvsnt expansion of (6.7) gives the interpre-

tation to momentae I A(k)nf(k)g(k)d~ in terms of differential polynomials in u(x), u'(x), ....

We briefly state basic facts on the resolvent expansion of (6.7)

following Drach [32] and [37], [15] and explain its relatlon with higher KortewegdeVries equations [37]~ [38], [15], [24]. The equation for the resolvent~(x,z) q C[[z'l]]:z 2 = ~ of the Sturm-Liouville problem (6.7) has the form [32], [37], [38]: (6.9)

-~"' + 4 (u+~)~' + 2u'~ = 0;

\c k=o

~ = ~[u,u',u",...]

whara

are d i f f e r e n ~ i a X polynomials in

u(x) ( i . s . ,

pol~omials

in UmUx,Uxx,O.. ). After the integEatlon of the equation (6.9) J. Drach obtains the equation of the second order

(6.i0)

-2~"

+ (~,)2 + 4(u+~)~2 = c(~)

for c(~) 6 C[[~--I]] with constant coefficients. For c(~) E i in (6.10) we obtain

(6.11)

R(x,z) -k~oRk[u]~ -k-1/2.

Here R ° = 1/2, R 1 = - I/4u, R2 = 1/16(3u2-u"),... . As a corollary of (6.9) we have one of the most famous identities ("Drach" Burchna11-Chaundy-Lenard 19 ) :

377

(6.12)

-R~' + 4uR~ +

4

! Rl~+l + 2u'R k

=

0.

These quantities ~ [ u ] are the grounds for the proof of the complete integrability of the Korteweg-deVries equation [37], [47] u

= t

6uu

-

X

u

XXX

According to [47] the KdV equation is equivalent to the followlng identity dL 2 ~ ' " [L3,L 2] for L 2 ffi -d2/dx 2 + u(x,t), e 3 ffi-4dB/dx 3 + 3(u ~ x + Tx u).

In general the functional

In = fRn[U,U' ,... ]dx defines a nonlinear (x,t) Hamiltonian system (6.13)

u t ffi-4 ~ x R [u] n

commuting with the KdV equation.

This equation (6.13) is called (n-l)th KdV equa-

tion (cf. n ffi l j2) and is equivalent to the Lax representation dL 2 dt = [L2n+l' L2 ] d for an operator L2n+l of order 2n+1 in d-~ " 6,.5.. We may now ask,when does the potential u(x) admit the representation (6.8): u(x) = -21of(k,x) g(k,x) dIJk for elgenfunctlons f(k,x), g(k,x) of (6.7): -y" + uy ffik2y. sentation in §7.

We consider this repre-

E.g., for the potential rapidly decreasing on infinity u(x) the

measure d~k is a combination of a Lebesque measure on [0,~o) with a finite number of 6-functions corresponding to

discrete elgenvalues

-xj2 [15].

For periodic C~-poten -

tlals u(x),d~k is singular with support at points of periodic or antiperlodic spectrum so-cal1~d ends of lacunae. The representation (6.8) or even

(6.14)

u ~ -2[ f2(k,x)d~k

for (6.15)

f ( k ) x x = u f ( k ) - k2f(k)

i s n o t u n i q u e ( o f . §7 for p e r i o d i c p o t e n t i a l s u ( x ) ) . However, the measure d ~

(corresponding to the choice l(k) ffik 2) can be fixed

if one demands certain norming conditions for representation of the coefficients Rn[U] of the resolvent expansion in terms of momentae /k2nf(x,k)g(x,k)d~k (cf. [26],

[14]). We change sllghtly the normlng of eigenfunctlons, from f(x,k)'g(x,k) to l(k)~(x,k)-~(x,k).

In this case by the classical moment problem [2] the following

378

representation

can be proved:

There exists

(for the general statement see §9).

measure d ~

a

(corresponding to ~(k)) such that

A

(6.16)

Rn[U] = 1/21nl(k)n¢(x,k)$(x,k)d~k

for all n = 0,i,2,... and @(x,k), ~(x,k) are the elgenfunctions operator corresponding (6.17)

of the Schr~dinger

to the eigenvalue %(k) and the potentials:

u(x) = -21 l(k)~(x,k)'$(x,k)d~ k.

d Moreover, for real u(x) and self-adJolnt - - - + dx 2

u(x) we can choose f E g, so (6.14)-

(6.15) are satisfied. Main Statement

([15], [26]).

For self-adjoint L and the spectral measure d~k cor-

responding to L~ ffi -k2~ there exists

a system ~(x,k) of eigenfunctlons

of

d2

(6.18)

(- "7"'f + u(x) ) ~ -k2~ =

dx

such that

R[u] = 1/2fk2=¢2(x,k) d~k

(6.19)

for all n = 0, i, 2, . . . . (6.20)

In particular,

i = f¢2(x,k) d~k;

(6.21)

_ u2 = fk 2 ~ 2 (x,k) d~k," ..

NOW the measure d~k and the system of eigenfunctlons conditions Remark:

~(x,k) is fixed by the

(6.19).

If w e

demand only R [u] = [PCk2)~2Cx,k)dp ' + C n - n' k n

for polynomials Pn (k2) of degree n, then the measure d i know only that supp d ~

is not unique at all (we

C_ supp d ~ ) .

6;6 ,. Now let us explain the important relatlon between higher kdV equations and the Hamiltonian

flows Hh(k) commuting with the Hamiltonian

(6.51 H I of the Russian chain. We consider potential u(x) in the Sturm-Liouville problem -Yk,, + uYk ffi k 2 Yk

and the spectral measure d~ k corresponding

to %(k) ffi k 2.

Rn[U] = I121 k2n~(x,k)~J(x,k)d~k for ~(x,k)xx = u(x)~(x,k)

Now by section 6.5:

: n = 0, i, 2 .....

- k2~(x,k); $(x,k)xx = u(x)~(x,k)

- k2~(x,k).

If now we

change k~(x,k) to f(x,k), and k~(x,k) to g(x,k) we Come to a Russian chain (6.4) with %(k) = k 2 having constraints

379

(6.22)

Rn[U] = I/2fk2n-2f(x,k)g(x,k)d~k:

n = O, i, 2 . . . . .

where

(6.23)

u(x)

- 2 I a f ( x , k ) g(x,k)d~ k.

=

Let us now consider the Hamiltonian Hh(k) = ~h(k2)C[k]d¢ k. time corresponding to the evolution according to Hh(k).

Let t = th be the

It is easy to compute the

evolution of u(x) in (6.23) when (fk,gk) evolves according to Hh(k):

u th = -2 ([ah (k 2) f (x ,k) g (x ,k) d~k) x" In particular, for h(k 2) = k 2n, Hn = fk2nc[k]d~k we get the evolution according tO H : n > 0 in the following form, taking into account (6.22): n

(6.24)

ut

= - 4 ~ x~ Rn_l[U]:

n -> i.

n

This is equivalent to an n-th KdV equation (6.13).

In other words, for u ( x , t )

in the form (6.23), the evolution of u according to an n-th KdV equation is equivalent to the evolution of (fk,gk) according to H I (ffiH o) in the x-direction and according to H in the t-dlrectlon. n 7. The Russian chain with constraints. 7.1.

While the Russian chain itself is a completely integrable system, the most inter

asting problem is the existence of Invarlant constraints naturally arlslno f ~ m resolvent relations with momentae (see section 6.4). We change (as in section 6.5) our notations, multlplying f(x;k).g(x;k) by %(k).

In this case we obtain the

following Canonical Russian Chain with constraints on f

= f(x;~) and g~ = g(x;~)

corresponding to ~(~) and Rilbert space L2(~;d~):

(7.1) g~,xx = -2g-l=j~Z(B)f°g°d~°p p p + Z(u)g . The simplest c o n s t r a i n t is the equation (f,g) = 1 for ~ = (f :a ~ ~), ~ = (ga:a e ~ ) : (7.2)

I fagadpa E 1. If we consider the potential u(x) defined by (6.17):

(7.3)

u(x) - -2/Q~(~)f g d ~

=

then the set of constraints on (f ,gu) is given by [15], [26]:

(7.4)

Rn[u]

=

1/2[Z(~)nf : _ g dP

380

n - 0, i, 2, .... The first two constraints (~ = O, I) are (7.2), (7.3). plete set

The com-

of constraints can be deduced from (7.4).

Then the constraints for the system (7.1) come from the equation for resolvent (6.10).

These constraints are exactly the following:

~o

~-E~(~)Ef~g~d~ +

(7.5) co

oO

£=--'~

J

~

~

4-0

Here are the first constraints: (7.6)

ff g d ~

- i;

(7.7)

]l(~)feg du e -

(7.8)

fl(u) 2f g e d ~

f(fexg ~ + f gc~x)d~ =- O;

ffc(xgc~xd]Jc~; _= IX(e )f xgc~xd~ + (If(e) feg d~)2.

It is possible to verify directly that, in the most interesting case,

our system of constraints imply that all the first integrals C[~] of the Russian chain are zeros! In particular, the Hamiltonian is zero. So far, one can call this system "a harmonic oscillator" with constraints; though constraint to the "zero ground state". In section 7.3 we examine in detail the system fk,xx = u(x)fk - l(k)fk together with the restriction (7.9)

[ f2gd~ - 1

(corresponding to (7.2) for fk - gk ) " We found that then the corresponding equations for fk are, in fact, equivalent to canonical Russian chain (7.1) with constraints (7.2), (7.3).

In other words, systems of Jacobi-Neumann type with constraints (7.9)

are indeed imbedded into Russian chain (7.1) with constraints (7.6)-(7.7). 7.2.

Let us show how the Russian chain arises from stationary solutions of G-models

(§§I-3). While O-models themselves are connected with the inverse scattering

method

for

381

Laplace operator or a general (~x,Tt) linear partial differential operator of the second order; the stationary solutions of o-models are connected with the "Russian chain" or inverse scattering method for Sturm-Liouville problem. Let us present the description of all stationary solutions of different Gmodels following the paper [26]. We consider the most general operator nonlinear o-model equation (3.1) for n,m in L2(G,do ).

We can write the stationary solution

of equations of O-model in the form (n'-)e = f (~+n)eP(e)(~-q);

(7.10)

(~)

= ge(~+n)e-P(~)(~-n);

~ e ~,

where (~,~)

- 1.

Substituting (7.10) in (3.1) we obtain a system of equations on f , g . Then our system of equations for f = (f :a 6 ~), ~ = (g :~ 6 ~) is equivalent to the following (7.11)

/ f~'~$ = p(~)2f~ + Vf ;

\ g~,~

= p(~)2g~ + Vg a

for V = -I fe,$gu,~due - I P(~)2fegedGe together with constraints (7.12)

I f g doe ~ i;

/ p(~)f g dG

E 0.

This is one of the versions of the "Russian chain" (7.1)

We can turn it into a

better known form by taking into account that (7.13)

V - -21 p(~)2feg dG~ + C. If now ~(e)2 = p(=)2 + C, then we have the following system (which is the

Russian chain (7.1)): -

2/ ~(8)2fgggdOg"fa;

(7.14)

- 2[n~(8)2fgggdGg"ge, togehher with constraints (7.12) and (7.7) e ~ I fa~ga~d°~ - I P(C:)2f~g~d°~ '

382

We note that a change of p(~)2 to ~(u) 2 corresponds to a gauge transformation (cf. [20]) of n and m in (7.10). The same reduction of the stationary solutions of G-models to a Russian chain is true if we take stationary solutions in the form (n)~ ÷ ~ f (~+~)e p(~)(~-~)"

(~)~ g~(~+~le-P(~)(~-7n) =

for ~ ~ 0. 7.3.

For 7 = 0 we get simply Instanton

solutions [20], [26].

Here we investigate the relation between the canonical Russian chain (7.1) with

constraints (7.6)-(7.7) and the Jacobi-Neumann systems of harmonic oscillator constraint to a sphere.

We show that these systems are indeed imbedded into the Russian

chain with constraints and correspond to the zero values of first integrals (e.g., of Hamiltonlan) of (7.1).

That is why we call the systems (7.1) or similar to [ 7.1)

"free motion" with constraints rather than "harmonic oscillator".

We consider the

Sturm-Liouville equations (7.15)

f(~;x)

XX

= u(x)f(~;x) + l(~)f(~;x)

restricted to a sphere of L2(~,d%): (7.16)

I f(~;x)2do E i. Of course, the condition (7.16) itself does not define an invariant manifold.

In order to write down equations for an invariant manifold which are the consequences of (7.15)-(7.16), it iS necessary to differentiate (7.15) once to obtain

[ f(~;x)f(~;X)xda = 0.

(7.17)

)

Differentiating (7.17) once more and using (7.15)p we come to

f

f(~;X) xdO + u

Io

f(~;x)2d~ +

I

X(~)f(m;x)2de ~ 0.

In other words, first of all we get (7.18)

u = - ! f(~;x)~do - I ~(~)f(~;x)2do

and the condition (7.16) is equivalent to the system (7.19)

I f(~;x)2d~ = i,

I f(~;x)f(~;X)xdO - 0.

Now the system (7.15), (7.18) takes the Jacobi-Neumsnn form (7.20)~

f(~;X)xx " {- I~ f(~ 1 ;x)2dG x I - [× A(ml)f(ml;X)2d°} ~

f(~;x) + l(~)f(~;x).

383

where the invariant restrictions (7.19) should be imposed. It is clear that if one denotes invariant submanifold (7.19) b Y ~ l ,

then

the system (7.20), restricted t O ~ l , is Hamiltonian with the following Hamiltonian H1 = ~1

(7.21)

+ 7

~p(~)2do _ ~

%(~)q(~)2do +

q (m) 2dO-1

+

Here it is assumed, of course, that q(~) stands for f(~;x) and p(~) stands for f(m;x) x. Equations of evolution for the Hamiltonian H I take the form: q(m)x = P ( m ) ( m ° d ~ ' ~ l ) ;

P(~)x ffi%(~)q(~) - q(~){i

%(~l)q(~l)2dOl + I P(~l)2dOl}(mOd~l ),

which o n ~ l is equivalent to the equations (7.20)%. Now the equations (7.20)% can be naturally imbedded into the Russian chain (7.1). Lemma 7.1. The system of equations (7.20)% o n ~ l with given % and given value of energy H 1 is equivalent to the Russian chain (7.1) with f E g: (7.23) 1,

f(~;X)xx = -2ff(~l;X)2dO~l'f(~;x) + %'(~)f(~;x)

for the function %'(~) = %(m) + 2HI on the manifold~l. Proof. Taking into account the form of Hamiltonian H I we conclude that o n ~ l :

In other words, [J p(~)2do _ (a %(~)q(~)2de - 2HiIaq(~)2de f . J a = 0 (mOd~l);

(7.24)

I p(m)2do- I%'(~)q(m)2do = 0 ( m o d ~ I) for

%'(~) = %(~) + 2H I. Substituting (7.24) into (7.20)% we obtain f(~;X)xx = -21

%(~l)f(~01;x)2dOl'f(~0;x) + 2Hlf(~0;x) + %(~)f(~0;x),

which is equivalent to the system (7.23)%, Remark 7.2.

It immediately follows from (7.24) that the system (7.20)% o n e 1

for

384

a given energy H I after imbedding into the Russian chain (7.235%, for ~'(~5 = k(~) + 2~ I, has zero energy in terms of the Hamiltonian (7.23)%,. After statements 7.1, 7.2 we find it necessary to consider the system (7.235 together with the following invarlant I f(~;x)2dO = I, ~

(7.25)

I f(';x)f(~;X)xdO = 0,

I f(;x)mdo - la ( )f( ;x52dO- 0 x The infinite dimensional manifold defined by (7.25) is denoted a s ~ 2

or (~2)%.

Corollary 7.3. The Russian chain (7.23) A considered on the m a n l f o l d ~ 2 is equivalent to the system (7.15) or (7.20)l, lf the latter one is restricted to the manifold ~X

C~2

and the condition H 1 - 0 is added.

Conversely, any system (7.155 o n ~ l

for a given ~ is equivalent to the Russian chain (7.23)~, on (~2)~,, where l'(~) ~(~) + 2~ l,

Of course, it is clear that for given I the manifold (~2)i is a sub-

manifold o f ~ 1 satisfying the equation H 1 = 0. Our simple results show that any system (7.15) on the m a n i f o l d ~ 1 is equivalent to some Russian chain o n ~ 2. itself than (7.20)~.

Certainly, the Russian chain is much more interesting

The system (7.20)%, as it is seen from the Hamiltonian H1 in

(7.21) describes harmonic oscillators with frequencies l(~) on the unit sphere of L2(~,do).

This interpretation of the system (7.20) as a sequence of harmonic os-

cillators on an n-dlmensIQnal sphere corresponding to the case I~l = n goes back to 3acobl and was explicitly proposed by Neumann [51].

In the infinite dimen-

sional case the system (7.205 with c o n s t r a i n t s ~ 1 appeared for the first time in the paper [26]. Of course, the system (7.1) o n e 2 the corresponding Hamiltonian o n e 2 .

corresponds to the trivial zero value of In such a situation it is not very natural

to speak about harmonic oscillators restricted to some manifold.

It is a free

motion restricted to some manifold. §8.

quantization of the Russian chai~.

8.1.

The most important feature of the reduction of two-dimenslonal completely

integrable systems to operator one-dimenslonal is the possibility of simple quantization.

First of all, instead of field equations we quantize the usual one-dimensional

Hamiltonlan systems,though in operator form.

Secondly, the simple form of Hamiltonian

and conservation laws gives simple rules for quantizatlon: there always is 8 typical kinetic part and we can always replace Px by ~/~qk' for example.

The rules for

quantizatlon are "classical": one replaces Pk and qk in all the formulae by operators Pk and %

satisfying ordinary commutation relations. [ek,Pk,] = [Qk,Qk,] = 0,

[Pk,Qk,] = ~(k-k').

As it was explained before one is allowed to take the representation Pk " ~/~qk °

385

NOW the main result obtained from the existence of infinite numbers of conserved quantities for the Hamiltonian is the existence of infinite numbers of commuting differential operators in Pk' Qk that commute also with H considered as an operator. in Qk"

As it is supposed to be, these operators are quadratic in Pk and quartlc

One of the interesting observations here is the followlng:

formation of H is the canonical transformation o n , a n d

ferential operator anti-commuting with H and other Hamiltonlans. to

BHcklund trans-

can be represented as a difThis way we come

certain forms of Bethe ansatz. Now there is no problem to translate this quantizatlon together with com-

mutatlvity and antlcommutatlvlty relations for operator from one-dimenslonal multicomponent systems to two-dimenslonal one component (here one component may be one operator component).

At the same time we have direct connection with reflection

coefficients because our first integrals are immediately expressed in terms of reflection coefficients.

This way we immediately reconstruct all the commutatlvity

conditions between different reflection coefficients that were considered by Honerkamp, Faddeev, Sklynln, Bergknoff, Thacker, TakhtadJan, and Kullsh. However, in the same way we obtain slmilar formulae for arbitrary operator (or multlcomponent) nonlinear Schrodinger, multlcomponent modified KdV, matrix sinGordon or any equation of "second order" in two dimensions. the possibility to quantize the systems in three dimensions.

Our approach also gives The best candidate

in three dimensions is so-called modified two-dlmenslonal KdV or two-dlmenslonal nonlinear Schrodlnger equation , but, these equations are very complicated and probably do not have any physical sense. On the other hand, we want to mention certain ambiguities in quantlzatlon of nonlinear $chrodinger.

While eigenfunctlons for the second quantization of nonlinear

Schr~dinger are determined by Bathe ansatz, the eigenfunctions of the quantized stationary nonlinear Schrodinger are eigenfunctlons of the quartlc oscillators.

We

can guess that in order to get consistent quantlzation of two-dimenslonal completely integrable systems and their stationary counterparts additional "non-elementary" functions (e.g., elgenfunctlons of the quartic oscillator) should be introduced. Remark:

In connection with future complete integrabillty of full quantum four-

dimensional Yang-Mills it should be noted dimensional quartlc oscillator.

immediate appearance of multi-

As it had been noticed by Zakharov i four-dlmenslonal

classical Yang-Mills implies the following two-dimenslonal system: U

xx

-U

tt

=UU+U.

If you change Utt by iUt you get operator nonlinear Schrodinger that is completely integrable as we had shown.

If you add to the right hand side mU for m ~ U~ then

the systems will not be completely integrable~ but what to do with this system itself, we don't know. quantum systems.

Our guess is complete integrability for both classlcal and

386

8.2.

We start first from the classical Hamiltonian 1 2 2 H.~l {fp~d~k + fX(k)q~ + 7(fqkd~) }

where q(k), p(k) are considered as elements of L2(~,d~k ) , ~(k) is a measurable function (from L2(~,d~k)). Now we naturally quantize this system.

We replace Pk and qk by operators Pk'

Qk' assuming the following basic commutation relations take place [Pk,Pk,] " [Qk,Qk,] = 0,

[Pk,Qk,] = ~(k-k').

The main result concerning operators commuting with H is the following observation, which is based on the corresponding classical fact. Theorem 8.1.

Let us define for any k 6 ~ the following operator Gk = Pk2 + ~(k)Q~ + Qk2 IQ ~ d ~

Then Gk commute:

l--!---{Q~P k - QkP~ }2d~ ' - ~~(~}-~(k~

[Gk,Gk,] = 0 for all k, k'.

Using linear transformation

G ÷ IJ

h(k) Gkd~ k

we can represent these operations in a more isotropic form, e.g.,

H= F1

[a0kd~

is our Hamiltonian 1

H" 7

Now replacing Pk' Qk by Pk' qk we obtain from ~ Hamiltonian H.

2

2

first integral C(k) of the

These C(k) are the motivation for the introduction of Gk.

This theorem is proved just by taking commutators of the corresponding operators.

However, one can use an analogy with the inverse scattering as C(k) is

expressed in terms of 2k/~ ~nIR(k) l2. The same quentization is applied to the Russian chain (6.4), when one replaces fk by Qlk' ~ by Q2k' and ~ x by Plk' fkx by P2k" This way there appears new commutation relations between operators C[k], K[k], C2[k], C3[k] 0 when the canonical variables are substituted for operators. [K[k'], Cj[k]] = 0 [K[k], K[k']] = 0;

In particular, we have

for k' @ k, [K[k], Cj[k]] = 2Cj[k]: j = 2,3.

One can combine this quantization of the Russian chain with our reduction of KdV-type equation to a combination of the Russian chain and commuting Hamiltonian flows.

These new relations between K[k], C[k], C2[k] , C3[k] after quantization give

387

rise to a nice commutation relations between quantum Hamiltonians and different quantum scattering coefficients b(~)p b*(~) for quantum modified KdVp nonlinear Schrodinger, sin-Gordon, massive Thirring models obtained by Honerkamp [40] I Thacker [57], Faddeev [56], [39]° §9. Completely infest able two-dimensional systems associated with linear differential operators of lthe f.!rst order. 9.0.

This chapter is devoted to the description of the class of equations solvable

by the inverse scattering method and associated with linear differential equations of the first order with matrix operator coefficients.

In 9.1, this class of equa-

tions is described in terms of certain integrodifferential operators following the method of AKNS [1] and Newel1 [52].

In 9°2, the same class of equations is charac-

terized in terms of coefficients of expansions of elgenfunctions in powers of spectral parameter, see [33], [34].

This enables us to associate with the coeffici-

ents of the expansion certain spectral measures dE E .

Then the coefficients of the

linear differential operator of the first order are represented as integrals over the spectral measure dE E of the product of elgenfunctions of the given and conjugate spectral problem,

Such representation immediately leads us to a certain version of

the operator Russian chain. 9.1.

Let us followlng [52] present nonlinear (x,t)-dimensional partial dlfferentlal

equations associated with inverse scattering method for matrix (operator) linear differential equations of the first order with matrix (operator) coefficients. We start with the first order matrix (operator) equation on a Hilbert space H: 49.1)

f

x

- - P(x)f + EAr

for f from H and diagonal A.

(9.2)

We assume, as usual, that

P - [A,V].

We assume in 9 . 1 the knowledge of asymptotics of P as x ÷ + ~.

Actually all that is

needed is the existence of a "monodromy" operator (cf° [52]~ [33]). llell ÷ 0 as x * ~ ® . e

have a "monodro~

operator"

as a s c a t t e r i n g

considers a fundamental matrix (operator) solution ~(x;E) of

In the case of

operator.

(9.1).

One

We can define

the scattering matrix R(E) of (9ol) as

49.3)

~(+ m;E) = ~ ( - m;E)RCE). The inverse scattering transform is a mapping between P(x) and R(E) including

(if necessary) information on the analytic extension to Im E > 0 or Im E < 0 of the principal corner minors of R(E). For the corresponding nonlinear (x,t) partial differential equation associated with (9.1) the dependence of R(E;t) on t is described by a llnear partial differential

388

equation as in Appendix to §4.

We have:

P(x;0)

÷ R(E;O) ~ R(E;t) ÷ P(x;t). direct linear inverse spectral evolution spectral transform transform

This class of (x;t) equations is defined as in §5 usin E an integro-differentlal operator. To make our notations clearer we adopt matrix ones. W on R sometimes in a matrix form W " (Wi, j) corresponding of H.

We present operators to soma choice of basis

Let A be a diagonal matrix:

(9.4)

A = (ai~ij)

If B " (bij), than we put B A - [A,B] = ((ai-aj)bij).

We also set B D = (bD,iJ) where

bD,ij " 0 if a i - aj and bD,i~ = bi~ otherwise; B K - (bK,ij), where bK,ij = bid if s i - aj and bK,ij = 0 otherwise.

So B D + B K = B.

NOW let B D be known (e.g., B A is known). some B K.

We determine in a canonical way

We put K(B) - [[P,BD]KdX I. "X

If B - B D + B K for B K = K(B) and [BK,P] K = 0, then we define D'BD = ~ x "B + [P,B]. NOW let B D be known and B = B D + K(BD).

Let us set

DROBA = D.(BD). Then the following assertion is true: Theorem 9.1 ([52]).

For entire functions G, ~, F of E the evolution equation for

P(x,t):

G(PR,t)-P t = ~(~R,t).[C,P] + F(Pa,t)-x [A,P]

(9.5) is equivalent

to: C(Z,t) -R e = n(E,t) - [C,R] - F(E,C) -R z.

We consider below only equations with F ~ O, in other words, only those equations that appear in a traditional AKNS [1] or Zakharov-Shabat

scheme [59].

Equations with F ~ 0 correspond to equations of Painlev~ type (cf. [18]).

As we

see in §i0, these equations can be reduced to a joint action of two commuting Hamiltonlanflows.

These Hamiltonians

(analogical to the Russian chain) arise from

the representation of the potential in (9.1) in terms of the integral over the spectral measure of product of eigenfunctions of (9.1) and the conjugate problem.

389

9.2.

Now the structure of equations of evolution associated with (9.1) is analyzed

in terms of certain expansions in powers of spectral parameter E at E = ~. fore we consider operators on a Hilbert space H as (infinite) matrices. V are considered as operators on H.

As be-

Potentials

For a diagonal matrix A (9.4) on H we define

the potential P in (9.1) the same as in (9.2), denoting P as VA: (9.6)

V A = [A,V].

Lemma 9.2.

The following equation 8-~- [~, V A - EA]

(9.7)

has a unique solution ~ = %A(E,x) as a formal power series from C[[I/E]] such that ~A = A + II,A/E + . . . . .7- I n ' A

(9.8)

nmo

En

with A0,A " A, Ii, A = - V A and for which (9.9) Proof. minina

IA ~ A

if

V A = 0.

We use the equation (9.7) and get the following system of equations deter-

~n,A = ~n,A (x): 3lnl A '~x = [~n' VA] + [A, An+l]:

(9.10) n = 0,1,2, ....

Then the conditions (9.8)-(9.9) together with lo,A = A determine

all the ~n,A from (9.10) consecutively. The quantities ~n,A are functlonals in VA,(VA)x,(VA)xx, ....

Nonlinear (x,t)-

equations of evolution associated with the linear differential matrix (operator) equation of the first order (see 9.1 and Theorem 9.1) can be expressed in terms of the functlonals ~n,A"

According to results of Dubrovln [34] or the results of

[33], [36] these equations can be represented as (9.ll)

[A, v t + ~ , A ] - o; o r

- [ A . v t] = [A. ~n,A] : n =

o, 1. 2 . . . . .

The ganeral ( x , t ) - e q u a t i a n s have the form [33], [34], [36]: N

(9.12)

[A, V t ÷ ~ i

An,ACn] = 0

for constants Cn(n = I,...,N).

Equations (9.11)-(9.12) are Hamiltonian systems in

VA and they admit Lax representations [36]: dL d'~ = [M,L]

390

d Here M is a linear differential operator In ~ x of

for L ffi ~..r-+dx BVA and AB = I. order N.

We relate An, A with the expansion of elgenfunctions

of a linear operator

naturally. We consider now the linear operator associated with the potential VA: (9.13)

L A - ~ x + V A - EA

and E is considered as (complex) spectral parameter.

In order to s p e c i f y the de-

pendence L A on E we put d LA, E = d-~+ V A - EA. The operator conjugate to LA, E is t

(9.14)

d

t

LA, E ffi d-'~- VA +

EAt.

Here B t is a matrix transposed to B. (9.15)

LA,EF = 0

(9.16)

t , LA,EF ffi 0.

t We consider eigenoperators of LA, E and LA,E:

and

If we define A

(9.17)

- F, F - F 't

then conditions

(9.15)-(9.16)

can be written as equations for rlght and left eigen-

functions of LA,E: a)

_~ddxFE = - VAFE + EAFE;

b)

__d dx FZ

(9.18) ffi

FzvA _ FE.EA.

v

Remark 9.3.

If F E is a solution of (9.15) or (9.18),a), then FE-C is also a solution

of (9.18),a) for an x-independent

C.

In this case

c. (~E)-I

is a solution of (9.16)

v

or (9.18),b) whenever F E is non-slnsular. For example, if one takes F E as any non-singular solution of (9,18),a), then amy other solutio~ of (9.18) can be written in the form

(9.19) ^

F E ffi D" From the class of solutions of (9.19) we want to slngle out some speclal ones that determine spectral measure.

We want to consider spectral problems algebraically

but not analytically because we want to consider the most general class of potentials

391

V A.

In order to determine spectral measures the method of the matrix moment problem

is used. First of all a few auxiliary results are presented; v ^ Let FE,F E be solutions of (9.18). If

Lemma 9.4. (9.20)

k E " FE'FE,

then k E satisfies the equation (9.7). Precisely, dk E dx " [kE' VA - EA]. Proof.

We have by (9.18), V 'V ^ w ^ V ^ kE, x = (-VAFE+EAFE)F E + FE(F E A-FE EA)

=

[kE,V A] + E'[A,kE].

The converse assertion is also true: Lemma 9.5.

If k E satisfies the equation (9.7):

dk E d x = [kE'VA-EA]' v ^ then there are solutions FE,F E of (9.18) such that the representation (9.20) is valid: v ^ k E - FEF E . Proof.

We choose the following system of fundamental solutions of (9.18):

^

and

) 1.

Let us put ZE

o -i.

- (FE)

v v . o . o -i . -i k E FE (FE) kE(F E) .

Then

o (~E)x - -(FE) v

_

l (-VA+EA}kEFv ~.

v Thus %E = const and k E . FEO.~E.FE."

+

(F~)V _

V Z[kE,VA_~A]F~

+

~'~)W ° _lk .__

{_vA+~A%. ~

According to (9.19) this means that ~

O.

admits

the representation (9.20)t and the lemma is proved. w A Now we want to find the solutions FE, F E of (9.18) such that for k E = PE'FE one has the asymptotic expansion (9.8) exactly for E -~ o o with conditions (9.9) satisfied.

In order to explain more precisely what we mean we should recall the

Bore1 definition of asymptotic expansions [61].

The series (9.8) is deflnitely an

asymptotic expansion for an unknown function that may be non-analytlc.

Nevertheless

if one assumes that AA is a Hilbert transform of some, maybe complex measure, then the coefficients in the asymptotic expansion (9.8) are

nothing but momentae with

respect to the spectral measure [2]. First of all in order to get spectral measure we have to show that asymptotic expansion (9.8) can be really realized as a function k E from (9.20) for some solutions

392 v

^

FE, FE of (9.18).

For this, one needs canonical expansion of solutions of (9.18) in

descending powevers of E as E ÷ m [34]. The lemma below is a classical statement on asymptotic expansion of solutions of linear differential equations in the neighborhood of a regular slngularity. Lemma 9.6. (cf. [33], [34]) v

There are fundamental solutions of the system (9.18)

^

~E and ~E cnat can be represented as a formal power series in E when E ÷ = in the following way:

(9.2l)

~Z =

~m z

exp(EAx)

(9.22)

~E "

exp(-EAx)"

T0

l,

with (9.23)

l

TO

m

can be reconstructed from elements of ~, V, VxP V

and elements of Tm P T m

and XX i'''

they are such that v

(9.24)

T

m

= T

m

= 0: m > 1

if V = 0.

- -

Moreoverp

^

v

~e = ($Z)

(9.25)

-1

or

I,

Proof.

Let us consider an asymptotic expansion (9.21) together with the initial

condition (9.24).

Substituting (9.21) into (9.18) we obtain the following equations

for the matrix coefficients ~

: m > 0: m

V

T (9.26)

o

= I

Tn ~ x

= [A, Tn+ I] - VAT n

for n " 0, i, 2, . . . . %E1x def ffi

and

The equations (9.26) follow from (9.18) because

Tm,xE-m÷T ~ -m+l A × exp(ZAx) =

If we denote the matrix ~T by n \/

= (tn.ij) then the equations (9.26) are equivalent to:

m z -m

exp(~Ax).

393

(9.27)

(tn,ij) x

(ai-aj)tn+l,ij - k ~

=

(al-ak)Viktn~kj

for

(9.28)

A- (ai~lj), v - (vi~). V

Equations (9.2) determine Tn completely by induction in n. If a i ~ aj, then

tn+l ,iJ " (al-aj)-itn,ij ,x On the other hand, let a i - aj.

tn+l'iJx = - ~

+ ~__ -i k (ai-aj) (ai-ak)Viktn ,kj •

Then

(ai-ak)Vlktn+l'kj = k,a~.a (aj-ak)Vijtn+l'kJ '

j k So tn+l,ij is determined in terms of tn+l,rs for ar ~ as.

N~w tn+l,ij can he

determined in such a way that tn+l,ij = 0 for V - 0. In order to get (9.25) one gets (9.22) with (9,23)-(9.24) and then use the v " / -i One simply notices that (~E) satisfies all the conditions

uniqueness of the Tm.

^

(9.22)-(9.24) so it is identical with ~E" V One of the most interesting problems is the locality of Tm in terms of V and A only.

We can get from 1emma 9.6 immediately representation of IA as k E for some

solution, ~E' ;~ of (9.1S.). Lemma 9.7.

Let ~E and ~E satisfy all the conditions (9.21)-(9.25) of the lemma 9.6

Let FE, FE be solutions of (9.18) such that

FE- ¢F.ci, vE= c2~~ and CI-C 2 - A.

Then for def~

we h a v e

(9.30)

~

nA = FE'~ z

(9.29)

the

v =

^

~EACE

same asymptotic

expansion

nA-~A+ ~ +

....t ~"'A. hie

for

E ÷ ~ as

XA:

En

Here XnpA are differential polynomlals in V,Vx,Vxx ,. .. and such that AO I A i A Proof.

and

~nsA " 0

for n > 1 if V = 0.

We have

~'~T m

nA " E~=O

exp(EAx)A x exp(-EAx)

-m

=

394

mmo

Here ~m,A are differential polynomials ~m,A " 0 for m ~ 1 if V - 0.

l~M'O

in A, V, Vx, Vxx,...

Because ~A satisfies

and by (9.23)-(9.25)

(9.7) and %A is a unique asymp-

totic expansion, we have

§I0. Matr lx spectral measure and representation of ~otential as a quadrati9 form i~n ei~enfunctions.

C~mmutin~ Hamiltoni@ns

associated wlth linear differential operator s

of the first order. I0.O.

We consider in this chapter the spectral measure dE E associated with the v ^ linear problem (9.18) in such a way that for elgenfunctions ~E' ~E we have the solunV ^ tion of the,/moment^ problem~ j~E ~IEdEE~E = ~n,A (n = 0, I, 2, ...),cf. (9.8). In particular, f~EdEESE = A, fE#EdZE~ E - -V A,

This way the linear problem (9.18) is re-

placed by a nonlinear Hamiltonian system, which is completely Integrable. corresponding

The

(operator) Hamiltonian system can be considered as an analogue of the

Russian chain (see §ii for the operator generalization of the Russian chain). Analyzing this Hamiltonian system and its first integrals we come to the following conclusion.

All the (x,t)-dimensional

"completely"

integrable systems solvable via

the inverse scattering method for the linear operator of the first order can be decomposed into two commuting one-dimensional

Hamiltonians,

These Hamiltonlans

are

naturally associated with spectral measure dE E and moment problem 10.1. lO.l.

Let us define BOW the spectral measure dZ E associated with the eigenvalue

problem for the linear differential equation of the first order (9.1) or (9.18) with operator (matrix) coefficients. powers of spectral parameter E. realizing the momentae An, A

We use for this the expansions of 9.2 in

We define the spectral measure as the measure

from Lemma 9,2.

spectral measure is the spectral matrix.

Of course, in this matrix case the

We formulate the definition of the spec-

tral matrix measure dE E " (doE,i~) in the following way: Spectral Problem 10.1.

Let __~E' ~E be solutions of (9.18) defined in Lemma 9.6 and

satisfying all the conditions

(9.21)-(9.25)

of the Lemma 9.6

The measure dE is

called a spectral measure dZ E - (doi, j) if it is a solution of the following moment problem: v (I0.i)

^ "

~E

n = 0 , i , 2 , ....

^ Of course, in (I0.I) ~E' ~E can be chosen as arbitrary solutions of (9.18). In this case dE E is multiplied by a constant matrix. The problem (I0.i) is looking like a classical matrix moment problem [2], [6], [7], [45].

First of all, an n x n moment problem is always reduced to n 2 scalar

395

moment problems.

The peculiarity of the moment problem (i0.i) lies in the following.

Following Stieltjes-Borel

studies we can always solve the moment problem with

momentae {An,A! n - 0,i,...} in the following way: f

(10.2)

EndUE - An,A:

u - 0,1,2, ....

E

The interesting feature of the moment problem (I0.I) is the representation

of

the weight function dE in (10.2) in the form dT - ~EdEE~E, i.e., with separation of E-part and x-part.

This representation of dT is achieved because one can treat

(I0.I) or (10.2) as a moment problem in E depending on an additional parameter x. We ignore for a moment the problem of uniqueness of the measure dE in (10.2) referring the reader to the discussion of exceptional

cases in [6], [2].

Another

problem treated in detail in [2], [6], [45], [61] is the positivity of dl (or dEE) in self-adJoint cases. Proof of the existence of a spectral measure dE E in 10.1.

Let us consider an ar-

bitrary solution (weight matrix or matrix measure) dE = dT(E,x) of the moment problem (10.2).

We want to show that dT(E,x) can be represented in the form

~EdEE~ E for ~E* ^~E Being solutions of (9.18).

For this we use equations (9.10)

satisfied by %n,A: f En(d~) x - [f End~,V A] + [A,] En+lda] E E E f En{dTx - [d~'V4-EA]) ~ 0: E In the uniqueness case (cf. [2]) we have dE x ,,

f ~n-[dE,VA-EA] E

or

n = 0,1,2, . . . .

=

[dE,VA-EA),

By lemma 9.5 one obtains

p

dT - ~E.dZE.~E for measure dE E independent of x and any solutions ~E* ~E of (9.18).

This way the

measure dZ E corresponding to solutions of (9.18) satisfying conditions of 1emma 9.5. Now the solution of dE E o£ the spectral problem 10.1 can be combined with (9.18) in order to separate variables in an (x,t)-dimenslonal

"completely"

integrable

systems (9.11)-(9.12). 10.2.

(10.3)

Let us consider the system of linear differential

~Ex "

- VACE

+

~Ex "

~EVA

SEEA"

-

equation s (9.18):

EASE;

Then one can consider according to i0. ! the eigenfunctions

' ~E such that for some

spectral matrix measure dZ E ~ (doi~ E) depending on V and A we have the solution of the moment problem~

(lO.4)

Xn,A:

n-

o,1,2 .....

396 The most important are the first two momentae f~EdZE'¢E - A;

(10.5)

V

^

SE¢EdZE'OE " -V A. The representation for the potential V A in (10.5) allows us to treat linear differential equation (10.3) as a Hamiltonian system with constraints (10.4) (or only (i0.5)).

In other words we have, e.g., the following system of equations:

10

"

•

EACE ;

~

A

(10.6) ^

^

x/

~E,x " -~E~EI EI~EId~EI~E I - ~EEA " On the other hand one can use the first of the formulae in (10.5) and replace A by a quadratic form in ~E' ~E'

In this case instead of (10.6) we obtain

~E,x = ~E (EI+E)~EI'dZEI'~EI'~E ; 1

(lO.7)

E1

1

•

Both Hamiltonlan systems (10.6) and (10.7) are completely integrable systems v

and both systems have first integrals quartlc in ~, $.

~owever, we consider here

only the system (10.7) because for this system the description of Hamiltonlan structure

and first integrals are the simplest.

10.3.

We now study the following Hamiltonlan system that is derived in I0,i - 10.2

via a matrix linear differential operator of the first order: V

x/

A

(10.s) ^

m

^

V

^

¢~,x -*x'S (X+~l*ud~u*u: B v

A

X E ~, where operators (matrices) _ ~A, ~I are defined on H.

Here d~ (B 6 ~) is

a

certain matrix measure and the triplet (~Aj~q,d~x) is considered on the space eL2(~,dE )H' The system (10.8) possesses a lot of symmetries and many first integrals. once we present the corresponding results. Theorem 10,2.

For any A ~ ~ the following quantities are the first integrals of

the system (10.8):

At

397

~$~

K[M

(ZO.lO)

c[x[ Sx" ~-j *~'dS~'¢~'*X'

Proof.

=

and

(10.9)

-

The proof is actually extremely simple: one Just differentiates with re-

spect to x and uses (I0.8). This way one gets: ~K[I]

^

~

^

v

^

v

^

v .

In order to prove the same assertion for (I0.I0) we use the following auxiliary statement: d

^

%"

^

'/

"./

^

V

^

-"

^

\?"

~

N~

^

"/

d-~{$ASndEn*n*A } " SX~ (n-X)$udE~*~'$ndEnSn$1 + SxSndZn*n~(X-n)$~dE~*U'*A" In other words we have

d C[A] = 0 for any I E ~. dx We d e n o t e

where^ one can consider ~l,iJ and ~l,iJ as canonical variables and one assumes that

~ A,Ji

is the canonlcal coordlnate-impulse conjugate to canonical coordinate variable

~MiJ"

v

In conjugate variables $ and (~)t the general ~amiltonian equations have the form

v

~_~H

^ t

8H

Here for P = (Pij)i,j one denotes for simplicity"

(io.i1)

~H ~'Y

"

( ~j)

i,j

"

Then we consider the following Hamiltonian:

(10.12)

HII tr{(~Md~l]$p)"(~$ "' -

d~ ~$p)} ^

This Hamiltonian Hiiis exactly the Hamiltonian of the system (10.8). let

dEp = ( d O i j ) , then

Indeed,

398

~II 1 ~ ×

ili2i3i4i5,i 6 /~1~Iili2 dOgli2i35~li3i4 x

¢

¢

[~2~2vD2i4i4d°~2i5i6^~2i6 i i ' Then, e.g., in the case of dE H - Id~ we have

~RII

V

V

^

^

V

^

Zk*Xjk(f~P* dE *~) + XZKCZjk(f~CudZu*~)ki;

v

•"

^

- r.k(f~o dE ¢~)jkeXk i + lEk(f~¢~dE~¢~)Jk~ki •

x/

^

V

V

^

V"

6¢X,ij Now by the definition of symplectic structure we have ^

a~i~

(¢x,ij)x " -

aHi1

v

v

;

(¢~,ij)x =

aCx,ji

^ aCX,ij

•

These are exactly the equations (I0,8). Let us consider the following auxiliary Hamiltonian

H~-

tr

v,, $ (X_~)_l~ dr $ }. c[~] = tr{¢~¢~"

Let t% be the time corresponding to the Hamiltonian H I. ~

_i v ^ w

at--~" Cn" (x-n) ~¢X~n a

v

Then we have:

f

_i v

: n ~ ~, and ^ w

l^v^

"

Sn " -<x-n)" *n*X*x ~ n $ x,

a . ^

_$ ~u(~_~ 1- O dZ, u .

Now for

we have ~I 1

~

^

nv

^

The most important objects are the following commuting Hamiltonians arising from C[A]:

399

(10.13)

Hm " tr {fXxmc[x]dZx}

: m = O, I, 2, 3, ....

These Hamiltonians can be rewritten in the explicit form:

Hm = t r { ~ x ' ~ xm(x-p) - ICxdEx¢ . . . . x. x CvdEv¢^ } : m ~ O,

(Io.14)

H

~

0,

H 1 ,,

tr

e,g,,

o

v {

(fxCxdr.x$~)2},

% = tr { (J'X¢;dZX%).(IaX'~xd~x$;)}

Now for •y

^

I 1 - fu~¢wdE~¢~ and evolution according

to

Hamiltonian H

we have m

at8H

11 - [ ]e1.t ¢12d~1,1~b¢'

" f~ m"¢dZuCu]: m_>0.

m

If as before the spectral (matrix) measure dE x is chosen according to

i0.1

in

such a way that

f m e dZ ¢ -

m " Op i, 2, ... , then

Now

-W~ H

vA -

[A, ~,al

m

which is the nonlinear

(x,t H ) evolution equation (9.11) associated with matrix m

linear differential equation:

dF d x " -VAF + XAF for a diagonal A. §ii. The operator Russian chain (stationary - nonlinear Schr~dinger equatipn). iI.___~0. In this chapter we consider an operator generalization of §6.

of the Russian chain

This system is dealt with in Ii.i end 11.___~2where the relationship between

the operator Russian chain and operator equations of order two introduced in §5 are

400

considered.

This relationship is similar to the results of §i0, where the systems

of order two associated with linear differential equations of order one were decomposed into common action of two commuting one-dlmenslonal Ramiltonlan In [14]~ [15] we proved that (x,t)-dlmenslonal

flows.

equations of order two connected

with the Schrodlnger operator can be represented as the common action of the Russian chain in x-dlrectlon and of one of the Hamiltonlen

flows commuting with it in the

t-dlrectlon. ii.I. We derive the operator Russian chain generalizing

(R) or (7.1) as a stationary

case of the operator nonllnear Schrodlnger equation (of. §5). Let us consider the coupled Operator Nonlinear Schrodinger of §5, (5.6):

_i~t i

~**

and l o o k a t its s t a t i o n a r y

- *xx solutions:

~(x,t) = ~(x)etAt; $(x,t) for a constant A.

- e-fAtS(x)

Then we have the coupled stationary nonlinear Schr~din8er:

~_¢-A~ ¢~ - ~xx; A'~ *~ *xx ' 2 Now we consider H and L ( ~ d ~ ) - e x t e n s i o n

L2(~,d~) such that ._f~a(~)dZB(~) -

1.

o

we take

and define operators #(~), ~(~) over H O. followlng form: for f 6

of H : H - @ ~

H

L-Z'-(~,dEJ" o

and a(~), 8(~) on

~(~) a s a m e a s u r a b l e

function

~ ÷ ¢

We c h o o s e operators ~ and ~ over H, in the

H,

(#.f)(~) - a(~)" / ~ ( m l ) d Z

l'f(~l)

(~.f)(~) - ~(~). fnS(~l)dZ~l"f(~ I) (Af)(~) = ~(~)f(~).

l.e., ~ and $ on H are of rank 1 over H o.

Then the stationary nonllneer Schr~dinger

equation can he reduced from H to H O in the following multlcomponent

form:

401

In the particular case ~ ~ ¢, ~(~) = -w 11.2.

2

we get Just an Operator Russian Chain.

We now consider such an aggregate: infinite component operator Stationary

Nonlinear Schrodinger equation

or

In other words, ~X and ~% are (left and right) eigenfunctions corresponding to the eigenvalue ~2 of the operator Schrodinger equation with the potential: U(x) - f¢~(xldZ Cp(x). However, from now on we consider (OR) as (an inflnite-dlmensional) Hamiltonlan system.

Here we assume of course that @~, ~

belong to (a Banach) algebra of operators,

dE

is an operator (e.g., matrix) measure on ~ or ¢ and we may impose rather restrlc~ tlve conditions on U and ~ , ~ . E.g., we demand

U"

~dE~

to be a bounded operator (for example, in analogy with the classical inverse scattering U(x) can vanish exponentially on infinity).

and

f,~*u,

f*udZ~¢~,

/¢d~@~

be bounded

to

Also we demand all ~ ,

~

operators.

All such restrictions wlll be important if we want to write down all the solutions of (OR).

But from the point of view of the proof of the complete integrability

and the correct choice of action-angle varlables it is necessary to have only a formal scheme.

We have conservation laws for (OR) of the type of ~6:

Theorem ii.i [14]:

For the system (OR) we have the following first integrals

1 ;X-/%dZ;f,X + (11.3)

+

1I x=_-'77 1 [~Xx% ¢~¢nx]d~n[~nx~X ~n~Xx1' -

-

Moreover, all the first integrals C[X] are in involution.

Here we consider

elements of @Ax (~Ax) as the conjugate variables to the corresponding elements of ~A (CA) (if one views @, ~ as infinite matrices).

In particular we have first inte-

grals of the system (OR) in a traditional form: for an arbitrary constant operator S(A) the following Hamiltonian is in involution with (OR): H s = f trH{dZ~C[~]S[X]}

402

In particular, on the solutions of (OR), HS m const. Also the Hamiltonlan of (OR) has the form

H = ~ftrH{dE%C[%]}. The most important class of Hamiltonians H S is the class of "momentae" Hamiltonians. We define,

Hn = f trH{xndE%C[%]} for an integer n. In general Hamiltonians H S define a rather complicated evolution for

(~'~x; * ~ x )' and of course the evolution of

under the Hamiltonian flow H

may be nonlocal in U. However, the evolution of U S under the Hamiltonlan Hn for even n = 2m N> 0 is always local in the sense that the equation of the evolution of U under Hn takes the form of the m-th operator KdV equation (§5): Ut2m = P2m[U,Ux,...,Ux...x].

E.g., Ut2 = 6(UU x + UxU ) - Uxxx

etc.

2m+l Now we come to the main result concerning the (OR) [14].

Any equation of order

two presented in §5, theorem 5.1, and corresponding to ~(L) and 8(L) for the potential U(x,t) = f*~(x,t)dZ ~ ( x , t ) is equivalent to the evolution of (~(x,t), ~ ( x , t ) )

according to

in x-dlrection and according to the Hamiltonian of the form H S in t-direction. in the case ~(L) = O, ~(L)-arbitrary, M = I, we have HS =

ftr{~(-12)dZ%C[%]}

and

together with U t = Rn[U,Ux,...,Ux...x ] for g(X) " % n 2n+l

E.g.,

403

11.3.

One of the questions that arises in connection with the Russian chain (OR)

is the possibility of the representation (11.4)

u(x)

fCdZ¢~

-

for the large class of potentials U(x) in terms of (left and right) eigenfunctions ~,

$~ corresponding to the eigenvalue _ 2 of the operator Schrodinger equation -

the potential U(x).

with

We refer to §i0 where such a problem is reduced to the matrix

moment problem. Another important problem: how is the measure dE measure naturally associated to U(x)?

connected with the spectral

We know already answers to these questions

in the scalar case, so we can conjecture the answer in general.

In certain cases

this conjecture can be proved (see [14], [15], [26], [30]). In [15], [23]-[25] we considered the representation of the C -periodic (say, finite band) potential u(x) through the sum of squares of its eigenfunctions.

For

the C -periodic potentials u(x) there are representations (cf. [49], [23]): 2 u(x) = - / f21+l °E2i + C I i=o

u(x)

-

and

~-- 2 + C2 f2i+l'~2i+l i=o

where ~2i are periodic and ~2i+I are antiperiodic eigenvalues - end of lacunae - of -y"+ (u+~)y = 0 and for ff(x)

being eigenfunctions corresponding to ~i (~J = 0 if

~j is a degenerate eigenvalue). As we know (see §i0 and §6) there is, however, the 2 2 one canonical way to present u(x) through fi when all eigenfunctions fi are present. Here 2 X2n+l

~

~2n

~ n2 ~

7+

0(l) +

0(__12 )

....

n

Naturally singular solutions of the form

do k = ~ . ~(k-k i) i-I with arbitrary ~i may correspond to an arbitrary quasiperiodic potential u(x).

How-

aver, for the general quasiperiodic potential we have no idea what the spectrum is. Probably we need some diophantlne conditions for quasi-periods of u(x). We have a complete reduction of the inverse scattering problem %~

the infinite-

component Russian chain in the case of u(x) rapidly decreasing on infinity (cf,§§6,7) Let us define all spectral data in this case. f(x,k) be the solution of -f" + uf = k2f with f(x,k) and

~

e ikx :

X

++~

Let -F lu(x) l(l+x2) dx < ~ and

404

f(x,k) = a(k)e ikx + b(k)e-ikx: x -~

oo

The reflection coefficient r(k) can be defined as

r(k) = - b*(k) a(k) ' It is known that u(x) may have only a finite number of bound states _82 n <.. •< -812 < 0. Then the potential is uniquely determined by the reflection co2 efficient r(k), n bound states 81,... ,82 n and n norming constants

cj

=

: J

-

l,...,n.

It is possible to express u(x) in terms of f(x~k) 2 in the form of (7.23).

This

was done for the first time by M. Kruscal, C. Gardner, J. Greene, and R, Miura for purely soliton solutions, for which r(k) -- 0: n u(x) = -4 ~

c~8~f2(x;iSj).

However, for an arbitrary potential with r(k) ~ 0 we have in the representation (11.4), both slngular and smooth parts. There is, however, one case when we have a smooth representation only. is the case of u(x) without bound states treated by Delft & Trubowltz [30].

This In this

case, a very simple formula exists:

2i U(X) = -~

-~=o kr(k) f2 (x,k)dk

However, we found that a formula containing both reflectlonless and bound stateless potentials exists.

This corresponds to (11.4) with both singular and

smooth parts: n

kr(k)f2(x,k)dk - Z ~ c j S j f 2 ( x ; 1 8 j ) .

uCx) = T

J-1

Moreover, according to our previous results we have a more general representation for

~tu].

Theorem 11.2.

In the most general case we have ~[u] = ~

r

(-l)Kk2K-ir(k)f2(x'k)dk

n

+ ~

(-l)K+ls~K-icjf(xliSj)2: K = 0,1,2 .....

J-I We would like to mention that such a representation of u(x) as (11.4) is not unique! E.g., a finite band potential u(x) can be represented as a linear combination

405

of any n from the 2n+l squares of eigenfunctions which correspond to the end of the bands.

An analogical assertion is true for arbitrary C -potential, which is

periodic.

The proper normalization of the type 11.2 is proposed in [14], [15],

[26], and in §6. Let us consider an operator ease: U ( ~ asymptotically

is hermitian, U = U + and U(x) vanishes

exponentially or faster, i.e., for some E > 0, lim

[exp(clxE)u(x)] = 0.

We assume also that U(x) doesn't have negative eigenvalues

(this restriction may be,

however, removed). Theorem 11.3.

Under the assumptions above on U(x) we have the representation "I[ U(x) -.-~.

kOl(X,k)~l(x,k)dk ,

where 01(x,k) , ~l(X,k) are eigenfunctions

corresponding

to U(x):

$1,xx(X,k) = U-$1(x,k ) - k2~l(X,k):

~l,=(~,k) - ~l(X,k).U - k2~l(X,k ) These functions ~l(X,k), ~l(X,k) are closely connected with Jost eigenfunctions: ~l(X,k) ÷ A(k)eikx:

x ÷+=

~l(X,k) + A(k)eikx:

x ÷+=

and A(k)A(k) = R(k), where R(k) is the reflection coefficient. The same type of result is true for periodic potential U(x), U(x+T) H U(x): However, in the last case dZ

T @ 0. should be a singular measure with the support at the

set of ends of the lacunae of the Bloch spectrum. E.g., let £o~ d2 (---+ dx 2

U - k2)~ - 0

we have nontrivial forbidden zonae, and U is periodic. Then for eigenvalues 2 .. ~2 ~I'" ' n'''' : n < = being the ends of the forbidden zonae, we have the representation

u(~) -~--$1~i__ + i=l

c

406

with a constant operator C and eigenfunetions ~i' ~i corresponding to eigenvalues 2

%1: 2 ~i,xx = U~i - li~i ' ~i,xx = ~~iU - li~i: 2~

i = I, . . . .

Of course, for real U = U+' li are real and v~ ~ = ~k+ , i.e., U = ~ $i~i+ + C. i=l §12. Three-dlmensiona! completely Inte~rable systems of the Kadomts@v-Petviashvili type. 12.0.

There have already been for ten years several examples [59] of (x,y,t)

systems that arise from conditions of eommutativlty of two linear differential operators in 8/8x, ~/Sy, ~/~t displayed in §i, 1.4, (1.15).

These equations were

called "completely Integrable" because a large class of solutions was known.

How-

ever, not too much was known about these systems in general. One of the reasons why three and more dimensional systems hardly can be called completely integrable is the absence of the consistent Hamiltonian structure on

them.

On has here the following problem: we consider certain systems in variables

Xl,...,x n.

Then one of these variables, say Xl, is singled out as time and we try

to build a Hamiltonian structure using conservation laws.

We find immediately

that if n > 2 these conservation laws are nonlocal and it is even unclear what canonical shape they have.

On the other hand, if we choose another x. as time, l do we have a guarnatee that conservation laws will change into conservation laws? We also want to project Hamiltonlan structure to the stationary case, when a solution does not depend on Xl,X2,... etc.

If conservation laws in the whole sys-

tem were independent, Involutive and complete, will it be preserved after projection?

We fJnd one way to answer these questions.

This way is to separate variables

Xl,...,x n in a given completely integrable system and present it as a Joint action of n commuting Hamiltonlan flows ~ n the usual sense)on infinite dimensional slmplectic manifold. Now, for the first time, we can call some three-dimensional system (1.15) completely integrable because these systems can be: a) represented as Hamiltonlan systems; b) can be indeed reduced to actlon-angle variables. These phenomena will be discussed below and we shall see how several three-

dimenslonalcompletely integrable systems of the type (1.15) or [59] can be reduced to

one-dlmenslonal operator systems as a result of motion via three

Hamiltonian flows that commute.

Of course, as you may already guess, this one-

dimensional operator system is nothing but the operator Russian chain:

407

(OR)

dx 2 = _2~(~)./ 2~(~)dZn~(n ) + ~2~(~) dx 2

Our way of thinking can be explained in a very simple way.

We know (by §7 or §ii)

that all the higher KdV equations and, generally speaking, all two-dlmensional completely integrable systems of order two, can be reduced to an infinite component Russian chain (an infinite component nonlinear stationary Schrodlnger).

In other

words, the introduction of a new spectral variable k allows us to get rid of variable t.

Now it is quite natural, that if we add in the (x,y,t) system one

variable k we can get rid of variable t, and if we introduce variable X we can get rid of variable y.

This way we pass first from a two-dimensional KdV equation to a

multieomponent nonlinear Schrodlnger and then from a multieomponent nonlinear Schrodlnger to an operator stationary nonlinear Schr~dinger (an operator Russian chain).

Basically the rule here is the following: If you want to construct the

solution u(x,t) of a two-dlmensional system of KdV type, then you write it in the form u(x,t) = -21k2f(x,t,k)2d~ k where f(k) are elgenfunctions of fxx(k) = u(x,t)f(k) + k2f(k), with the evolution in x determined by this Russian chain and the evolution in t governed by any Hamiltonian flow commuting with the Russian chain (or the Hamiltonian HI).

Now if one wants to find the solution u(x,t,y) of an (x,~,~)-

dimensional system of KdV type, we present this solution in the form u(x,t,y) = - 2 / t r [ ~ 2 ~ d Z ~ ] where

+ $~, ~

arises from a stationary Russian chain d2~(~) dx 2 = .2~(K)f 2~+(n)dZn~(n) + ~2~(~)

and two Hamiltonian flows in t and y commuting with the given one in x. 12.1.

In order to consider three-dimensional systems we take an example of the

Kadomtsev-Petviashvili equation (so-called two-dimensional KdV) in (x,y,t)-dimensions on a Hilbert space H . o For an operator 0 6 o n H o , ~ = ~ ( x , y , t (12.1)

3 ~ yy = ~

(4~t + ~xxx

) this equation has the following form:

- *~x

~

+ ~ x

) + 6*[f~ydX',~]).

408

In the scalar case (H

o

- Qne-dimenslonal) one gets the usual Kadomtsev-Petviashvili

(see §i): 3Uyy " ~ x (4ut + Uxx x - 12UUx )' Now this equation can be represented as a result of common action of three commuting Hamiltonian flows arlsing from the operator Russian chain. 12.2.

In order to reduce (12.1) to three commuting operator Hamiltonians we first

reduce (12.1) to two two-dlmenslonal operator systems.

They are: the operator non-

linear Schrodinger (5.7) and the operator modified KdV (5,5) equations.

Both of

these equations are particular cases of equations of order two (§5) or the equations discussed in §9.

We decided to deduce these equations from the Dirac equa-

tion of the first order (see [14], [17], or [52]).

If we consider the Kadomtsev-

Petviashvili equation over H, then the Dirac equation is considered over H ° x H o , where H O is an infinite-dimensional L2(~,dM~) extension of H. We proceed from the Dirac equation on H

x H .

For the better representation

o A d~f ~ ~[I of the Dirac equation we adopt notations of o§9 for 0 -~I'

Then we consider

the fundamental (operator) left and right elgenfunctions ~(~), $(~) of the Dirac equation on H

o

x H : o d ~y( 0

--dx

"~

~

= -v.¢(O + ~ 0 3 . ¢ ( 0

,

(12.2) ddx $(0 = $(~).V - ~ $ ( 0 " % , for

123

v i::l 31::I

and the operators Q, R on H o.

We can pass from the Dirac equation (12.2) to the

operator Schrodinger equation on H ° x H o .

(12.4)

then

v v #(~) - ~(~),

~(~) - $(~).03,

iterating equation (12.2) we obtain the Schrodlnger equations .d2 v

v

~(~) = u~(o + ~2~(~)

(12.5)

dx 2

d2 ^ ~(O

=

2^ $(~)U + ~ ¢(~)

dx 2 for the potential

(12.6)

If we put

U - V2 - V . x

,

409

According to §§9-10, we can find a certain spectral measure d M

such that V

and U and some canonical system of polynomials in V, Vx, Vxx , ... is represented as a system of moments of ~(~)dM~(~).

We can use for this 10.1, the formula (10.1)

for the Dirac equation and the eigenfunctions Schrodinger

(12.5) and eigenfunctions

~, ~ of (12.2) or §ii for the operator

¢, ¢ in (12.4).

We present in Theorem 12.1

the formula for the lowest moments that are obtained from the expressions for Hamiltonian densities An, A of Lemma 9.2 for A - 03 .

(I0.i)

The corresponding ex-

plicit formulae were presented in [42], [52] for a general class of potentials V. %/ Theorem 12,1. For the operator solutions ¢(~), ~(~) of(12.2)and the operator solutions ¢(~), ¢(~)

(12.4) of (12.5) there exists a spectral measure dZ~ such

that

(12.7)

-V =

¢(~) ;

-

-(V2-Vx)°3 = I 2 ~ ¢v( ~ ) d ~

;

¢^( ~ ) ; - U

,,

I

2~¢(~)dE~¢(~)

and

(12.8)

2V 3 - V

XX

+ W

X

- V V X

Iv ^ $(~)( 2~ ) 3dM~__~ $( ) .

Here Vo 3 =

2V3

12.3.

-

Io

-

, U = V2 - Vx =

+W VXX

X

-VV= x

I

QRx-QxR

; 2QRQ-Qxx 1

I,Rxx+2RQR ; RQx_Rx Q

Now we can consider nonlinear two-dimensional

ponding to the Dirac equation

j

evolutionary equations corres-

(12.2) described in (9.11)-(9.12)

for A = 03 .

By

§10 these equations are equivalent to the common action of two one-dimensional Hamiltonians

(10.13)-(10.14)

commuting with HII in (10.12).

are H : m - 2,3,4 corresponding to the equations

The simplest Hamiltonians

(9.11) for n - 2,3,4.

Let us pre-

sent these three equations as evolutionary Hamiltonian equations on Q and R with three time variables to = x, t I - y, t 2 = t: Rt 0

(12.9)

=

gx D

Qt0

" Qx (i.e., t 0

iRy " Rxx - 2RQR;

=

x),

-iQy = Qxx - 2QRQ

(i.e., t I - y)

R t - 3RxQR + 3RQR x - Rxx x, Qt " 3QxRQ + 3QRQx - Qxxx

(i.e., t 2 - t).

410

According to §i0, these three flows arise naturally from commuting one-dlmensional Hamiltonlan flows on the Infinlte-dimensional symplectic manifold corresponding to the Hamiltonian HII:

;

--dx

= ~(~+~)¢(~)d~u0(U)'¢(O;

(12.1o3 dx

)~

Of coursed Theorem 12.1 and the results of §Ii allow us to represent all equations (12.9) in terms of Hamiltonlan flows commuting with the Operator Russian Chain system (ii.i) or (OR). equations (12.9).

Let us deduce the Kadomtsev-Petvlashvill equation from the

We do know that the equations (12.9) are consistent because they

correspond to commuting Hamiltonlan flows. Thus we can write the conditions of consistency (12.11)

Rt

" Rty'

Qyt " Qty"

It is impossible to verify immediately that the conditions (12.11) can be reduced to the following ones.

Let

Y " QR, W " QRx, Z so Y

x

= W + Z.

"

Qx R,

Then (12.9) takes the form: iRy - R xx - 2RY;

-iQy " Qxx - 2YQ;

R t m 3R xY + 3RW - Rxx x,• Qt m 3yQ x + 3ZQ - Qxxx' Then the conditions (12.11) take the form: (12.12)

R x ( 3 Yy+ 3 1 Y x x+ 6 i xW )

+ R(61Y Y+61[Y,W]+3W -21Y x

y

xxx

+3iW

xx

-21Y ) - 0 t

and (12.12)'

(61Zx-3Yy-31Yxx)Qx + (31Zxx-21Yt-6i[Y,Z]-3Zy+61YYx-21Yxxx)Q = 0.

important

One makes the following Corollary 12.2~ system

If rank (R) and rank (Q) are larger than rank (Y) + rank (W), then

(12.12) or (12.12)' is equivalent to the following: 3Y

y

- 3iY

y

+ 61W

x

8 0;

6iY Y + 61[Y,W] + 3W - 2iY + 31W - 2iY = 0. x y xxx xx t These last equations can be written in a more convenient form. Yx - W + Z, Yy = (W-Z)x or (12.13)

W ffi~ Y x

+

YydX;

Z = ~ Yx - ~

YydX.

We have

411

The second equation in the Corollary 12.2 can be rewritten taking into account (12.13) as a single (x,y,t)-operator Kadomtsev-Petviashvili equation (12.1):

3Yyy = ~ ' x (4Yt+Yxxx-6(YxY+YYx)+6i[

(12.141

ydX,Y])

We must remind the reader that in order to get the equation (12.1~) we demanded rank (R) > rank (Y). For this it is natural to take R and Q as operators with Range (R) = Range (Q) = H with codim H in H

to be infinite.

can look st L2(~,dM )

(~,dM)

o H o as a L2-type space over H: H ° = % 2

Now we choose an operator

As a model we

H, for a certain space

R,Q in the following matrix form over H:

(R) il = Riej ; (12.15)

(q) ii = BiQi ;

i,J e

for operators R i, Qj and ~j - ~(J), Bi - ~(~) on Ho over L2(~,dM ) such that (12.16) Here (12.15)

f C~(J)dMj B(j) = I. fl means t h a t

R~

R acts

on e l e m e n t ~ = ( ~ ( ~ ) :

SeHo,

J E ~) o f H° a s

- Rj a

Analogically Q~I = 91 if ($I) ~ = 8~! QidMi~(i): J e a. Now (12.17)

(Q'R) ij = ~i~J f~QldMlRl"

We define c'matrix (an operator on L2(~,dMI))

~s

c = (Bi~j), and potential (an operator on H)

(12.18)

[ C% = ]aQxDM~RX.

Thus QR = C.~= ~.C. stant operator.

Taking into account (12.16) we get C2 ,, C and C is a con-

Substituting Y=

QR=

C-~

into the equation (12.13) we obtain a single equation for ~ (12.19)

3~yy "

only"

412

v

^

12.4. We now consider operator solutions ~o(~), ~ (~) of the operator Dirac equal. v v ~ ,~ O ,. tion (12.2) or solutions ~o(~) - ~o(~), ~o(~) - ~o(~)O 3 of the Schrodinger equations (12.5) for the choice of operators R, Q on H ° as in (12.15)-(12.16). the spectral measure d ~

depends on ~

and on (~,dM) as well.

In this case

According to

Theorem 12.1 or §11 we have (12.5): ~

v

v

2

v'

d2 ~o(~ ) - U~o(~) + ~ ~o(Z:); dx 2 ^ d2 ~o(~ ) - ^~o(~).U + ~2 "~o(~) dx 2

for (12.20)

U " V2 - Vx m

.

EQ X

Then by Theorem 12.1 (cf. §Ii) we have for a spectral measure dE~:

(12.21)

~ (~)dE~ ^~o (~)' U = -2 I 2~ ~o x R . We take trace with respect to H 2 "

This is an operator identity on H O

H O

O

O

x R /R and obtain O

(12.22)

trR2/H(U)--2

2v

^

trH2/H( ~ ~ o ( ~ ) d ~ o ( ~ ) ) .

o

o

Now

trH2/H(U) = trHo/H(QR) + trHo/H(RQ) = 2trHo/H(QR). O

But by (12.16)

tr~ /~(QR) = ~

.

e

Thus from (12.22) i t fellows (12.23)

~--/trH~/H(~2~(~)d~

~o(~)) .

This formula gives the most general expression for the solution of the Operator (x,y,t)-Dimensional Completely Integrable System of the type (1.15) or (12.1): Coro!lar~ 12.3.

In order to get the solution~(x,t,y) of a matrix (operator)

three-dlmenslonal systems (12.1) (or commuting with it) over the Hilbert space H~ it is necessary to consider the Hilbert space

H2 - RO × Ho

for H o inflnite-dlmen-

aloha1 over H (being L2-extenslon of H) and the corresponding operator Schrodlnger equations over H2:

2v v 2~ d ~(n) . U~(n) + n~(n); (OR)

dx 2

dx 2

413

being also the operator Russian chain (II.I) for (12.24)

U - -2 IE 2v¢(~)

d~O~(~). V

^

NOw we take the solutions ~(~;x,t,y), ~(~;x,t,y)

of the system (OR), (12.24)

satisfying simultaneously two Hamiltonlan systems in t and y co--.utlng with (OR). These two Hamiltonians can be chosen in a standard form such that, e.g., U t - -2 ~ x {J~2n~(~)d~ ~(~)}

for

Then one can represent the solutlon ~

s

for $(0 - ~(~;x,t,y); adJoint06,~

~(0

+ - ~and

~(~)

t - t2n.

as:

Hz/H°

" ~(~;x,t,y).

Moreover, in t h e c a s e of r e a l and s e l f -

self-ad~olnt problem (OR) we have

= ®(0 + , U + - u.

Then ~(x,t,y)

= -21trH2/no{~+(~;x,t,y)d~#(~;x,t,y)}

for

d2

-~(0 ~ -2~(0 • dx 2

I*+(~)d~

~(~) + ~2~(~).

n

In general, coefficients of the linear differential operator L -

d~

~_U~

j=o

~

can be expressed in terms of products of elgenfunctlons of L: L~ = X~ and eigenfunctlons of the conjugate problem L*~* - A~*.

This representation for n - 1 is de-

scribed in §§9, I0, and for n = 2 in §11.

Such a representation and its connection

with spectral measures, allows us to reduce any two- (or three-) dimensional completely integrable system of the Zakharov-Shabat type (see §I for definition) to a common action of two (or three) commuting one-dimenslonal completely integrable Hamiltonians.

We postpone the general discussion for further publications and

now briefly consider only the case n - 3.

As the case n - 2 is connected with the

nonlinear Schrodinger equation (5.7), thecase n = 3 is connected with the coupled modified KdV equation (5.5) (see [13], [14]):

I

~t

=

~t

" -~xxx + ~x ~

-~xxx + ~x ~

+ ~x; + ~x

"

Again we can consider stationary solutions ~ = eAr,o; ~ = ~o e-At for ~o' ~o depending on x only and for a constant A.

We now consider operators over the

L2(~,dZ )-extenslon of a Hilbert space R.

For the sake of simplicity, for the

L2(~,dZ ) sequence of operators ~ = (A), B = (B)

(~ ~ ~) on H, we denote

414

÷

(A,B) =

I ~d~B.

Then the stationary operator modified KdV equation can be represented as the following coupled system of equations of the third order in ~k' % :

f-+k'x=+ (*'*)*k'x+ (L'O+>+k: k *k -Ok,xxx + Ok,x ($'~) + ~k $'Ox) = -k30k" Now we understand that the moment problem of the type of Problem I0.I, leads to the representation of the coefficients U and V of the eigenvalue problem of the third order

d3 (- " ~ + dx

U~x + V)*k = k3~k ,

in terms of the eigenfunctions Ok and eigenfunctions Ok of the conjugate eigenvalue problems.

Namely, we ha~e the following representation in the case of U and V being

operators on H: U = I ~kdEk0k;

V-Ifl4~kxdEkOk

for a certain spectral measure dEk.

,

E.g., as in §I0-Ii (see [15]), such a repre-

sentation leads to the decomposition of the Boussinesq equation

u yy=(6 ~Ux-Uxxx)x

into the common action of the Hamiltonian HIII (scalar case) and one of the Hamiltonians commuting with HII 1. We mention here only that the Boussinesq equa~u tion corresponds to t h e ~ = 0 case in the Kadomtsev-Petviashvili equation; thus there are interesting relations between HI11 and the Russian chain.

Bibliography. i. M.J. Ablowitz, D.J. Kaup, A.C. Newell, H. Segur, Stud. Appl. Math. 53(1974), pp. 249-315. 2. N.I. Akhiezer, The classical moment problem, ~afner Publishing Company, 1965. 3. G.G. AppalFot, Mat. Sbornik v. 16 (1892), Prec. Moscow Univ. Sect. Phys.-Math., No, II (1894); N.B. Delona, Algebraic integrals of motion of a rigid body about a fixed point, Petersburg, 1892; V.V. Golubev, Lectures on integration of the equations of motion of a rigid body about a fixed point, Published for NSF, Jerusalem, 1960; D.N. Goryachoff, Math. Sbornic., 21 (1900), No. 3; P. Nekrasov, Math. Sbornik., 16 (1892); V.A. Steklov, Transactions of the physical sciences of the society of amateurs of science, v. 8 (1896). 4. H.F. Baker, Prec. Royal Soc. London, Set. Al18 (1928), pp. 584-600. 5. A.A. Belavin, V.E. Zakharov, Pisma JETP 25(197~, No. 12, pp. 603-607. 6. Yu. M. Berezansky, Trudy Moscow Math. Soc., 21 (1970), pp. 47-102; Yu. M. Berezansky, Salf-adJoint eiganfunction expansions, Providence, Rhode Island, 1970 7. Yu. M. Berezansky, Self-adJoint operators in spaces of functions with infinite number of variables, Naukova Dumka, Kiev, 1978. 8. A.S. B~dagov, L.A. Takhtajan, Dokl. Acad. Sci., USSR, 235 (1977), No. 4, pp. 805-808. 9. A.S. Budagov, Proceedings of LOMI, 1970, pp. 24-56. 10. J.L. Burchna11, T.W. Chaundy, Prec. London Math. Soc., 21 (1922), pp. 420-440; J.L. Burchnall, T.W. Chaundy, Prec. Royal Soc. London, Ser. Al18 (1928), 557-573.

415

11. F. C a l o g e r o , A. D e g a s p e r l s , Nuovo C ~ e n t o 32B ( 1 9 7 6 ) , p . 201; F. C a l o g e r o , A. Degasperls, Nuovo Cimento 398 (1977), 1. 12. I.V. Cherednik, Funct. and Appl. 12 (1978), 45. 13. D.V. Chudnovsky, C.E. Acad. Scl. Paris, A289(1979), pp. A-429-A-432. 14. D.V. Chudnovsky, Phys. Lett. A, 74A(1979), pp. 185-188. 15. D.V. Chudnovsky, Saclay preprint DPh-T/79/80, J. Phys. (to appear). 16. D.V. Chudnovsky, Left. Nuovo Cimento, 26(1979), pp. 423-427. 17. D.V. Chudnovsky, C.R. Acad. Scl. Paris A289(1979), pp. A-438-A442o 18. D.V. Chudnovsky, Riemenn ~onodrom7 problem, isomonodromy deformation equations and completely integrable systems. Lectures. Cargese Summer School on Mathematical Physics, June 1979, D. Reidel Publishing Company, Dordrecht, Holland (to appear). 19. D.V. Chudnovsky, The generalized Riemann-Hilbert problem and the spectral interpretation, Proc. Intl. Meeting "Nonlinear evolution equations and dynamical systems", Lecce, Italy, July 1979, Lect. Notes Physics, Springer-Varlag (to be published). 20. D.V. Chudnovsky, Saclay preprlnt DPh-T/79/38, Nuovo Cimento B (to appear). 21. D.V. Chudnovsky, International Congress of Mathematicians, Helsinki, Finland 1978, Abstract of short communications, p. 183; Notices of the AMS, 26(1979), No. I, Abstract 763-35-10, p. A-83-A-84. 22. D.V. Chudnovsky and G.V. Chudnovsky, Nuovo Cimento, 408 (1977), pp. 339-353. 23. D.V. Chudnovsky and G.V. Chudnovsky, Seminaire sur lea equations non-lineaires I (1977-1978), Centre de Mathematiques, Ecole Polytechnique, pp. 300. 24. D.V. Chudnovsky and C.V. Chudnovsky, Lett. Nuovo Cimento 22(1978), pp. 31-36; pp. 47-51. 25. D.V. Chudnovsky and G.V. Chudnovsky, C.R. Acad. Sci. Paris A286(1978), pp. A-lO75A-I078. 26. D.V. Chudnovsky and G.V. Chudnovsky, Phys. Lett. A., 72A(1979), pp. 291-293. 27. D.V. Chudnovsky and G.V. Chudnovsky, Saclay preprint, Dph-T/79/llO. Z. Phys. D (to appear). 28. D.V. Chudnovsky and G.V. Chudnovsky, Seminar on Spectral Theory, Riemann Problem and Complete Integrability, INES 1979, Paris (to be published); G.V. Chudnovsky, Pad6 approxim~tlon and the Riemann monodromy problem. Lectures. Cargese Summer School on Mathematical Physics, June 1979, D. Reidel Publishing Company, Dordrecht, Uolland (to appear). 29. G. Darboux, C.R. Acad. Sci. Paris, 94(1882), p. 1456. 30. P. Delft, E. Tzubowitz, Comm. Pure Appl. Math. 32(1979), 121-200. 31. A. Degasparls, Lecture Notes in Physics, v. 80, Springer, 1978. 32. J. Drach, C.R. Acad. Scl. Paris 167(1918), pp. 743-746; C.R. Acad. Sci, Paris 168(1919), pp. 47-50; pp. 337-340. 33. B.A. Dubrovin, V.B. Matveev, S.P. Novlkov, Russian Math. Survey 31(1976), 59. 34. B.A. Dubrovin, Funct. Anal. 11(1977), No. 4, 28-41. 35. R. Gamier, Circolo Math. Palermo, 43(1919), pp. 155-191. 36. I.M. Gelfand, L.A. Dikij, Funct. Anal. 10(1976), No. 1, pp. 17-25. 37. I.M. Gelfand, L.A. DiklJ, Russlan Math. Survey 20(1975), No. 5, 77. 38. I.M. Gelfand, L.A. DiklJ, Funct. Anal. Appl. 13(1979), No. 1, pp. 7-20. 39. H. Grcsse, Institut of Theor. Phys., Univ. Wien, UWThPh-1979-19 (to appear). 40. J . Honerkamp, Frelburg, 12p. Preprlnt THEP 7 9 / 8 . (see this Les ~ouches volume). 41. C. Jacobi, Vorlesunger uber Dynamic, Berlin, 1843. 42. D. Kaup, J. Math. Anal. Appl. 59(1976), 849-864. 43. W. Koppelman, J. Math. and Mech. (1961), pp. 247-277. 44. S. Rowalevski, Letters to Mittag-Leffler, Letter from 21 November, 1881. Correspondence, Published by the Academy of Sciences, USSR, 19~8; S. Kowalevski, Acts Math., 12(1889), pp. 177-232; Acta Math. 14(1890). 45. M.G. Kreln, M.A. Krasnoselsky, Uspekhl Math. Nauk 2(1947), No. 3, 60-106. 46. JoM. Kritchiver, S.P. Novlkov, Funct. Anal. Appl. 12(1978), pp. 41-52. 47. P.D. Lax, Commun. Pure Appl. Math. 21(1968), pp. 467-490; P.D. Lax, Comm. Pure Appl. Math. 28(1975), pp. 141-200. 48. B.M. Levltan, Math. Sbornlk 76(1968), pp. 2~9-270. 59. H.P. McKean, P. van Moerbeke, Invent. Math., 30(1975), pp. 217-300. 50. H.P. McKean, E. Trubowitz, Comm. Pure Appl. Math. 29(1976), pp. 143-226. 51. O. Neumann, J. Relne Augew. Math. 56(1859), pp. 46-63. 52. A.C. Newell, Proco Royal Soc. London, A365(1979), pp. 283-311.

416

53. L.P. Niznik, Nonstationary i n v e r s e s c a t t e r i n g problem, Naukova Dumka, Kiev, 1973. 54. K. Pohlmeyer, K.-H. Rehren, Frelburg Preprint 1979, J. Math. Phys. (to appear); Comm. Math. Phys. 46(1976), 207-221; Freiburg Priprint THEP 79~6. 55. L. Schleslnger, J. fur Eeine Angew. Math. 141(1912), pp. 96-145. 56. E.K. Sklvanin, L.A. TakhtedJan, L.D. Fadeev, LOMI preprlnt P-I-79, Leningrad, 1979, pp. 35. 57. H.B. Thacker, D.Wilkinson, Fermilah, pub. 79/19-THY, February 1979 (to appear); H.B. Thacker, Phys. ReVo D 17(1978), 1031. 58. V.E. Zakherov, A.V. Mikhailov, Zr. Eksp. Teor. Fis. 74(1978), No. 6, pp. 1053-1973 59. V.E. Zakharev, A.B. Shabat, Funct. Anal. Appl. 8 (1974), No. 3, pp. 54-56; Funct. Anal. Appl. 13(1979), No. 3, pp. 13-22. 60. J.L. Gervais, A. Neveu, E.N.S. Preprint (to be published). 61. E. Betel, Lectures on divergent series, Paris 1928 (English translation ed.by J. Gamins1, Los Alamos, LA-6140-TR, 1975). 62. D.V. Chudnovsky, Prec. Natl. Acad. Sci., USA, 75(1978), pp. 4082-4084; Lstt. Nuovo Cimento 25(1979), pp. 263-265.

Quantization of Exactly Integrable Field Theoretic Models - The Operator Transform Method -

J. Honerkamp Fakult~t fHr Physik der Universit~t Freiburg Hermann-Herder-Str. 3 7800 Freiburg

Whereas

/ Germany

the construction of an exactly integrable model or the discussion of whether

a given model is exactly integrable is a more or less mathematical interesting question for a physicist - What are the ~onsequences

problem,

a very

for the quantization

of the

is immediately:

of the exact integrability

theory - which relations of the classical canonical

structure survive in the quantization

procedure - which benefit can be derived for the quantization

of the theory from the knowledge

of the exact integrability? I would like to discuss these questions

in the context of the nonlinear

Schr~dinger

model, where we have already a complete answer. But of great interest are also other models like: a) The massive Thirring model b) the Heisenberg model c) the Toda Lattice model because pursuing the same strategy one may uncover new aspects which become trivial in the nonlinear

SchrSdinger model because of its simplicity.

In the following I am going to recapitulate nonlinear

the classical canonical

structure of the

Schr~dinger model.

Then I translate the classically

defined quantities

into quantum operators

and I

demonstrate a) that some of these operators b) that these eigenstates

can be used as generators

can be characterized

very special structure known as "Bethe-wave c) finally that some other quantities translated

into quantum mechanics,

of the quantum eigenstates

by a wave function which is of a function".

of the classical canonical

structure,

can be regarde d as generators

quantities. One obtains in this way an infinite set of commuting observables.

of the conserved

418

1. The classical canonical structure of th~ nonlinear SchrSdinger equation. The nonlinear

SchrSdinger

i~ t ~ =

-~ x

2

~ + 2c

~ ;

(1)

From Zakharov and Shabat guarantees

equation reads X

(1)

~x

we know t h a t t h i s n o n l i n e a r e q u a t i o n f o r t h e f i e l d

(v) (:)

the compatibility

L

of the two linear differential

v

~(x,t)

equations

(2)

=

2 (2b)

where

L

=

; ~cc ~'

I~ [Vll

• Iv2/

M

=

-i~ x

is a solution to ( 2 a ) w i t h

I

Wlth elgenvalue

-2~ 2 + c ~

eigenvalue ~ = ~ +

in

~ = ~ - iD,

~e may define special

solutions

asymptotic behaviour.

We assume ~ ÷

u(~,x)

4/~x)

~c(i~+

and w(~,x) by

to the equation o

for

~

'

thenl v~2~l is a solution %11

-v

(2a) by reouiring a special ÷ oo and we define the

Jost-solutions

(,~=i~.

u¢~,~)

>

(~)

e-i~ x

(4)

X ÷--~

Then u

t'l u2J

and

X

->+~

-u

: =

~.uU

represent

a complete

set of solutions,

like w and w,

and expanding u ( ~ , x ) i n terms of the w, w we o b t a i n u(~,x,t)

=

a($,t) w(~,x,t)

+

b(~,t) W(~,x,t)

(5} ~> (a(~,t) e-i~x1 x ÷ +

b(~,t)

e iSx J

419

~he coefficients of this expansion, the Jost-functions a(~) and b(~), can be obtained by looking at the asymptotic behaviour of the solution u(~,x), which may be constructed by converting the differential equation (2a) into an integral equation anff by Solving this Volterra equation by iteration. One obtains in this way b(~;~(x,t),~x,t))

=

~c ( fdx ~ x , t ) e - 2 i ~ x +

(6) + c fdXldX2dx3 @(x I > x 2 > x3)e-~gi~(xl + x2 - x3)~(x])~(x2)~(x3) +

•

.

.

)

and

a(~; ~(x,t), ~(x,t)) + c

f dx|dx

=

(7)

1 +

2i~(xj-x 2) @(x 2 >

xl)e

~(x I) ~(x 2) + . . .

The structure of the higher terms of the series (6) and (7) is obvious. The remarkable fact now is, that if ~(x,t) and ~x,t) evolve in time in accordance with the nonlinear Schr~dinger equation (I), then the time evolution of the Jost-functions a(~,t) and b(~,t) is very simple:

a(~,t) b(~,t)

=

a(~,O)

(8a)

b(~,O)e 4i~2t

(Sb)

This means: a(~) and b(~) represent another set of coordinates, instead of ~(x), ~(x)}

, with a very simple time behaviour. This transformation can be used

to solve the nonlinear equation (I), the time evolution will be calculated in terms of the coordinates a(~) and b(~); the step which remains is the transformation from the a~,t) and b(~,t) back to ~x,t),

~(x,t), which can be done by an inverse

scattering method, which gives this whole procedure its name: the inverse scattering transform method.

420

By the Poisson brackets {~(x), ~(y)~

=

i6(x-y)

(9)

and the Hamiltonian H

=

fdx(-~(x) $2~(x) x

+

c(~(x)~(x)) 2 )

(I0)

the field equation reads

~t ~ =

{H,~

(11)

and the equation (8) can be written as {H,a}

=

0

;

{H,b (~) }

=

4i~2b (~)

(12)

One may calculate also some other Poisson brackets

{a(~), a($')}

=

{a($), b(~')}

{b(~), b(~')}

{b(~), b(~')}

c ~-~ "i~

=

=

0

(13)

a(~) b(~')

(14)

2~ica(~)a(~)~(~-~

)

(15)

421

2.

The quantized version of the nonlinear

SchrSdinger model

In quantizing we interpret

the field ~(x,t)as an operator

conjugate as the hermitean

conjugate

[~(x), ~+(y)]

The Hamiltonian

=

~ (x-y)

~

+

c

operator

~+(x)~+(x)~(x)~(x))

which we have taken as normal ordered. and the eigenvalues

rule we take

(16)

(IO) then goes over to a Hamiltonian

: /dx~2x

field, its complex

field ~+(x,t). As a quantization

(17)

Now the question is: What are the eigenstates

of this Hamiltonian

operator?

It will turn out in a moment that the Jost-functions when translated into quantum operators,

defined in (5),

(6) and (7),

play a central role in the quantized theory

namely: - The odd Jost function b($) generates,

as an operator,

- The even Jost function a(~) generates,

as an operator,

the quantum eigenstates

the conserved quantities.

Let us make precise these statements. a) The eigenstates

of the Hamiltonian

We define an operator

B+(q)

= ~

b(~ = %

, ~ + ~op '

fax ~+(x)e -iqx +

c

(is)

j(dx 1ax2ax3u~Xl>X2>x3)e .

.

.

.

.

liq (Xl+XmlX3) +]

~txi)~(x2)O(x3 )

+ ° . .

B+(q) contains an odd number of ~ or

~+'s. For c = O, B+(q) is a creation operator

,of a free particle with momentum q and energy q2, where the ground state or reference state

hence

l~>

is defined by

¢(x) l a >

=

o,

BI~>

=

o

422

and

l

fdx+(x)e_iqx]~>

(19)

Proposition: + B+(ql ) . . . B (qN) I ~

is an exact eigenstate of H with eigenvalue

N

E = ~ qi 2" (-~
=

q2B+(q)

(20)

i.e. that the classical Poisson bracket (12) {H, b(q)l

iq2b (q)

may be taken over to the quantum level by omitting a factor i as in going from (9) to (16). The proof of (20) was given by Faddeev and Sklyanin (3)and by Thacker and Wilkinson~4)r Sklyanin pointed out that the linear differential equations (2a), (2b) can be translated into differential equations for operator valued components G(q,x) and F(q,x) of the Jost-solution u =

~xG (q, x)

=

( 112

, interpreted as an operator:

i~cc F(q,x) ~(x) - i~/~ G(q,x)

(21) ~xF (q, x )

= -i~cc ~+(x) G(q,x) + iq/z.F(q,x)

Note that this equation is normal ordered and F and G come out to be normal ordered. In a similar way one may translate the equation (2b) and the compatibility of these two linear operator equation requires

i~t~

=

[~,H]

(22)

We obtain the operator analogue to u(~,x,t) if we require G(q,x) ~ (F(q,x)])

-iqx~. X _~_ I ~> ( e O

(23)

423

Then we obtain exactly -iq~z~ G(q,x) (24) x

~q)e lq_

÷

The time behaviour of the Jost operators can be calculated in the same way as in the classical case (2), but now with help of the equation (21) and the operator version of (2b). Hence one obtains also (q =~/~)

A(q,t)

=

A(q,O)

B+(q,t)

=

B+(q,O)elq t

• 2

(25)

On the other hand we have from (22)

i$ tB+(q't)

=

[B+'H]

(26)

=

q2 B+(q)

(27)

=

0

(28)

Therefore by (25) [H, B+(q)]

and likewise

[H,A(q)]

(b) The structure of the eigenstates.

The operator B~q) is represented as an odd series in the elementary fields ~(x),~+(x), but the eigenstates B~ql) ... B(qN) + I ~ > when expressed in terms of the elementary fields are monomials: !

+ + + B+(ql ) ... B(qN) I ~ > = /d xl...dXNX(Xl...x N ; ql..,qN)~(Xl)...~(XN)l~ >

(29)

The reason for this simple structure is that there exists an operator - called "particle number operator" -

N

=

Idx J

~+(x) ~(x)

(30)

424

with [N,H_]

=

0

(31)

and [_N,~+(x)]

~+(x) ; [N, B+(q)]

=

B+(q)

(32)

Hence the eigenstate (29) is also eigenstate of N with eigenvalue N and is written as a superposition of such other eigenstates of N, namely of

¢+~X;) ...~XN)[~ >

(33)

The superposition-coefficients

are given by the wave-function X(Xl..°xN, ql...qN)

which has to be chosen so that the N-particle state (29) is just an eigenstate of H. This wave function can be calculated (5'6)to be

~(xI...XN, ql...qN )

=

e

i

K=~IqKXK__" = II(1+ i%(qK, qj)e(xK-xj) ) ]
(34)

where -

c

(35)

(qK' qj )

[qK-qj j

E(XK-X j)

[~|| for for

and

=

XK>X j x K<xj

Formula (34)means that for X K # X j ; K,j = I,...N, the wave function is a free one. At xK = xj the wave function makes a jump. The height of it is dictated by the dynamics and this jump is just the whole consequence of the dynamics. It is well known that these wave functions are the quantum mechanical eigenfunctions of the N particle Hamiltonian N

N

2 H = -

Z i=1

~ ~x. 2 i

+

2c Z ~(xi-xj) i,j=l

i#j

(36)

425

These eigenfunctions were found in 1963 by Lieb and Liniger(7)and already in |931, Bethe (8) had shown that for the isotropic Heisenberg model on a lattice the eigenfunctions have such a quasi-free structure as in (34). Here one is able to derive this Bethe wave function by looking at the states

B+(ql) ...B~q N) [fl> The Bethe structure of the wave function has immediate consequences for the S matrix Choosing ql < q2...< qN and x1> x2...> XN we get (Xl...XN ; ql...qN)

= H ( | + iA(qK,qj))e i E q K x K

(37)

I
H(l k<j

By choosing Xl< x2< ...< xN X-(Xl...XN , ql...qN)

+ i%(qKqj))

X

<38)

we see that an outgoing state can be defined by =

~<j(l_iACqK,qj))

-I X

(39)

Hence I + i~(qK, q j) ;(Xl'X2''''XN

' ql'''qN )

I-I k<j

X+(Xl...XN

, ql...qN

)

] - i~(qK'qj)

and ¢

m

S (q°l...qN ; ql...qN )

= •-

#

; ql...qNI + , ql...qN>

I + i%(qx,qj) = H I k<j w

where 6

i%(qx,qj)

q'l...q~ ql...qN

NN'

f

q l " " "qN

is the symmetrized product of Kronecker-symbols.

ql " " "qN

Hence we have as consequences of the Bethe structure of the wave function:

(i)

Particle number conservation, where particles are created by those fields, in terms of which the eigenstates are monomials so that a wave function can be defined.

426

(ii)

The set of incoming momenta is equal to the set of outgoing momenta:

{ql...qN} ={ql'''qN} (iii)

Factorization

of the N particle S-matrix in terms of the two particle

S-matrix ic

I

I + i%(ql,q 2)

lql-q21

S (ql,q2)

(42) I - i%(q|,q 2)

; +

ic

lql-q2L The question arises whether also in the other models mentioned eigenstates

at the beginning

the

can be constructed by this strategy and whether they have a Bethe-struc-

ture with all these consequences. Up to now we know that d~is is true in the massive Thirring model anisotropic Heisenberg model (9). In the Toda-lattice in this way the eigenstates of which the eigenstates

model(]O)one

is able to construct

But the field coordinates

can be written as a superposition

defining a wave function, such coordinates

of the Hamiltonian.

(5)and in the

of monomials,

in terms thereby

are not yet known to me. It may even be the case, that

do not exist and that the S-matrix for the Toda-lattice model is

of a more general structure. (11) It should be noted that the S-matrix which is calculated

in this way within the

massive Thirring model is not yet the physical

Because in this case the

reference

state

Io>is

as a superposition

S-matrix.

not the ground state, one has to construct the ground state

of the eigenstates.

states and by this construction

The excited states are then the physical

one is also able to calculate

the fermion-antifermion

bound state spectrum. (12)

c)

The conserved quantities

By (25) and (28) we know that A(q) is a conserved quantity,

coefficient

therefore so is each

in an expansion l i k e A(q) =

E

q

-n

I

or

A(q)

n

n=o

=

E n=o

qn j

(43) n

The J

are nonlocal quantities, we do not know whether they are useful. n For the In(n = 0 , 1 , . . . ) we o b t a i n :

I°

=

1

;

I!

=

-icN (44)

"

-icz

-


-

427

13

=

icH - (ic) 2 (PN - P) + (ic~6

( N 3 _ 3 N 2 + 2N)

etc. where N and H are given by (30)'and (17) and

E

=

~fdx(C°x®

- ~x c®)

(45)

Hence: The even Jost-function in its quantized version generates an infinite set of conserved quantities, to which the Hamiltonian belongs. In order to discuss the eigenfunctions and the eigenvalues of these conserved quantities one takes advantage of the con~nutator

q'-- qic - ie B+(q) A(~)

[A(q') , B+(q)]

(46)

From this one deduces that B~ql) ,,,B+(qN)]O>is also an eigenstate of A(q') and therefore of all I 's: n

N A(q)B(ql)'''B(qN)IO>

=

H i=l

qi - q°- ie

""

By expanding both sides in q' one obtains the eigenvalues of the I ' s . n

Note that the commutator (46) is the translation of the Poisson bracket (14), but on the right hand side of (46) one has to take care with the ordering of the operators. However, the classical Poisson bracket (15) does not have such a simple translation. For the quantum operators one obtains 2 C

[B(ql) ' B+(q2)]

=

~(ql-

q2)A~ql)A(q2)

+ B~q2)B(ql)

2 (ql - q2 + is)

Recently Faddeev(13)developed a method of calculating the commutator

[A(q)

,

B(~)]

in every integrable model with boson fields. Here A(q) stands for an even Jostoperator, B(q) for an odd operator. For the massive Thirring model however this method seems not to work. (|5) In conclusion one can ascertain that the canonical classical structure provides us

428

with a powerful tool for the construction of eigenstates and conserved quantities in the associated quantum field theory. The remaining problems in the various models may be clarified certainly very soon. For relativistic

theories as for the massive Thirring model, however, these eigen-

states and the corresponding S-matrix are not yet the physical quantities,

as I

mentioned above. Whereas there exists a well established procedure to calculate the physical energy spectrum,(12~he- calculation of the physical S-matrix from these eigenstates remains obscure, though there exist some intuitive arguments which, however, have not yet been made precise.

It is a very challenging problem, whether

the Green's functions of these integrable field theoretic models can be calculated (16) by e x p l o i t i n g t h e methods d e v e l o p e d by Sato and h i s c o w o r k e r s . References: I)

V . E . Zakharov and A. B. Shabat Zk. Eksp. Teor. Fiz. 6!I, 118 (1971) (Sov. Phys. JETP 34, 62 (1972)

2)

V . E . Zakharov and'S. V. Manakov Teor. Mat. Fiz. 19, 332 (1974) (Theor. Math. Phys. 19, 551 (1975)

3)

L . D . Faddeev and E. K. Sklyanin Dokl. Akad. Nauk, USSR 243, 1430 (1978) E. K. Sklyanin, Dokl. Akad. Nauk USSR 244, 1337 (1979)

4)

H.B.

5)

J. Honerkamp, A. Wiesler and P. Weber, Nucl. Phys. B 152, 266 (1979)

Thacker and D. W. Wilkinson, Phys. Rev. D 19, 3660 (1979)

6)

A. Wiesler, Freiburg preprint Okt. 79

7)

E.H.

S)

H. Bethe, Z. Physik 71, 205 (1931)

Lieb and W. Liniger, Phys. Rev. 130, 1605 (1963)

9)

see e.g. P. P. Kulish and E. K. Sklyanin, Phys. Lett. 70 A,46| (1979) J. des Cloiseaux and }!. Gaudin, Journ. Math. Ph---ys.~, 1384 (1966)

IO)

see e.g. M. Toda, Phys. Reports 18 C, 1 (1975) H. Flaschka, Phys. Rev.--B9, 1924 (1975) and Progr. Theor. Phys. 5_~I, 703 (1974) K. M. Case and S. C. Chiu, J. Math. Phys. 14, 1643 (1973)

II)

see e.g. the lecture by D. lagolnitzer,

12)

this colloquium.

see e.g. H. Bergknoff and H. B. Thacker Phys. Rev. Lett. 42, preprint Fermi-Lab Pub-78/84-Thy

135 (1979)

13)

L° D. Faddeev, preprint Akad. Nauk USSR Lomi-P-2-79

14)

H. Grosse, preprint Wien 1979/~thPh-1979-19

15)

P. Weber, private communication

16)

M. Sato, T. Miwa, M. Jimbo, a series of papers entitled " Holonimic Quantum Fields ": I, Publ. RIMS, Kyoto Univ. 14, (1977) 223-267, II ibid. 15 (1979), 201-278 RIMS-preprint--s 260 (1978), 263 (1978) and 267 (1978) See also a series of short notes "Studies on Holonimic Quantum Fields" I-XVI in Proc. Japan Acad.

Aspects of Holonomic - - Isomonodromic

Quantum Fields

deformation and Ising m o d e l - -

By Michio Sate, Tetsuji Miwa, Miehio Jimbo Research lnstitute for Mathematical Kyote University,

Sciences

Kyoto 606, JAPAN

- Text written by M. Jimbo and T. Miwa The present article is based on a series of lectures delivered at the international colloquium at Les Houches,

September 5-13, 1979.

We intend to give here an expository

survey on the works done in the past three years 1977-1979,

in collaboration

Professor M. Sate and partly with Dr. K. Ueno and Dr. Y. MSri, concerning of linear differential

with

deformation

equations and quantum field theory.

We did not try to put everything

in this paper.

Rather, we restricted

ourselves

to the following four main topics, divided into four sections one for each: §i. §2.

Field theoretical

approach to the Riemann-Hilbert

General theory of isomonodromic

deformation

problem on monodromy

of linear ordinary differential

equations §3.

Scaling limit of the two dimensional

§4.

The one dimensional

XY

model and its scaling limit.

Each section is written rather independently, without much referring

Ising model

to other sections.

so the reader can go through with it

Since the second topic is mathematically

a bit involved, we suggest those who are mathematically

less oriented to skip it for

the first reading. Among the topics not included here are the density matrix of impenetrable gas and the variational

formulas

in higher dimensions.

to the following review paper along with the bibliography M. Jimbo, T. Miwa, M. Sate and Y. MSri, Rolonomic ipated link between deformation

theory of differential

preprint RIMS 305 (1979), to appear in the Proceedings on mathematical

physics

(M0~), Lausanne

We would like to acknowledge

quoted therein:

quantum fields

--The

unantic-

equations and quantum fields - of international

conference

1979.

our indebtedness

to Professor D. Iagolnitzer

inviting us to the Les Houches Colloquium and providing us with an opportunity give lectures.

bose

For these we refer the reader

for to

480

§i.

i.i

Field theoretical approach to the Riemann-Hilbert

The classicalRiemann's

problem

problem

We begin our story of monodromy preserving deformation with a prototype in this branch:

the classical Riemann's problem.

A particular simplicity consists in that

it is concerned with linear ordinary differential equations admitting only regular singularities• Let us write them in the form of a first order system with mxm matrix coefficients dY d~ = ~ i x - aA~ ~ Y,

(i.i.i) having

x = al,..-,an ,=

A : mxm constant matrices,

as its regular singularities.

Let

fundamental matrix solution of (i.i.i) normalized as is an arbitrarily chosen base point. its endpoint

Y(x 0) =i, where

For any closed path

x0, the analytic continuation

yY(x)

Y = Y(x)

of

y

Y(x)

in

denote the x0~C-{al,.-.,an}

C-{al,...,a n}

with

induces a linear trans-

formation

(1.1.2)

Here

Y(x)

,~ ; yY(x) = Y(x)My

My = yY(x 0)

class

[y].

.

is a constant matrix depending on

It is clear that

MyiY2 = MyiMy2

y

only through its homotopy

Thus to each system (i.i.i) (+ its

solution matrix) there is associated a representation of the fundamental group, the monodromy representation: P (1.1.3)

~l(C-{al,...,an};X

0)

[y]

-

, GL(m,C)

r

> M Y

Since

~l(£-{al,...,an};X0)

is a free group with

n

generators

YI""'Yn

(see

Fig. i. i) a2

"

. .

Y l ~ / ~ n

.

~''~ an

x0 Fig. i. i the monodromy group My~ (~=-i,... ,n).

@(~l(C-{al,...,an};X0))

Viewed as functions of

are known to be entire (holomorphic). of these entire functions. (1.1.4)

Given

al,-..,a n

is generated by the

AI,... ,An

n

matrices

these monodromy matrices

M

=

M~

The Riemann's problem amounts to the inversion

Namely it states as follows: and

Ml,...,MnC GL(m,C), find a system of differential

431

equations

(i.i.i) whose monodromy representation

p([yV]) : M

p

is the prescribed one:

(v=l,...,n).

Actually in order to specify the solution uniquely it is necessary to refine the data {al,...,an,Ml,...,Mn}

slightly.

With an additional condition n

(1.1.5)

the eigenvalues of

the normalized solution

(1.1.6)

Y(x)

(0)

Y(x 0) = i,

(i)

Y(x)

A1,...,An, A

= - [ A~ ~=i

do not differ by integers,

is shown to have the following properties:

is (multivalued)

analytic and invertible for

x # al,...,an,

a~(~), (ii) at

x = a~

it has the form (*)

Y(x) = Y(V) (x) (x-av) LV , where

Y(~)(x)

is holomorphic and invertible at

x = av (V=l,---,n,~). 2~iL~

Here the exponent matrices

LV

are related to the monodromy through

so that they constitute a refined notion of the latter. the branch points

a

as well.

In this case condition

We admit

M~ = e

x0

(1.1.6)-(0)

to be one of

should be replaced

by

(1.1.6)-(0)'

Y(V)(x O) = 1.

Conversely,

as has already been noted by Riemann,

such a matrix

Y(x)

(if ever

exists) should necessarily be a solution of a differential equation of theform (I.i.i) dY y-l. The argument is standard. Consider dxx It is single-valued (the monodromy cancels out), holomorphic except for

x = al,...,an ,~

Y(~l(xlx ~] Y(~)(x) -I + holomorphic at must be a rational function

~ A~ ~ix-a~

It is also unique, for the ratio

x = a~ (resp. holomorphic at with

Yl(X)Y2(x)-i

(*)

For

z ; -, g

= -~ Av Y

x = ~

v~ix-a~

replace

of two such matrices

Yl(X), Y2(x)

Thus the following equivalence holds: Multivalued matrix

Differential equations

with singularity data AI,-..,A n.

~), and hence

A~ = Res d~Y y-I = ~(~)(a )Lv~(~)(av)-I. x=a~ ax

is seen to be constant = Yl(Xo)Y2(x0 )-I = i.

dY dx

by (i), has a local expression

al,''',a n,

x-av

by

i

Y = Y(x) with monodromy data

L1,''',L

n.

al,''',a n,

432

From now on we pose the Riemann's problem in the following strict sense:

(1.1.7)

Given

al,...,a n

L1,...,Ln,

and

find

Y(x)

satisfying

(1.1.6).

Y(x) = Y(x0,x) = Y [x0,x; La~'" ," " i'an ,Ln ] . We remark that a different choice of normalization point x 0 gives rise to a similarity transIts unique solution is denoted by

formation for

(1.1.8)

A

= Av(x0):

Y(x6,x) = Y(x0,x6)-iy(x0,x) A (x6)

1.2

= Y(x0,x6)-iA

(x0)Y(x0,x~)

.

Schlesinser equations Riemann proposed to study

x0,x,al, ....,an: variant?

Y|xr ,x;al'''''anj~ 0 LI,.. ,Ln

how does it depend on

as a function of

x0,al,..-,a n

n+2

when the monodromy is kept in-

This has been worked out after his death by L. Schlesinger.

the following.

In terms of

Y(x)

variables

His answer is

the necessary and sufficient condition (*) for the

monodromy to be preserved is that

Y(x) should satisfy a system of linear partial

differential equations ~= ~x0°~ =

(i.2.i)

n

A

A

I x0-a~ ~ Y' -~=i

In terms of the coefficients

A

~Y = (- x-a~ ~ + x0- ~a~)Y ~av

A V = A(x0;al,...,an)

(~=l,...,n).

the condition is expressed as

the non-linear partial differential equations ~A ° =

(1.2.2)

[A,A

1

x0-a~

(#v) ~A = [A ,A ]( ~a

=

~

-i a-a

-

The Schlesinger's equations Equations

~Y .y-l. ~a

If

x0 = =

(1.2.1) '

(*)

! )

(~=v).

x0-a~

(1.2.2) are the integrability condition for (1.2.1) and

(1.2.1) are derived by a similar argument as in §i

x0 z a,

(1.2.1) and (1.2.2) are modified.

simplifies them into ~y ~a~)

(~#v)

i x0-a ~)

[A ,A ] (a la -

~(#v)

(i.i.i).

+

A x-aj

We assume (1.1.5).

Y

applied to

For instance the choice

433

~A [AH,A ] ~--= a-~

(1.2.2) '

_

~

(~#~)

[A,A]

(~=~).

=

H(#~) aH-a~

1.3

Field theoretical formulation The Riemann's problem is put into various formulations, among which the l]ilbert's

approach by boundary value problem is of relevance here. F

passing through the branch points

al,...,an ,~

M(~)=I M(~) = M I M 2 " ' ~

a

l

an ""~

^

~

D+

Choose any simple contour

(Fig. 1.2).

)

=~In

~ a n -

~

I

D

O=Mn-zMn an-2

Fig. 1.2 Let

Y+(x), Y_(x)

contour

F

be the branches of

Y(x)

inside

(D+)

or outside

(D_) this

respectively, which are analytic continuations of each other through a

portion of

F, say between

a and ~. Then the monodromy property of y(x) n equivalent to the following relation between the boundary values of Yi(x):

(1.3.1)

Y+(~+) = Y_(~-)M(~), + Y±(~-)

where

in Fig. 1.2.

~ c F-{al,...,an,~},

= lim y±(x) and M(~) is a step function whose values are tabulated D±ox÷$ Thus the Riemann's problem amounts to finding two matrices Y±(x),

holomorphic and invertible in and that are related on

F

D±, growing at most polynomially at

through (1.3.1).

it works, assuming that the branch points F = ~i. ~*(J)(x), ~(J)(x)

al< ...
(j=l,...,m, x e ~ )

[~*(J)(x),~(J')(x')]+

= 6jj,6(X-X')

[~*(J)(x),~*(J')(x')]+ = 0,

(*) [x,Y]+= xY+YX.

Let us show how

all lie on the real line

denote free fermion operators in one

dimension, satisfying the following anti-commutation relations:

(1.3.2)

x = al,...,an,~ ,

This boundary value problem of Hilbert

admits a nice description in the language of quantum field theory.

Let

is

(*)

[~(J)(x),~(J')(x')]+ = 0.

434

What is meant by the term "field operator" will be made clearer in later paragraphs. At the moment it is sufficient to know that they are operators parametrized by in a suitable sense, acting on some Hilbert space space

~.

~*) has a distinguished vector, which we denote by

lowing the bra-ket notation of Dirac. vector

~Ivac>

denoted by

with

For an operator

Ivac> (reap.
fol-

~, the inner product of the


< ~ > = .

x~

The latter (reap. its dual

~, and is

In our case of free fermion operators we assume

the following: '

(1.3.3)

<~*(J)(x)'~ (j)(x')> = <~(J)(x)~ *(j <~*(J) (x)~ *(J") (x')

The main point is the

+iO

~, the expectation values as functions of x plane

=

~

i

(x')>

i

6jj, 2~ x-x~+i0

O, < ~ ( J ) (x)~ (j ') (x')> = 0.

prescription in (1.3.3).

This guarantees that, for any

<~(i)(x)>,

(reap. <~(i)(x)~>, <~*(i)(x)~>)

<~*(i)(x)>

are analytically prolongable to the lower (reap. upper) half complex

Im x < 0 (reap. Im x > 0). Now let

')

We shall come to this later.

be a f i e i d operator satisfying the following commutation relation

with the free fields: m

~ .~(k) (x)

(1.3.4)

=

.~,(k)(x) =

~ ~(J) (x) " ~'M(X)jk j=l ~ ~*(J)(x).~ .(tM(x)-l)jk j=l

where

(1.3.5)

M(x)

=

Ml(X) ".. Mn(X) 2~iLv0(_x+a~ )

M (x) = e Let

Y±(x)

m×m

x0 e~

"

2~i(x-x0)<~*(J) (x0)~

is a normalization parameter.

The reason is simple. is prolonged analytically to Im x > O.

(k)

(x) ~>I< F>,

2~i(x_x0) <~,(j) (x0) ~ ~(k) (x) >/< ~>.

Hilbert problem satisfying

tion to

(x > av) (x
matrices whose (j,k)-th elements are given by

Y+(X)jk= Y_(X)jk=

(1.3.6)

Here

be

[i le2~iLv

=

Then these are solutions to the Riemann-

Y±(x0) = i. As has been mentioned above, each matrix element of Im x < 0.

Similarly

Y+(x)

has an analytic continua-

This latter statement is seen by rewriting (1.3.6) as

= -2~i(X-Xo)<~(k)(x)~*(J)(x0)~>/<~>

Y_(x)

Y+(X)jk

(the remainder term,being proportional

to

435

~(x-x0)

by (1.3.2),

is cancelled by the factor

commutation relations

tion condition is seen from term at

x = x0).

Tt should be mentioned

bility

that the above heuristic argument is not quite sufficient. Y±(x)

constructed

through

(1.3.6)

(ii) including

relation of the type (1.3.4) corresponding

respectively,

ing matrix

then the product

M(x)M'(x).

~'

G (the Clifford group), and the correspondence

~i

In particular

it is sufficient

x = a, ~

1.4

I M(x) ~

~ I''" ~n"

admits an elementary

see this enables, us to construct explicitly

M(x)

satisfy and

form a multiplicative

to construct

general case of (1.3.5) is obtained as a product with two branch points

~ °

~, ~'

to matrices

~'s

G.

If

has the same property with the correspond-

In fact the totality of such

tation of

and its inverti-

For this we need a closer look at the structure of the operator

the commutation

group

satisfy stronger

x = ~

Before closing this paragraph we give one useful observation.

M'(x)

Finally the The normaliza-

i < ~ > ÷ holomorphic x0-~+i0 how to find such an operator ~ ?

(1.1.6), namely the local behavior

(i).

in front).

Y (x) = Y+(x)M(x).

<~*(J) (x0)~(k)(x) ~ > = ~jk 2~i

Our problem is then:

One should check that the matrix requirement

2~i(x-x0)

(1.3.4) ensure the relation

for

~

n = l, since the

But the Riemann's problem

solution

the operator

gives a represen-

(x-a) L.

As we shall

for the Riemann's problem.

Theory of Clifford group The operator

~

introduced

in 1.3 has the following remarkable property.

If

we write as

(1.4.1)

~(J)(x)

we see from (1.3.4) free fields.

= T(~(J)(x))~,

that

~*(J)(x)

T(~(J)(x)),

T(~*(J)(x))

To get insight into the algebraic

= T(~*(J)(x))~

are again linear combinations

structure of the problem,

this situation in the case of finite degrees of f r e e d o m - - t h e

of

let us study

theory of Clifford

group. Let

W

be a complex vector space of even dimensions N, and let

non-degenerate

symmetric bilinear form on

an algebra generated by

(1.4.2)

W

G(W)

Let

A(W)

< , >

be a

be the Clifford algebra,

with the defining relation

ww'+w'w = <w,w'> ~ C

The Clifford group

W.

for

w, w' c W.

is by definition

the set of invertible

elements

g c A(W)

such that

(1.4.3)

,

T (w) = gwg g

-i

e W

for any

Clearly the linear transformation {TcGL(W) I = <w,w'>

for

T

g

w ~ W.

belongs

to the orthogonal

group

O(W) =

w, w' E W}, and we have the exact sequence of

436

group homomorphisms

(1.4.4)

1 ----+ GL(1)

G(W)

g

In other words factor. Let

(V*,V)

(1.4.5)

V*

tively. called

is uniquely

W = V* O V

subspaces

Thus

g

To make explicit

>

T

--~

1

from

T up to a multiplicative constant g we need the notion of normal •ordering.

g

determined

this correspondence

be a decomposition

of

W

into a direct

sum of ordered

pair of

such that

= 0

for any

V*l,V2* • V*



= 0

for any

Vl,V 2 • V.

V

generate

in

A(W)

There exists a unique

the Grassmann

isomorphism

of left

algebras A(V*)-

A(V*)

and

and right

A(V), A(V)-

respecmodule,

the normal ordering

A(W) = A(V*)AA(V)

~

% such that ment in

: 1 : = i.

I

For

and is called

Example.

For

the term of degree

to the grading

the (vacuum)

Wl,W2,..-

(1.4.7)

wl...w k =

c W

expectation

we have

0

of the corresponding

ele-

A(W) = ¢ @ W • A2(W) @ ..., is denoted value of

a.

w I = :Wl: , WlW 2 = < W l W 2 > + :WlW2:, WlW2W 3 In general

(Wick's

theorem)

> [sgnrll 2 . . . . . . k ]<w w > ..- <w w \ ~l'''~r ~l'''~s j ~i ~2 ~r-i ~r ×

:

W

-..

Vl (summed over partitions ~i < "'" <~r'

Each decomposition algebra"

,

:~:

a cA(W),

A(W), with respect

• C

A(W) = A(V*)-A(V) ÷

= <WlW2>W 3 - <WlW3>W 2 + <w2w3>w I + :WlW2W3:.

A(W)

.



and

(1.4.6)

by

---+ O(W)

i

of free fermion.

W

:

~s {l,...,k}

~i < "'" < ~s' r = k-s

W = V* @ V

allows

= {~i' • -.,~r}U{~l,..-,~s }, even).

us to realize

For this purpose we introduce

A(W)

as an

two vector

"operator

spaces,

on which

in ~

(resp. ~*),

acts from the left or from the right:

(1.4.8)

~

= A(W) mod A(W)V

3"

= A(W) mod V*A(W).

If we denote by

Ivac> (resp.


the residue

class of

i EA(W)

437

we have

(i. 4.9)

= A(W) Ivac> = A(V*)Ivac> ~*

:
A(W) ~ Endc(~)

~ Endc(~*).

These vector spaces are dual to each other through the (well defined) map ~*× ~ --+ ¢, ( ) e---+ . M e A(W)

with the properties

(resp. ~*)

Furthermore, there exists a unique "number operator" [N,v] = -v (vcV),

[~,v*] = v* (v*£V*), = O.

is decomposed into a direct sum of eigenspaces

sponding to the eigenvalues

k=0~l,...,N/2.

these "k-particle sectors"

~ k (resp. ~ )

~ k (resp.

~)

corre-

By choosing a basis

v@eV* (resp. vjcV) J are spanned by the "state vectors" of

c~]l"''Jk v....Vjk) , I Cjl'''Jk v@JZ ...v@Jk Ivac> (resp.
the form and

called the creation (resp. annihilation) operators of free fermion. (**)

What we get

here is a miniature of quantum field theory. Now we proceed to a formula which recovers basis

Wl,...,w N

(1.4.10)

of

J = (<wj,wk>)j,k=l,..., N = K + t K , E+ = j-IK,

Formula i.

If

geG(W)

from

T cO(W). g

Choose any

W, and set

K = (<WjWk>)j,k=l,...,N (***)

E_ = j-i tK "

E+ + E_Tg

is invertible~ we have N

(1.4.11)

g = :eP/2:,

p =

RjkWjW k e A2 (W) j ,k=l

where

(1.4.12)

2 = nr(g) det(E++E_Tg)

(1.4.13)

R = (Rjk) j,k=l,...,N = (Tg-I)(E++E_Tg)-IJ -I.

Here we have set

nr(g) = gg* = g*g ~ ¢

for

g ~ G(W), with

antiautomorphism of A(W) such that w* = -w (w £ W) . Note that 1 +-g-T(o~)2+''" is always a finite series in A(W).

* denoting the unique e p/2"

(*)

~'-1 = 0,

(**)

The term "fermion" refers to the anticommutation relation (1.4.2).

~'N

~+l

1 +~-- ~ +

= 0.

(***) The notation is a little bit confusing. <wj ,Wk> stands for the inner product (1.4.2), and <WjWk> for the expectation value of a product wjw k.

438 More generally the closure of

(1.4.14)

G(W) . {c:w . . (I) .

G(W)

in

A(W)

is characterized as

w(%)e p/2 :I ccC, w (I) ,...=w(1)eW, pcA2(W)}. N

We briefly indicate here how (1.4.13) is derived.

Let w =

~c.w. be an arbitrary

j=l] element of of

W.

Setting

g : :eP/2:

_I

we calculate the normal product representation

wg

and gw by using the rule w:w(1)..-w(%): = :ww(1)...w(%): + ~ (-)k-l<ww(k)> k k=l ×:w (I)... ^'''w (~)'.. The result is of the form

(1.4.15)

w~ = :w'eP/2:,

w' :

gw

w"

N [ c[w. j=l ] ] N

:w'e p/2

c'Jw j=l ] ]

where the coefficients ÷c' = t(c i' .. ,c½) ' 7- = t (C'~,'"',c'~) are determined from + t R and c = (Cl,.-.,c N) as .+

(1.4.16)

c' = (l-RtK)c

~"'= To attain

to setting

(I+ILK) c

g = const.g ÷

it is sufficient to choose

~" = T c ' ,

i.e.

g

Products

of e l e m e n t s

I+RK = (I-RtK)Tg.

of t h e form ( 1 . 4 . 1 1 )

venience we employ the vector notation

Formula 2.

Rl (1.4.17)

R =

(1.4.18)

T

Solving this for

g

= T~. This amount: g R we get (1.4.13)

as follows.

For con-

w = (Wl,,..,WN). (v=l, ..- ,n), and set

_t K "

, Rn

Assuming that

so that

are calculated

gv = :e PV/2 :, pv = w+ R t÷ w

Let

R

I-RA = t(I-AR)

A =

0 _t

itK

is invertible, we have

gl'''gn = :exp 2 (WRl2...n t~):

with

(1.4.19)

= " " "det (I-KA) 1/2 n

R12...n

=

~

[(1-RA)-IR]>v

p,~=l where

[

]D~

denotes the (~,V)-th block according to the partition in (1.4.17).

439

For physicists it is perhaps clear that formulas (1.4.11) ~ (1.4.19) are no more than a Wick's theorem applied to the Clifford group.

1.5

Construction of operators We now go back to the continuum and construct an operator

~

in 1.3.

To go from discrete to continuum we first proceed most naively; replace sums by integrals and "apply" the theory in 1.4.

As a vector space W of free fermions we t choose the space of 2m tuple of functions w = (w~(x),.-.,w~(x), Wl(X),--.,Wm(X)) equipped with the inner product

(1.5.1)

<w,w'>

= ~ [ +~dx(w~(xlw~ (x) +wj (x)w~' (x)). j=l J

By the identification lent to (1.3.2). (1.5.2)

where or

3

-~

w =

We set

~,(3)(p) =

3

[+~°dx(w@(x)**(j) (x) +wj (x)@ (j) (x)) J=iJ'~

dxelXp~ *(j)(x),

~*(J)(-p), ~(J)(p)

(1.5.1) is equiva-

3 ~(J)(p) =

dxe-iXp~ (j)(x)

are creation or annihilation operators according as

p <0

p >0, respectively, satisfying

(1.5.3)

[~*(J) (p) ,~*(J ') (p')]+ = 0,

[~(J)(p),~(J )(p')]+ = 0

[~*(J) (P) ,9 (j')(p')]+ = ~jj,6(P"P')

(1.5.4)

In the basis

$*(J) (-p) Ivac> = O,

~(J)(p) [vac> = 0

(p > 0)



(p < 0).

~*(J)(x), ~(J)(x)

the matrices (1.4.10) are now integral operators

with the matrix kernels

(1.5.5)

E±(x,x')

i ±i 2~ x-x'±i0"12m

To put it differently we have chosen a decomposition V)

consists of

2m

tuples, whose components

W = V* • V

w~(x), wi(x)

such that

V* (resp.

are boundary values of

holomorphic functions in the upper (resp. lower) half plane

Im x >0

We are to find an element ~

corresponding to the

of the "Clifford group"

G(W)

"or thogonal transformation"

(1.5.6)

: (~*(x),~(x)),

. . . .~ (0*(x),~(x)) itMx)-l 01 M(x)

(resp. Im x < 0 )

440

~*(x) = (~*(1)(x),...,~*(m)(x)) ~(x) = (~(1)(x),...,~(m)(x)).

Lemma.

+i E_Y$ E+ : 0,

(1.5.7).

Then

R

Suppose there exist invertible matrices

E+YZ+i E_

: 0,

Y±

satisfying

Y_ = Y+T.

in (1.4.13) is given by

(1.5.8)

R = (YTI-y-I)(E.Y_+E_Y+)J - I . ~ _ _

In our context we choose ItY0(x)-i

0

]

Y±

to be multiplication by matrix functions

C°nditi°ns (1"5"7) then state that

Yi(x)' Y± (x)-I

sh°uld be

Yi(x) . boundary

values

of

holomorphic

functions

(x EI~).

We have thus shown a converse to a statement in 1.3; once the Riemann-Hilbert

problem is solved, an operator

(1.5.9)

~

~ = < ~>'exp(

on

Im x <> O,

and

that

Y_(x)

= Y+(x)M(x)

satisfying (1.3.4) is found explicitly as

dxdx' ~ ( x ) R ( x , x ' ) t+~,(x,)) : ~-oo~-oo

R(x,x') =

(y+(x)-l_y ( x ) - l ) ~ i Y (x')+ -i Y+(x')) (x-x'+i0 x-x'-i0 "

As remarked in 1.3, the case of only two branch points mentary solution

Y±(x) = (x-ali0) L.

for

(with the normalization

~=

~(a;L)

(1.5.10)

~ (a;L) = :exp(

a, oo admits of an ele-

Correspondingly we have. the following formula <~>

= i)

dxdx' +~(x)R(x-a,x'-a;L) t+ ~*(x')): ~_oo~-oo

R(x,x' ;L) = -2isin~L.x'Lx 'L ~

( ix_x,+±0 e-'~iL + x-xT-i0-i e~iL) . (*)

By applying the product formula (1.4.18) we may derive an infinite series expression for Y in the general case. Using (1.4.15)-(1.4.16) , <w (I) :w,e p/2 :> = <w(1)w,> and the Neumann series expansion

(I-RA)-IR = R+RAR+..n

(1.5.11)

Y(x0,x) = l + 2 ~ i ( x - x 0)

~ Z ,(xn,x), ~,~=i ~v u

where

(*)

xL = 0

(x > 0), = Ixl L

(x < 0).

we find

441

0

0

z (x0,x)= ~f J_ooJ_oo f d~ldX2 2~1

(1.5.12) +

.

.

.

n

1010 ~

.

i

x0-xl-aU R(xI'x2

;L~)~{ x2+ai -x

1 "dx2£+2 2-~x0-~l-a UR(xl'x 2 ; L U ) A l(x2,x 3)...

1 i R(x2%-l'X2£;Lv£ I)Av£ l~(X2%'x2%+l)R(x2%+l'X2£+2;L~ ) 2~ x2%+25a -x

We have set Auv(x,x') = ~(l-6uv)x+a _x,_ ~ v_ie~v 0 , EU~ = sgn(U-V). At this stage our formal argument is justified with mathematical rigor.

Making

use of the lemma below one can show that (1.5.12) is convergent and analytic for complex

(x0,x) ~ (¢-Fu)x(c-F)(*),

provided that

L

s

are close to

0

(Fig. 1.3).

aUJ

Fig. 1.3 Lem~a.

If all the eigenvalues of

L

lie in the strip

]ReAI < 1/2, the inte-

gral operator f(x)' is bounded in

i ! [0 dx' ixle x_x,+igl x I-Lf(x ') > J_2~

(g > 0)

L2(-~,0;dx) m.

Local behavior of

Y

at

x = a

is verified by rewriting (1.5.11) as

-L Y(x0,x) = (x0-a U)

(1514)

~ (Xo,X) = ~ -2~i(x xA)( dx_f dx2 2~1 x0~xli(xl_a)L(~ x ~x (l_~ .) U~

+ Here

x0

~(x0,x)(x-a

L v) ~

(1.5.13)

and

x

o JC

~

~

±Jc~

are supposed to lie inside the contours

of Fig. 1.3.

Local behavior at

the estimate

IZ(x0,x) I = O(

x = ~ 1 )

l- 2

U

-L i i Z~o(xI'X2)) (x2-av) 2~ x2-x CU

and

CV, respectively,

is checked through a little argument using

(Ixl ÷ ~), which follows from (1.5.12).

Cfxl In this way, for small

LV, quantum field theory provides a new method of solving

the Riemann's problem.

(*)

To be precise we argue first for the case Im al> ...>Ima n. 2 is attained as a limiting case (L -convergence).

The original case

442

1.6

T function Let us turn to the vacuum expectation value

~ = < ~(al;Ll)... ~(an;Ln)>

itself.

It is shown that its logarithmic derivative is expressible in terms of a solution to the Schlesinger's equations (1.2.2) as follows: da -da d logT = -traceA A 2 ~ ~ "9 a - a v

(1.6.1)

The right hand side represents a closed 1-form for any solution of (1.2.2).

This

fact will be generalized in §2 to the case admitting irregular singularities of arbitrary rank. We give below several arguments to derive (1.6.1). (i)

The formula (1.4.19) enables us to express

log •

as a Neumann series

co

1

-i log det(l-RA) = 2

1 ~ trace ~ (RA) . Unfortunately the kernel (RA)%(x,x ') is .%=1 singular on the diagonal x=x', and its trace is meaningless by itself. Nevertheless its derivative does make sense as a convergent series:

(1.6.2)

d logT = - 2i

~ ,co''" dXl'''dx2~ .%=2 ~i' " "" ,~=i --oo

x trace(LVlsin2~L ida llX IIL~I-IA l~2(x l,x2)R(x 2,x 3;Lv2) ...

-L~)I-1 • • • R (x 2 %-2' x2 Z-I ;LV%) AV%~ 1 (x2.%-1'x2 %) Ix2~ 1

)•

Comparing this with

a~- Yvv(av 'x) Ix=a

(1.6.3)

(0 (0

1)

,J_oo1_ool dx_dx_sinl 2

2

= - 2--i I~,

-L-I

]Lv-I HEv x_i

p(#V)~ C(~>) zpC(xl+a~)'x2+a~)Ix21

we find that n

(1.6.4)

dlog T =

(V) =

trace (L~Y~) da~ )' "9=1

Y1

~ A

~ Yv~(av'x) Ix=a

On the other hand, L (1.6.5)

Yv(x) =

lim (XO-aV) x0~a~)

VY(Xo,X) L

= ~w(%,x) (x-%) is the solution of Riemann's problem with the normalization (1.1.6)-(0)'•

If we set

443 dY ~x

(1.6.6)

n

A (v)

Y~

x-a

~=I we have (1.6.7)

A (~) = L ,

Y{~) + [Y~),L

] =

A (~) ~ ~(#~) a - a

so that (1.6.8) Since

trace(L Y A's

~V)da )

=

with different normalizations are similar to each other (see (1.1.8)),

trace "(V)A (~) = trace A A (2)

( ~ da ~ trace(A ~)A V)) a- ---a ~(~) ~

is independent of

x0, and we have (1.6.1).

The above procedure is carried out at the level of field operators.

Dif-

ferentiation of (1.5.10) yields

(1.6.9)

d (~)eP(a;L)/2

d~ ~(a;L) = :daa

:

+ t÷* (a;L)/2 = 2~i:~_L(a)L ~tL(a)eP :. Here ~(a;L) = :e p(a;L)/2", . and the row vectors ~* (a) = (~*(1)(a),-..,~*(m)(a)) are given by tL LL tL ÷~_e(a) = ~-i

(1.6.10)

[a

÷@-L (a) = (~(1)(a),...,~(L)(a)), -L

÷

dx~ (x)sin~L. Ix-al -L-I

~--oo

÷

i

a ÷* t I dx~ (x)sin~ L" ix_al tL-I

~ e (a) = F ~-_~

Moreover we have the formal operator expansion (1.6.11)

~(x) ~(a;L) = (~_L(a;L) + ~_L_l(a;L)-(x-a) ~_L_k(a;L) = :~_L_k(a)eP(a;L)/2-

+ ..- )(x-a) L

(k=0,1,2,---).

This implies (1,6.12)

Y^

(a ,x) = 1 + YI(V)(x_a) + ...

)~(~)(a )eP(a~;L~ )/2 :''" ~(an;L n) (Y~))jk = 2~iT-l< ~(al;Ll)'''" -~*(J)(a tL - ~ Hence (1.6.4) follows from (1.6.9) and (1.6.12). (3)

It is also possible to start from the formula (1.4.12) in 1.4.

Since

444

nr(gg') = nr(g)nr(g')

(1.6.13)

we have

2 2... 2

det (E++E_T) T = T g~ ,

n

T = TI-..Tn .

det (E++E_T~) ~=i The effect of a variation

6T

is calculated as

(1.6.14)

n ! trace((E++E_T)-iE_6T_ [ (E++E T )-IE 6T ) 2 " 9el

~ log '''

Choosing decompositions (1.5.7) of YII(E+Y_+E_Y+)E_ = T-Iy+IE_y+ Y ±

T = y~iy_, T

and likewise for

are multiplication operators by Assuming

(1.6.15)

dM(x).M(x) -I = 2wi [ Lg6(x-a )da~ 9=i

we get

(E++E_T)-IE_ =

In our case, T, T , Y±

M(x), M (x), Y(x±i0)

tively. (*)

al<'''
= Y~Y~_ T .

and

and

(x-a ±i0)LV, respec

we find from (1.3.5) that n

Hence the right hand side of (1.6.14) is interpreted as

f (1.6.16)

n

dx 2~iv=l [ 6(x-a)da

trace((Y(x) -I

- (x-a) =

[da ~=i

trace(Y~V)L + [ ~ ) , L ~ ] L

x-xX-i0-iY(x') -L~I -i (x,_a) L 2~ x-x'-i0 ~)L~)Ix=x'

+O(x-av))Ix ~ a

'

which is (1.6.4).

References Field theoretical approach to the Riemann-Hilbert problem was initiated by [i]

M. Sato, T. Miwa and M. Jimbo, Holonomic quantum fields. II, Publ. RIMS, Kyoto Univ. i_~5(1979), 201-278.

Classical works in this branch including Schlesinger equations are cited in the references of [i]. The theory of Clifford group is expounded in [2]

M. Sato, T. Miwa and M. Jimbo, Holonomic quantum fields. I, ibid. 14 (1978), 223-267.

Main results of [2] are summarized in the Appendix i of [3]

M. Jimbo, T. Miwa, Y. MSri and M. Sato, Density matrix of impenetrable bose gas and the fifth Painlev~ transcendent, preprint RIMS-303, Kyoto Univ. (1979).

(*)

More precisely those of double size

ItM(x) -I (

~, etc. ~

I

[

M(x)

J

445

§2.

Monodromy preserving deformation and

T

function

2.1 Singularity data Let us consider the following system of linear ordinary differential equations on ~i, for an

mxm

matrix

Y(x)

(2.1.1)

~xY(X) = A(x)Y(x).

Here

is a rational

n+l

A(x)

distinct points

(2.1.2)

mxm

matrix given by the following partial fractions at

al,...,a n

and

aoo--~.

A(x) = A~(x) + Al(X) + ... + An(X) ,

too - ~ A

I

j=l

Av(x)

(2.1.3)

=

.xj-I

(v=~)

-J

rvT

i

(v#®).

jLoAV'= -J (x-a)) j+l

If

r =0, a V

is called a regular singularity for the system (2.1.1).

If

rv>0 ,

it is called an irregular singularity of rank We assume that G(~) "

A

(v=l,''',n, ~)

r . v is diagonalizable by an invertible matrix

V,-r

(2.1.4)

A

(2.1.5)

T(V) = (6 ~t(r~) ~ :diagonal. -r ,B)~, B=I, "" • ,m

We normalize

~, - r

= G(V)T(V)G (v)-I -r

(2.1.1) so that

the gauge transformation from diagonalizations

A

o~-r

is diagonal and choose

a~ to

of highest poles at

o%

G(°°)=I.

We call

G (~)

It represents the difference between two

a~ and

oo.

Along with (2.1.1) we consider the following transformed system

(2.1.6)

dy(~)(x) dx

= A (~)(x)Y (~))(x)

(2.1.7)

A(x) = G(~)A(~)(x)G (~)-I.

We denote by

A! v) (~=l,...,n,°°; J

expansion of

A (~))(x)

at

x=a .

j=-r),-r +i,...)

the coefficients of the Laurent

446

j=-r (2.1.8)

A(~)x -j-I 3

(~=~)

.(~)(x_a )j_ 1 A.

(~).

A (~) (x) =

J

3

The systems (2.1.1) and (2.1.6) are specified by the following table, which we call the singularity data.

(2.1.9)

~ ;

A~<

=

Tj~, A~)+I,

al; A(1) = (I), A(1) -r I T-r I -rl+l' an; A (n) = T(~ ), A (n) , -r -r +i n n n

°..

• (~) ~ ~k0

G (°°) =

• ..

, A ~ I),

G (I)

• ""

(n) G(n) , A0 ,

i

•

The above construction implies the following constraint. (2.1.10)

A~ ~) + G(1)A~I)G (I)-I + "-"

(n). ( n ) ~ ( n ) - i

+ ~

~0

u

= 0.

Further we assume the following.

(2.1.11)

t (~) -r,~

(~)

# t (~) -r,~

(~#~)

if

r

if

r ~) =

> 0,

L(~)

t0,~ - ~0,~ ~

Z - {0}

0.

We denote the space of singularity data (2.1.9) with constraints (2.1.10) and (2.1.11) by ~ .

2.2 Monodromy data Under the assumption (2.1.11) we can find uniquely a formal solution of (2.1.6) of the following form. (2.2.1)

Y(~)(X) = ~(V)(x)e T(~)(x), oo y(OO)x_j, j~0 j

(2.2.2)

(oo) = i Y0

(~=~)

~.D)(x_aM)j (V) = i j=0 3 ' Y0

(~#oo)

y (~) (x) =

•

447

(2.2.3)

Here

roo j~iT_(j)(x_~.) + T~)log(I)

(~=~)

r (x-a) -j j~IT_j~ (~) (-J)

(~#oo).

T(~)(x) =

T(j!

(v=l,...,n,O~; j=0,...,rv)

+ T~)log(x_a )

are diagonal matrices.

exponent matrix of formal monodromy at

x=a).

T~ ~)) is called the

The constraint (2.1. i0) implies

• . 0(~) +T O(i) + ... + Tg n)) = 0, crace(T

(2.2.4)

which is called the Fuch~ relation.

_(~)

We note that

T. J

.

and

(~)

Y. J

are rational

functions of the singularity data

(2.1.9).

In order to describe the global properties of solutions to (2.1.1), we prepare some notations on topology. (2.2.5)

~

and denote by

=

~i

~

map, respectively.

and

'''

, an,~}

~: ~ ~ ~

the universal covering of ~

We choose a small neighborhood V of a -l(vv_{a }) v ~v) in such that ~

e(~) ~ ( ~ )+i ~o I ' " .. ' "-k

sectors

~(~))=~(~il )~ small

_ {al,

We set

and

~(%__~I~V))=V -{a }.

6 > 0, a set of sectors

I {x~V

and the covering in ~i and a set of ~(~) #~, N "'%+1

As a special choice, we choose, for

~(~) .~g,@ (%=i,-.. ,2rv+l):

(v=~)

~(r%il)-@ < arg(1)< r~ }

(2.2.6)

(v#~).

~(~,@,~(V)~=I IxEvvl~ . ~(%-l)r-~~ < arg(x-a ) < ~-%} r

For

r =0

we set

~(~)=V -{a }.

We choose a point x , which encircles

a

~(~)~

x ~(.~ 1

j

and a closed path

anticlockwise.

We choose a path

y~

in

V , with its endpoint

¥~oo in

-i ) -i ) X o to x such that (YnooYnYnoo ... (ylooYiYloo is homotopic to If r~)=0, the infinite series (2.2.2) or equivalently

~(~) ( x ) x (2.2.7)

y(V)(x) =

_T(~) 0

T(9) 9(V)(x)(x_a ) 0

~ -i ~oo•

(~=~) (~#oo)

starting from

448

represents a local solution to (2.1.6). If

r

> 0, the infinite series (2.2.2) is, in general, divergent.

for a sufficiently small solution

Y ~).x)(

(2.2.8) Since

Yi(V)(x)

and

~i ~(V)

having the asymptotic expansion

in

(v) (x) Y~+I

satisfy the same equation (2.1.6), there exists a

such that

. (V) rx ) = Y£ . ( v ) ,kx)az ,~(v) . YZ+I"

(2.2.9)

s(V)

to (1.1.6) in

*% " (V)(x) m Y(V)(x)

constant matrix

Nevertheless,

8 > 0, there exists a unique holomorphic and invertible

(i=l,...,2r)

are called the Stokes multipliers at the irregular singularity

We also note that

,(~) , +, r~V)(x)e 2~IT ~2r +i ~x ) = Y±

(2.2.10)

Here, for a point

x E

~), we denote by

x+

the point in

'O2r +i

satisfying

~(x+)=~(x). From ( 2 . 2 . 9 ) ~i ( V ) ,~x )

and ( 2 . 2 . 1 0 )

we see t h a t

the analytic

continuation

Y ~ ) (x+)

of

is given by 2ziT~ v)

Y~V)(x+) = .Y1( V ) ( x ) e

(2.2.11)

( V ) - I " ' " ~~1( V ) - I S2r

Now, let us compare the analytic continuation of Y1(V) (x)

in

~~ (i V, ) 6"

Since

(~)

YI

(x)

_(v). (v),

and

~

(2.1.1), there exists a constant matrix

(2.2.12)

We call

YI

C (~)

Y

,

~x)

~)(x)

along

Y~oo and

satisfy the same equation

such that

,i I(cO)(x) = G(V)y}V)t ± (x) C (v) C (V)

the connection matrix from

a

to

o%

From (2.2.11) and (2.2.12) we have

(2.2.13)

"l 1(~)(x +) = "I 1(~)(x)M (v)

(2.2.14)

M (~) = C(V)-Ie2~iT~V) s(V)-i 2r V

M (v)

is called the monodromy matrix of

The homotopy equivalence

(x E ~

v)

,

x + ~_a @~ 2( v )r +i' ~ ( x ) = ~ ( x + ) )

,

~(v)-l~(v) "'" ~i y ~m)(x)

u

.

corresponding

-i to the path yv~y~yv ~

a .

449

(2.2.15)

".. (ylly1Y1 m) ~ T~1

(~nlYnTnm)

implies 2~iT~ m) e

(2.2.16)

1 2ziT0n)s(n)-I ( " o(n)-lo(n) S(~)-I ... g(m)-i (c(n)- e "'~i u ) 2r -I 2r n

• "" × (C(1)-le2giTgl)s(1)-l~ 2r I

..- S~I)-Ic (I)) = I.

Thus for every singularity data we obtained solution matrices ~=i, • ..,2r +i)

with the following monodromy data:

(2.2.17)

TT:

";

~ (co)

_(co)

"'"

' ~0

' al

'

... '

s (~), c(~)=, koo

_(n)~, 51 _(n)_ ,

, TN

...

_(n)~, c(n)

, 5~

n

n

In the present case

k v =2r .

2.3

A(x)

to

Now we consider the problem in a converse way; namely we start from an matrix

Y(x)

,n,~;

• .. , Ski

an: T(n)-r, " ""

Y(x)

(v=i,...

(i), c(1)

al; T (I) -r I '

From

~. ( v )

mXm

with the following monodromy properties and show that it solves a

system of linear ordinary differential equations of the form (2.1.1) with a rational coefficient

A(x).

Monodromy properties. i)

Y(x)

is holomorphic

(~=l,...,n, ~)

(2:3.1)

in ~

, and there exist

constant matrices

mXm

S(~) k (~=l'''''n'°°)

~ (v)+1' ( x ~ l~ ( v ) ' x+~ "~k

constant matrices such that

~ (x) = ~ ( x +) )

C (~) (~=l,-..,n,°°;

Y(x)C(~)-I,y(x)C(~)-Is (v),-. .

C(~)=I)

_(~) Y(x)C(V)-Is~M)... 5~

%

G(~)~(V)(x)e

T(~)(x)

"

and

-(v)

51

.

.

, • ,

. , .Y(x)C(~)-IsI . . . (v)-~

2 (~) ''''' >~k ~(~)+i , respectively: have the same asymptotic expansion in --i v

(2.3.2)

M (v)

such that

Y(x+) = Y(x)M(v)

ii) There exist

m×m

5k

450

for some

G (~) (v=l,...,n,~;

and (2.2.3), respectively.

G(~)=I]

and for

Y(~)(x)

and

T(~)(x)

given by (2.2.2)

Here we assume that the Fuchs' relation (2.2.4) holds.

The above monodromy properties imply the following. iii)

Y(x)

is invertible in

iv)

A(x)-~xY(X)-Y(x) - I ~

~. ~i.

is rational on

We also note that v)

Y(X)

is uniquely determined by the monodromy data (2.2.17).

We shall give a sketch of proof. First (2.3.2) implies that

det ~ -(~) = i .

(2.3.3)

We set -trace( (2.3.4)

d(x) = det Y(x)-e

(2.2.14) with

2r

holomorphic in (v=l,...,n). at

x =~.

replaced by

~.

~

k

and (2.3.3) imply that d(x) is single-valued v Moreover (2.3.2) implies d(x) is also holomorphic at x=a

Finally the Fuchs' relation (1.2.4) implies that

Thus

d(x)

is a non zero constant, hence

The differentiation with respect to (2.3.1).

T (~) (x)-T~°°)log(1) 1

~=l,...,n,oo

Hence

--~Y(x).Y(x) -I

v), let us assume

and

satisfy monodromy properties i)

Since

is single-valued in

normalization

YI(X)Y2(x)-I=I.

(2.3.5)

M (v)

~.

We note that the partial fractions (2.1.3) for

<x)~xl

~.

Since the exponential factors in

Y2(x)

and ii) with the same monodromy data (2.2.17).

imply

is invertible in

~i.

Yl(X)

monodromy data, Yl(X)Y2(x)-i G(~)=I

Y(x)

is holomorphic

does not change the monodromy property

is single-valued.

(2.3.2) cancel out, it is rational on To prove

x

d(x)

is determined by the

Then (2.3.2) and the

A(x)

is given by

(x)Y (°°)(x) -I mod x -2

A(~)(x)

Y(~)(X)d~T(m)(x)Y (v)(x)-I mod (x-a) 0.

This fact follows from the identity

(2.3.6)

2.4

A (~)(x) = d~(V)(x).~(V)(x)-i

Deformation equations for Now we consider a family

-i

Y(x) {Y(x't)}te~

monodromy properties in the sense of 2.3. t=(tl,...,tN).

+ ~(~)(X)~xT(~)(x)~(~)(x)

of

mXm

matrices

We assume that

Thus the monodromy data are holomorphic in

Y(x,t)

Y(x,t) t.

having

is holomorphic in

Moreover we impose

451

the following isomonodromy properties on partial monodromy data.

,~ (~) (2.4.1)

al 0

,~ (~)

~) = 0,

dS

dT~ I) = 0,

= 0, .-. , a~ko ° = 0

dS~ I) = 0, "'" '

dT~n) = 0,

dS (I) = 0, dC (I) = 0 kI

dS~ n) = 0, ... , dS (n) = 0, dC(n) = 0 n

Namely, we assume that formal monodromies, are independent of

t.

by a system of linear total differential

(2.4.2)

Stokes multipliers

Then what is the consequence on equations

and connection matrices

Y(x,t)?

for

The answer is given

Y(x,t),

dY(x,t) = ~(x,t)Y(x,t). N

Here

~(x't)=jYlRj(x't)dt'~3

(j=I,.-.,N). (2.4.3)

is a i form of

t

with

mXm

(2.4.2) means the following simultaneous

coefficient matrices

R,(x,t)

]

system.

~Y(x,t) ~t. = R.(x,t)Y(x,t). 3 3

We shall give an explicit form of

~(x,t).

following argument, we abbreviate

Y(x,t), ~(x,t),

Since

x

dependence is essential in the

etc.

to

Y(x), ~(x)

etc.

First (2.4.1) implies that

(2.4.4)

Hence

dM (~) = 0

(~=l,-..,n,~).

~(x)=dY(x)-Y(x) -I

is single-valued

in

~,

at

x=a

v

using (2.3.2) and (2.4.1)

we have

d~(°°)(x).~(°°)(x)-i + ~(°°)(x)d'T(=°) (x)~(°°)(x) -I (2.4.5)

(~--~)

~(x) = {

dG (M) "G(~)-I+G(~) (dY (~) (x)'Y(V) (x)-I+Y (v) (x)d'T (v) (x)Y (~) (x)-%G ~)-I (~#~), where r

m (~) x j -] (-j)

(~=~)

dl o •

j=l (2.4.6)d'T (~)(x) = r~

( x - a ) -j

I el . •

j =1

-]

(-J)

- ~(~) ± . (x_a)-J-lda j =0 -]

(M#~o).

452

Thus

~(x)

is rational in

x

and its, partial fractions

~(x)=~oo(X)+~l(X)+...+~n(X)

are determined by ^(oo ,T(OO) ^(oo) -I Y ) (x) d (x)Y (x) (2.4.7)

mod x

-i

(v=oo)

~ (x) -

G(~)Y (v) (x)a'T (v) (xiY (v) (x)-iG (v)-I

mod

(x-a) 0

(V#oo)

Namely, if we set

(~=~) j=lr ~ ] (2.4.8)

~(V) (x)d,T (v) (x)y (~) (x) -I =

x

o@

j=-r ~)-

(v#~),

i ~ ~) (x-a~) j

we have r

/

~

•

.

x

(~=~)

J j=0 -j (2.4.9)

~V (x) = | r +i

~ [ j=l

2.5 to

G(~)#(~)G(~)-l(x-a )-J -3 v

Deformation equations for

G (V)

If we subtract from

its singular part

~(x)

(~#~).

~ (x)

at

x=a , and restrict

av, we obtain a linear system of total differential equation for

(2.5.1)

dG (v) = @(~)G (~),

(2.5.2)

@(~)

Since

=

(~(x)-~ (x))Ix=a + G (~)- (~)"

~i

~(~))G (~)-I

aa~-w0

G (v)

"

~(~)-io(~) y(x)=G(~)y(V)(x)S v)-l. . "~i u , we obtain the linear system for

from (2.4.1), (2.4.2) and (2.5.1). (2.5.3)

dY(~)(x) = ~(V)(x)Y(V)(x),

(2.5.4)

~(V)(x) = G(V)-I(~ (V)(x)-@(~))G (v).

In the following we shall denote respectively.

Y(x)

and

~(x)

by

X

Y~"(°°~(x) and

~(oO)(x),

Y(~) (x)

453

2.6

Deformation parameters We have derived linear systems for

Y(V)(x)

and

G (v), assuming that the

partial monodromy data are independent of deformation parameters

t : (2.4.1).

next question is, what is the universal family of isomonodromy deformation.

The

The most

optimistic answer is that the rest of the monodromy data

(2.6.1) a I,

•..

T (~) -roo T (I) -r I

Ti) -r ' n

an,

,

T (~) -i '

. . .

...

T (I) -i '

...

n

can be chosen as independent parameters of deformation.

In fact this turns out to be

true. Since

d'T(~)(x)-

in (2.4.6) is written as a linear combination of- da

(~=l,...,n),

(v) dtj~

(j=-r ,...,-i, ~=l,...,m; ~=l,...,n, ~), the linear systems (2.5.1) and (2.5.3)

can be regarded as linear systems of partial differential equations with independent variables (2.6.1).

For the existence of solutions to these systems, the following

integrability conditions should be satisfied:

(2.6.2)

d@ (~) = @(~) A @(~)

(2.6.3)

d~ (~) = ~(~) A ~(~).

These are equations of 2-forms.

In the next section, we reduce (2.6.2) and (2.6.3)

to a completely inte~rable system of non linear total differential equations. implies that given an

m×m

matrix

Y(x)

This

satisfying monodromy data (2.2.7), for any

sufficiently small deviation in deformah~on parameters (2.6.1) there exists a matrix with the same partial monodromy data, i.e.

formal monodromies,

Stokes multipliers

and connection matrices.

2.7

Deformation equations for

A(~)(x)

In 2.3 we have seen that the monodromy properties for

Y(~)(x)

imply linear systems

(2.7.1)

~ x Y(~) (x) = A (~) (x)Y (~) (x)

(~=i,... ,n,~).

(~=l,...,n, ~)

454

Then in 2.4 we have seen that the isomonodromy properties imply further linear systems (2.7.2)

dY (v)(x) = ~(V)(x)y(~)(x)

Differentiating

(v=l,...,n,~).

(2.7.1) and (2..7.2) we obtain

(2.7.3)

d~--Y(~)(x) = dA(~)(x).Y(~)(x) + A(V)(x)~(~)(x)Y(~)(x)

(2.7.4)

~Y(~)(x)

= ~(~)(x)'Y(~)(x) $x

+ ~(V)(x)A (~)(x)Y (v)(X)

Subtracting (2.7.4) from (2.4.3) we have

,A (~) (x)])Y~) (x),

0=(dA(V)(x)-~(V)(x)-[~(~)(x)

or equivalently (2.7.5)

dA(~)(x) = ~x-~(~)(x) + [~(V)(x),A(~)(x)].

Note that

~(V)(x)

is rational in

x

and its coefficient matrices are rational

functions of singularity data (see Appendix 1 and (2.4.6), (2.4.7)).

Thus along with

(2.5.1) we have obtained a non linear system of total differential equations for singlarity data (2.1.9). Comparing formal expansions at number of systems.

x=a

of both sides of (2.7.5) we obtain infinite

Of course only finite number of them are independent.

Namely,

if we set (2.7.6)

dA (~) (x) - ~ x (~) (x) - [~(~) (x) ,A (~) (x) ]

=

I ~=(°°)x-J-i J-J

(~=oo)

~Z! V)(X-a )j+l j J

(~#o~),

then we have (2.7.7)

~(~) = 0 j

Moreover all the entries of ideal ~ (2.7.8)

if ~

j < -r

ej -(~) (~=i, .-.,n,~; j=-r +i, -r +2, -.) belong to the

of differential forms on ~

generated by

(~(~)) -j ~B

(~=l,.--,n, ; j=-r~+l ' -..,-i; ~ B )

(E~))~B

(~=l,--.,n),

(dG(~)-@(~)G(~))~

(~=l,..-,n).

'

455

Hence the unknown functions of our deformation

(2.7.9)

(Aj(~))~B

(~=l,...,n,OO;

(A~V))~$

( v = l , • • . ,n),

G (v)

(v=l,... ,n),

while the independent

+ i , "'',-i;

j=-r

variables are deformation parameters

Finally we can prove that the ideal

J

Thus by Frobenius'

J

theorem our system

equations is completely

The complete integrability arbitrary monodromy differential

data.

equations

integrable.

assures the existence

As for global properties

(2.7.11) it is conjectured

(2.7.12)

a

= a

(2.7.13)

t (~) -r~

v = t (v) -rS

of local solutions

of solutions of the non-linear

of the system itself:

for some

~ # ~,

for some

e # S

and some

~

with

function

We denote by

w

the following 1-form on ~2 :

(2.8.1) ~=l,...,n (2.8.2)

~ = -Res trace x=a

In terms of the coefficients

Y(~)(x) -1 ~Y(~)(x)d'T(~)(x).

•(v)

. in (2.2.2) J r v (0o) (~) trace L Z. dl j=l ] -J

w

is written as

_

(2.8.3)

(v=~)

w

r

for an

that their only singularities

poles except for the following fixed singularities

T

(2.6.1).

is d-closed:

= 0

of deformation

2.8

~),

dJ C ~ .

(2.7.10)

(2.7.11)

equations are

r +I

..(V)T(m) trace ~ Z.(V) dT (~) . + ~ jLj _j+laav j=l J -J j=l

(v#~),

r

-> i.

are

456

(2.8.4)

-(V) = y ~ ) zI ,

Z(~)

Z3

•

(~) 1 ,(~)2 = Y2 - 2 ~i '

2

(

)

. (~) 2. (~). (~) i. (~). (v) i. (~)3 = ~3 - 3~i ~2 - 312 ~i + 311

Z!~) = j

llj

p=li ~'1+.... ~ +%p j. ~'p(-Y~:))"'" (-Y~)p). %1,''',%p>__i

_(v)

We note that the coefficients

z. are polynomials of unknown function (2.7.9) with J rational coefficients in the independent variables (2.6.1). The fundamental property of

(2.8.5)

de ~ 0

mod ~

~

is the closedness:

.

Corresponding to the conjecture in 2.7, it is conjectured that on each solution leaf the only singularities of

w

other than the fixed singularities (2.7.12), (2.7.13)

are simple poles whose residues are positive integers. The closedness of

~

implies the existence of a function

T

of the deformation

parameters satisfying

(2.8.6)

w = d l o g T.

We call this function (with a constant multiple undetermined)

the T-function.

The

above conjecture implies that the T-function is multi-valued holomorphic except for the fixed singularities.

2.9

Schlesinger transformation and T quotient Given an

matrix

Y'(x)

expontnts

mXm

matrix

Y(x)

with monodromy data (2.2.17) we can construct another

with the same monodromy data except for integer differences in

t0~(v) (~=l,...,n,~; ~=l,...,m).

Since it was

L.

Schlesinger who considered

such a transformation systematically in the case of regular singularities, we call it a Schlesinger transformation. Choose diagonal matrices integer

(2.9.1)

entries

L (~) = (6 B%~))~,B=I,..., m ~

subject to the constraint

~°

(~) =

0.

~=l,...,n,~ ~=i

A Schlesinger transformation is called of type

(~=l,...,n,~)

with

457

oo

al

L (~)

L(1)

...

an

(2.9.2)

e(n)f

if it induces the following change of exponents: (2.9.3)

r (~) ~

~

T!u v) + L (~)

o

(~:l,.-.,n,~).

In general, a Schlesinger transformation is achieved by a multiplication by a rational matrix

R(x):

(2.9.4)

Y'(x) : R(x)Y(x).

The condition so that (2.9.4) gives the desired Schlesinger transformation is the following.

(2.9.5)

^ (~) e (~) ~ (~)' R(x)Y (x)x = (x),

(2.9.6)

R(x)G(V)y(~)(x)(x_a )-

Y(~) (x)

(2.9.8)

N

{

=

YO(~)' = i

(v=~)

(x-av)3'

I(~)'

(~#~)

0

: i

a I ''' the Schlesinger transformation of type NiL(~)L(I )... an L(n)} °f

We define the length

. L(h)

(~#~),

: j:O J

L(O~) L(1)

G(~)'y(~)'(x)

j:0 j

!

(2.9.7)

L (~)

by aI

...

e (°°) L (I)

an } L (n)

=

~ (~) .

~ ~ ~ ~=l,.-.,n, °° ~(~)>0

We shall give two examples of $chlesinger transformations of length

l.

We set

E~ 0

= (~0~B~0) Type

(2.9.9)

~, ~=i,... ,m {

~

al

-E 0+EB0

0

• • • a~ } : ..- 0

R(x) = E x + R 0 ~0

458

where

is given by

R0,~B

(2.9.10)

6=~0

~=~0

~#e0,5o

. (co)

-~2,a059

+

"(~)

X

"(~)

¥ (#~o)~I'~0~I'Y~o

~=~0

-Y

(co) i, ~OB0

-Y (co)

~0~

,~050 l/Y (~)

~=60

1 ,~0~0 -Y

~#~0,60

Type -E 0

(2.9.11)

where

(co)

. (co)

/~

I,~B 0

a I ...

a0

0

EB0

. . .

6~B

i,~0~ 0

.

..

a onl

" - -

R(x) = E~0(x-a O) + R 0

R0,~$

is given by

(2.9.12)

8=~0

(~)

8#e0

("0)

YI,~0yGyB 0

~=~0

Y(#~0)

-Y~) ,~06

(~0) G~OB 0

(Uo) (~o) -G g0 / C 050

~#e

W e denote by

q

~

aI

.-. a n

L (~) L (I) to

Y'(x)

}

6 ~

the quotient of the T function corresponding

L (n)

by the T function corresponding to

Y(K), and we call it the 7 quotient.

It is remarkable that contrary to the T function itself the T quotient can be expressed in terms of

()

G -V-

For example, we have

and

• (v)

~. J

.

459

~

(2.9.13) q

{ (2.9.14)

al "'" an I

-Eo+EBo

co

0

0

~ (~) = ~l,eoBo,

al "'" a~o "'" ~n} = 0

-E~o

In general, q

(DO)

q

~

G~O~0 ..- EBO

a I •.. an

L (~) L(1) ~ L(1) al ....L(n) an } N {L(~)

] f

is given as the determinant of a matrix of size

L (n) whose entries are polynomial in

with rational coefficients in

~(V) ~ B ' (G(V)-I)~ B ' y~V) ,~B

al'"''an"

References The general framewurk of monodromy preserving deformation with irregular singularities was given by [i]

K. Ueno, Master's thesis, Kyoto University (1979).

See also [2]

B. Klares, Sur une elasse de connexions relatives, C. R. Acad. Sc. Paris, 288 (1979), 205-208.

[3]

H. Flaschka and A. C. Newell, Monodromy and spectral preserving deformatoins I, preprint Clarkson college of tech. (1979).

The proof of complete integrability and the general definition of T function was given by [4]

M. Jimbo, T. Miwa and K. Ueno, Monodromy preserving deformation of linear ordinary differential qquations with rational coefficients I, preprint RIMS 315, Kyoto Univ. (1980). As for Schlesinger transformatoins and T-quotients, see

[5]

M. Jimbo and T. Miwa, Monodromy preserving deformation of linear ordinary differential euqations with rational coefficients II, preprint RIMS 316, Kyoto Univ. (1980).

In this chapter, we have not mentioned special examples such as Schlesinger equation and Painlev~ equations. Schlesinger equation is discussed in chapter I. As for Painlev~ equations, see [6]

K. Okamoto, Polynomial Hamiltonians associated to the Painlev~ equations, preprint Tokyo Univ. (1980).

460

§3.

Scaling limit of the 2 dimensional Ising model

3.1 Formulation of the problem The 2 dimensional Ising model with nearest neighbor interaction is a celebrated non-trivial but exactly solvable model in statistical mechanics. gular lattice of size

M x N.

To each site

Ojk ("spin") taking the values of o = {Ojk}j=l,.. .,M

+i.

Consider a rectan-

(j,k) there is attached a random variable

The probability that a specific configuration

takes place is given by

Z ~ e -BE(O)

(*), where

k=l, • • • ,N M (3.1.1)

N

M

E(O) = -EIj!I k =~ l O J k O j + i k -

E2j~I

N ~ ~.,Oo j m 0 k+l

k=l

(cyclic boundary condition)

with positive constants

(3.1.2)

El, E 2.

ZMN = ~ e -~E(O) O (summed over all

The normalization

constant

Ojk = ±i)

has been calculated for the first time by Onsager.

Here we are concerned with the n-

point spin correlation functions (3.1.3)

-i ! .... o e -BE(U)" Jlkl Jnkn = ZMN Ojlk I Jnkn

The Ising model admits of a variety of approaches, "good" problem.

as is always the case for a

One of the standard ways is to introduce the "transfer matrix"

and the "spin operator"

V

Sjk = vks.v -k, both acting on~ a linear space of dimensions 3

2M, and to rewrite (3.1.3) as

(3.1.4) Denoting by

= Trace(sjlkl.. .. S.jnmn1_ vN)/Trace(V N) Ivac>

the unique normalized eigenvector of

largest eigenvalue, we have for large

(3.1.4)'

corresponding to its

M, N

~ . Jlkl Jnkn Sjlk I Jnkn

Onsager observed in effect the following: operators

V

(kl~'''Jkn).

pj, qj

in such a way that

it is possible to associate free fermion

Sjk , V

belong to the Clifford group.

This

is the key to the solvability of the problem. In the limit of infinite lattice (*)

M, N + ~

there appears a critical temperature

B-I = kT, k: Boltzmann constant, T: temperature.

461

T c which sharply seperates the ordered

(T < T c)

and disordered

The behaviors of (3.1.2), (3.1.3) in the critical region interest.

(T > Tc)

T~T c

phases.

are of particular

In the sequel we shall concentrate upon the scaling limit

(3.1.5)

T + Tc,

j 2 + k 2~ -> co

1 2 const. ]T-Tc['(jv,k v) : (av,av)

fixed

(~= l,...,n).

It is possible to take this limit at the level of operators. factor proportional to

]T-Tc II/8, the spin operator

Sjk

Apart from a scaling tends to the following

two dimensional fields with relativistic covariance: (*) 0F(X)/2 (3.1.6)

T < Tc

~F(X) = :e

:

T kT

~F(x) = :@0(x)e0F(X)/2: c +~o +oo du

(3.1.7)

du'

~

-co -co

•

-

,

+

-i

-i(u-u') elm(x (u+u)+x (u +u

, -i

))~(u)~(u')

u+u'-i0

+~o (3.1.8)

In the above

C0(x) =

~(u)

f

~ du

e-im(x-u+x+u-I) ~(u)

(x± = (x0 ±xl)/2)

signifies the creation (u<0) --annihilation -i

free fermion carrying the energy-momentum

(u>0)

operator of

-i

(pO,pl) = ,k mU+U~ - - , mU--~U) .

It obeys the

canonical anti-commutation relations

(3.1.9)

[~(u) ,~(u')]+ = 2~ lu ]<~(u+u') .

Accordingly the n-point correlation functions (3.1.4)' (with a scale factor removed) tend respectively to The free fermion

< ~F(al)... ~F(an)>

pj ± qj

or

< ~F(al)--. ~F(an)> as

T ÷ T c~O-

are scaled to give the massive free relativistic Majorana

field +oo (3.1.10)

(,)

~±(x) = / f l

2~

0+/~lu±l e-im(x-u+x+u-l)~(u)"

Here we make the "Wick rotation" and consider the region where

= const. Ir - r c l ( - / ~ j , k )

is real.

x = (xO,xI)

462

From the explicit formula (3.1.6) it is apparent that ~F(X), ~F(x) "belong to the Clifford group". In fact they satisfy with ~±(x) the peculiar commutation relations (3.1.11)

~F(a)~_+(x) = s(xl - a l)~±(x) ~F(a) ~F(a)~±(x) = c(al - x l)~±(x)~F(a) for x- a

sapcelike.

_

_

+

-x +a

+

x -a

(-1) /

Fig. 3.1. In what follows we shall show how the scaled n-point functions < ~(al)--" ~(an)> (~=~F or ~F) are calculated by appealing to the monodromy preserving deformation theory. 3.2 Wave functions The monodromy problem comes in when we consider instead of the following objects as functions of x. (3.2.1)

<~(al)''' ~(an)>

WF~(X) = t(wF~+(x), WFv_(x)) <~±(X)~F(al)'''~F(av)'''~F(an)> WF~±(x) =

,

~ = l,.--,n.

<~F(al)''" ~F(an)> As an illustration first let calculated as

(3.2.2)

n=l.

In this case (3.2.1.) is explicitly

m -i =~-l[<~+(x)~0(a)> ] WF(X) ~ 2 [<~_ (x)~0 (a) >j

= f~ d u2~u / e 4 ~[£4vr~_ll 0 e-im((x--a-)u+(x+-a+)u-l)

463 1

-2m~-x-+a-)(x+-a+)

i

1

=2~

x 7 Y-a +)

Hereafter we allow the variables the Euclidean space

e

x, a,...

to be complex, in particular to run in

{x0(=-ix 2) e i~, xle~}.

z * = x +, aV =-aL, a~ =a~.

We often use the notation

z = -x-,

The point is that, due to the square-root in (3.2.2), (the

Euclidean continuation of)

WF(X)

is a double-valued function, changing its sign

each time when prolonged around the branch point

x = a in the Euclidean region.

This was to be expected from the commutation relation (3.1.11) by the same argument as in 1.3.

Since


(reap. ~±(x)Ivac>)

negative) frequency part, the expectation value

contains only the positive (reap. <~±(x) ~F(a)>

(reap. <~F(a)~i(x)>)

is analytically prolongable to the upper (reap. lower) half Euclidean plane <0

(reap. Im (x~-a ~) > 0).

For

each other, while for

xI < aI

monodromy

x = a.

-i

around

For general n,

x 0 = a 0,

x I >a I

Im(x~-a~

they are analytic continuations of

they differ by sign.

This shows that

WF(X)

has a

(3.2.1) is expressible as an infinite series (cf. Formula 2,

1.4)

~

(3.2.3)

-i

f~ du WF~±(x) = J 2~u 0

+

+

il 0+~7~u e-im((x -a~)u+(x -a~)u

-i

)

q~o +co

Ii. I

n %=0 ~0,...,~%=i ~0Vl ~iv2

X

0(u 0) 8(-c~0~lUl)''" e ( - s ~ £ v u ) + ~ 0

xexp(-im((x -a +(x

where

s

= -i (U
Likewise the T-function

(3.2.4)

i (Uo+Ul)... i(u%+u) ±I u0-ul+i0 u%-u+i0

+_ + -i + _ + -i . + + -i a 0)u 0 +(a 0 a l)U 1 + "-+(av%-av)u ))

= 1 (U >~)

and

6(u) = i (u>O), = 0

(u
itself has a representation

n

. ~ .2% . .~ . g~iV2 %=2 v I, • • • ,~%=i

,. s~%~l

du% "j2--~Ul 1

--co

i (Ul+U 2 ) × @(-S

du~

0)u0+ (a 0-avl)Ul+..-+ (a %-a )u

T F = <~F(al)-.. ~F(an)>

1 IOgTF .

duo

v%~

--oo

i(u%+u I )

~lUl )-..8(-s~%_l~%u%)ul_u2+i0 -'- u%-ul+i0

du

484

x exp(-im((a~fa~l)Ul÷(a

l-a 2)u2+..-+(a~£_l-a~ )uz

+ ( a+ -a~+ )u?l+(a + -a + )u]l+...+(a+ - + -i )). 1 I 91 ~2 z Ul -i a )u£

The series (3.2.3) , (3.2.4) are convergent for sufficiently large double valuedness of

Im(-x ~ +a.) >0

(~=l,...,n)

and

Im(-a~+a~),,.-, Im(-a ~ .+a ~) > O. (*) ~ The above argument on the ± z n--I n WF (x)

applies to the general case as well.

closer look at the properties of

WF~(X) = w

Let us take a

restricted to the Euclidean region.

Firstly it satisfies the Euclidean Dirac equation --~

m

(3.2.5)

_~

m

w = 0

outside

al,...,a n.

This is a direct consequence of the Dirac equation = -m~+(x) (3.2.6)

for the Majorana fields w

~±(x).

= m~_(x),

~-~+(x)

~_~

(x)

Secondly it has the monodromy property

changes sign when prolonged around

al,...,a n.

Finally we have the growth conditions (3.2.7)~

lwl = o ( ~ )

z ÷ a ,

(3.2.7)~

lwl = O(e -2mlzl)

Izl +~.

= l,''',n,

Properties (3.2.5) ~ (3.2.7) follow either from (3.2.3), or directly from (3.2.1) To see (3.2.7)

at the operator level, we need the following short distance operator

expansion abtained by applying (1.4.15) % (1.4.16) (3.2.8)

~-i

~(x) ~F(a) = ~iF ( a )

.w0+_~ml~ -~- ~ F (a)'w I+-..

+ 1 ~ F (a).w E + 1 ~-~ ~F(a).w~ +...

A

-1 =

i

w0+

-°

F(a)'wl +

o..

i -i ~ - ~ ~F(a).w E + ~ - ~ ~F(a)-w~ +''"

Here

w£ = w£(-x +a , x+-a+),

w~

= w~(-x-÷a-, x+-a +)

.

are solutions of the Dirac

equation with the local behaviors (*) a~'s

Using the covariance under are real.

a~ ~-+ e ±i8 a~

we may also cover the region where

465

(mz)

2 ((£11), + (%~)! m2zz---------* +'" ") 1

w~(z, z*) =

(3.2.9)

(mz)£+½(l + m 2z* +...) (£+~) ~ (~) !

i+l i (mz*) ((_~__ +

2 m zz*

1 (mz*)Z-~(~

m2zz*

½), (~+3),+'")

w~(~,z,) = +

(~-~), (~+})---7 +''') Explicitly they are given by the modified Bessel functions

i(~-})eI

e (3.2.10)

i ( ~ ) @~-2

w~(z, z*) = e

z =re i8/2 '

where

I

z* = re-iS/2.

having the monodromy

- i (~+}) 8

-i

e

l(mr) ]

I '

l(mr)

l(mr) ]

~+~

w~(z,z*) =

-i(~-})e

)

e

I

O(

1 +~)

w = ~N- N~cow°(z-a'z*-a*) ~ +i~-N c*w*(z-a,z*-a*) ~ ~"

l(mr)

£-~

It is shown that any local solution

and growing at most as

Iz-a[N as

I

w

J

of (3.2.5)

is expanded uniquely

Z

The following formula is sometimes

useful. (3.2.11)

~Tw~=mw~_l, ~ w i = l w £,

where

~

3

~ Z , , ~z*WfmW~+l ' ~--Tw~-mw~-l'

~---'W* ~z ~ = row*~+i'

~w~

= -Zw~

denotes the infinitesimal generator with spin

i

of the Euclidean

rotation around the origin (3.2.12)

MF=zf~-

z, ~

~+2

i[i

)

-i"

3.3 Linear differential equations for wave functions Now we show that the properties (3.2.5) ~ (3.2.7) are sufficient to characterize the wave functions in terms of a system of linear partial differential equations. First let us show that the vector space

W

of functions satisfying al,''',a n

466

(3.2.5) % (3.2.7) is finite dimensional.

(3.3.1)

IF(W,W') = ~

Set

2ff

idzAdz*(w+w$*

+ w_w'*)

= IF(W',W)*,

w, w' E W al, • . . ,an

Property (3.2.6) guarantees that the integrand is single-valued, while (3.2.7) ensures the convergence. product on

Hence (3.3.1) defines a positive definite hermitian inner

W

Using the Euclidean Dirac equation (3.2.5) we have al,-.',a n

midzAdz*(w+w~*

+ w_wi*) = id(w+wi*dz) , so that only the boundary term contributes

to (3.3.1):

(3.3.2)

IF(W,W' ) = ~

n~ lim v=l ~+0

f

w+wi*dz Iz-a l=c

o

~)=i where we have set oo

(3.3.3)

for

w =

,z*-av)

• ..

This implies 0) in

[ ~V)(w)-w~(z-a ~=0

for v = l , . . . , n then w =0. . In particular, if c O(~))(w) =0 ,an dim W < n and that there are no bounded function (other than al, •.. ,an =

weW al,

oo

~ C~))(w)'wZ(z-av,z*-a~)+ i=0

W al,

• ..

Actually

,an Wal'" .. 'an

is exactly

anslysis one proves the existence of

n n

dimensional. elements

Making use of some functional

wCl,... C n

of

Wal,...,a n

with

the following normalized local behavior:

(3.3.4)

WC~ = ~D~)w0(z-a ,z*-a~) + ~vw.(z-a± v'z*-a*)~ +... n + B~vw~(z-a ,z*-a ~) + ( ~ ~ o ~ ) w ~ ( z - a ~ , z * - a ~ )

+'..

o=1

(z ÷ av,

(The argument employed here is analogous tials on a compact Riemann surface.)

V,~ = l,...,n).

to the existence proof of Abelian differen-

In terms of the coefficients

B~V, the positive

definiteness of (3.3.1) is equivalent to (3.3.5)

B = (~v) for a unique

is negative definite and hermitian, H = tH*.

and hence

$ = -e

2H

467

[(

Moreover, by calculating

~wc~+

-)jid(wcN +

~z

fI.

~wc~_

d z ) = jJzd(wc~ - - -~z

dz*)

for

U # ~ we

ob rain

(3.3.6)

= (~U~)

is symmetric.

Likewise

(3.3.7)

gives BtB = I,

i.e.

H = -tH.

Let us derive the differential equations for Dirac operator (3.2.5) commutes with

MF

Wc .

in (3.2.12).

Note that the Euclidean Hence

MFwC~

satisfies

(3.2.5), as well as (3.2.6), (3.2.7) except for the growth conditions at On the other hand, growth condition combinations of

~zWc~

and

1 O(--~-). ~ 3z W c N '

~--~Wc~

z=al,-..,a~

share the same properties including the

By matching the singularities of

MFwC~

by linear

3 and Wc~ and applying the finite dimensioilality 3z*WC~

argument above, we obtain a system of linear differential equations of the form

(3.3.8)

(~Wcn) = (

+B

+F)

n

,

A, B, F : n ×n

Using the recursion formula (3.2.11) the coefficient matrices

constant matrices.

A, B, F

are

identified as

(3.3.9)

A = [ial'.

]j|,

B = -G-IA*G,

F = [~, mA]

"an

A The coefficients of

an] w$

and

G wI

le2H in the local expansion of (3.3.8) give respective~

(3.3.10)

GFG -I = -[~*, mA*]

(3.3.11)

[C2, mA] + ~ - F~ - mA* + G-ImA*G = 0

with

C2 = (C~V)(WF~))~,~=I,...,n.

(3.3.11)'

~UP = ~ ( ~ )

In particular the diagonal of (3.3.11) gives

f~f~ + ma* m(a~-a v)

(G-ImA*G)~.

468

Since the (Euclidean) wave functions are linear combinations of the basis

(3.3.12)

~-I

WF~ = ~i

WF~

of 3.2 belongs to

Wal,...,an,

they

Wcv:

~i = c~Wc~"

The short distance expansion (3.2.8) enables us to calculate the local behavior of WF .

We find that i

(3.3.13)

/~-ic~)(WF~)

(~ = ~)

=

(~ = ~) -i 0(~)(wFp) =

2

7-!2 where

TF^~

^~

denotes the Euclidean continuation of

F(an)>/< ~F(al)... ~F(an)>.

Setting

local behaviors of (3.3.12) we have

I-T

T = (iE = C,

< ~F(al)... ~F(a )... ~F(a )''" ~),

C = (c)

- I-T=C

B.

Since

and comparing the i-~=

I + G -I

is

invertible we obtain

(3.3.14)

T = ( I - G ) ( I + G ) -I = tanhH, C = 2G(I+G) -I = e-H(coshH) -I .

In particular

C

is invertible.

Hence

WF's

also span the vector space

W al,...,a ~

3.4. Deformation equations So far we have fixed the position of branch points regarded a , a~.

w CV 's as functions of

z, z*.

al, a~, • " ", an, a~, and

Next let us consider their dependence on

It is easy to see that their derivatives

Swc~

~Wcv

share the properties

~a~ ~a~ i with the growth condition at most 0 ( - ~ 3 , 3 ) (z + Thus /Iz-a%I a%). the same argument as in 3.3 (matching of the singularities) yields the following (3.2.5) % (3.2.7)

system of linear total differential equations for

(3.4.1)

(- dA'~z (dWcnJ

Here

d

Wc's

Gsz . + @)

:

. ~WcnJ

denotes the exterior differentiation with respect to

a , a~ :

469 n 8w dw = ~ l ( % d a

~da~),

+

and

@ = (@v)

is a matrix of 1-forms related to

F =

(f ~) in (3.3.9) through dapI

(3.4.2)

@

f~v

da~ a -a

(~¢ ~)

=

(~

0

=

~).

The Euclidean Dirac equation (3.2.5), equations (3.3.8) and (3.4.1) characterize (to within a finite number of integration constants) the wave functions as functions of total set of variables

z,z*,al,a~,...,an,a ~.

condition for them (i.e. cross differentiation etc.)

wCl,...,Wcn

As the integrability

M F = MF SaV '

~ap ~a

~av ~ap

we obtain a system of non-linear total differential equations, the "deforma-

tion equations" for the coefficients

(3.4.3)

F, G:

dF = [Q,F] + m 2 ( [ d A , G-1A*G] + [A,G-IdA*.G]) dG = ~G@ + e*G.

Here

-0* = (O~V)

is the complex conjugate of (3.4.2)$(*) and

F, G

are subject to

the symmetry

(3.4.4)

tF = -F,

tF*= GFG-I( = -F*),

tG = G-I = G*.

In deriving (3.4.3) we have used the following fact.

If

p~(~,~)

is a differen-

tial operator with constant coefficients and if p g ( ~ , ~ z , ) W c ~ = 0, then p ( 22 2 ~=i ) E 0 modulo ~z~z-----~ - m This is proved by examining the singularities.

,

~z*

As an example let us write (3.4.3) down in the simplest nontrivial case Regarding the Euclidean covariance we may set

mA = ~~[i -i ] = mA*

(3.4.5)

G=

[

cosh ~

i sinh ~

f* = f,

(,)

-i sinh ~

Our definition of

cosh

) '

d@ @=-

I -i)

f--ei

~* = ~.

@*

here differes from [5] by sign.

n = 2.

(e = m,al-a2, ) and

470

Then (3.4.3) reduces to 1 d~ f = ~O dS'

(3.4.6)

d~+ d@2

i d~ ~ d~ =

2 sinh2~

or equivalently to a Painleve equation of the third kind for

(3.4.7)

d2r] _ l(dn) 2

i dr] + 3

dG2

8 dO

-~d-8

~] = e-~

1 - ~ "

It is instructive to note that the deformation

theory developed here is includ-

ed in the scheme of §2 for ordinary linear differential equations.

Let

du [ vru (w÷ ] ] =w/Y~u ~ -I em(zu+z*ul)w(u)

(3.4.8)

be a formal Laplace transformation

to solve the Euclidean Dirac equation (3.2.5).

Then the systems (3.3.8) and (3.4.1) are rewritten in terms of an column vector

Wc = t(Wcl'''''Wcn) (G-1

__d~ = du C

(3.4.9)

m~*g

Fu

n

component

as

mA)~ c

u

(3.4.10)

d~c

= (

G-ImdA*.G u + @ - umdA)~c"

In other words we are dealing with the deformation ential equations u=~.

theory

of linear ordinary differ-

(3.4.9) having irregular singularities of rank one at

u=0

and

The system (3.4.3) are nothing but the deformation equations for (3.4.9)

(3.4.10).

3.5 Correlation functions By now it is relatively an easy task to obtain a closed expression for The key is the short distance expansion that the second coefficients

(3.5.1)

~-i

c I(~) (WF)

Z

-1 _(v) Cl (WFv)

This shows that

(3.2.8).

of the local expansions of

are given by

-i m Sa~ TF/TF"

d log T F = ~ ni / ~ ~ - i (_imc~)(WF~)da~

1 dlogT F = ~

WF~±(x)

= i ~a~ TF/T F

+ im~)(WF

and (3.3.14) that

(3.5.2)

d log T F.

From (3.2.1) and (3.2.8) it follows

n

~ c)~(~ ~,v=l

mda~ - (B~*)

mda~)

)da~), or by ( ~ 1 2 )

471

=

1 ~- trace (i - T)~mdA

Using (3.3.11)' and noting

+

complex conjugate.

1 trace T~mdA = trace mdAt~tT = - ~ trace TO, we have

(3.5.3)

1 1 d log %F = 4 trace (TG - 9*T) + ~ ,

(3.5.4)

1 ~ = - ~ trace (F~ - @*GFG -I) + m 2 trace (d(AA*)-G-IA*GdA-GAG-IdA*).

Finally the first term of (3.5.3) is integrated as (3.4.3).

(3.4.3).

Summing up we obtain the formula

if

(3.5.5)

~F = const'/det coshH exp(~ ~)

r

J~

denotes a primitive of (3.5.4).

ated with (3.4.9) ~(3.4.10) exact differential The ratios TF

by using

The 1-form (3.5.4) is shown to be closed for any solution of the deforma-

tion equations

where

i ~dlogdetcoshH,

~

term

We remark that the closed 1-form associ-

in the sense of §2 coincides with (3.5.4) except for the

m2d trace AA*.

of (Euclidean)

<~F(al)...

~F(aD).-. ~F(a~)..

~F(an)>

to

are obtained directly from (3.3.14) as

(3.5.6)

^D~ = -+(i tanhH)D~ ZF

(~ ~ v)

More generally we have

(3.5.7)

F.( a 2.

< ~F(al)-.. ~ F (a i) . = TF.Pfaffian

In particular the (Euclidean)

).. . .~ F (a m). " ~F(an) >

(i(tanhH)v~,)V,~,=Vl,...,~m.

correlation function

TF = < ~ F ( a l ) . . . ~ F ( a n ) >

has

an expression

i/

(3.5.8)

TF

For

(3.5.5) and (3.5.8) reduce to

n = 2

(3.5.9)

~F F T

const./det i sinhH

exp(~ ~).

cosh ~

1 exp [~

= const" ~-

t dt ~((

2 (t))

- sinh 2 ~(t))].

co

3.6 Field theory of Ising model in the scaling limit The continuum model

~F(x)

field theoretical point of view.

so far discussed is of equal interest from the It is at present the only nontrivial massive model

472

of relativistic field theory whose form.

n

point functions are known exactly in a closed

It is also the simplest model whose S-matrix has the factorization property;

there are no particle production, product of

N(N-1)/2

(3.6.1)

$2, 2 = - i.

~nd the S-matrix in the N-particle sector is the

pairwise 2-body S-matrices

Moreover the asymptotic fields

~in (x) out

of

~F(x)

are knownexplicitly.

neutral free bosons (*) whose creation - annihilation operators are expressed in terms of the auxiliary fermion

t

~t(u),

~t(u) (=~(-u)),

~(u)

They are ~t

(u)

as

t ~t(u).(_)N~(u) ~t(u) =

(3.6.2)

~t(U)

= (-)N~(u).~(U)

N~(u)

=

f

du' 8(±(u-u'))~t(u')~(u ')

0 (u > 0).

Since

~ n (u)~in (u) = ~t(u)@(u), the above formula is reciprocal, and it is out out possible to construct free fermion out of free boson. Actually we may mimic all the constructions by starting from neutral free boson ~B(X),

~B(x)

= t(~(x),~B(x))._

fermi statistics.

~(x)

to obtain analogous ~ fields

The latter is a 2-component spinor obeying the

By interchanging

~

and

~

in (3.6.2) we obtain the relation

between the asymptotic fields ~#(u),

~(u).

The

n

~ n (u), ~in (u) of ~B(x) and the auxiliary boson out out F point functions are related to T F, T through the simple

formula (3.6.3)

<~F(al)..-

~F(an)><~B(al)...

It is straightforward can be a matrix)

~B(an)> = ~det coshH .

to incorporate a general phase factor

in place of the simplest monodromy

-i.

e

2~i~

(£~ ~, or it

To achieve this one starts

from the free Dirac (i.e. charged) fields instead of Majorana fields.

The deforma-

tion theory of 3.4, 3.5 holds without any change (except for the symmetry conditions for

F, G).

The relevant fields are now related to the solution of the Federbush

model (3.6.4) (*)

~) "~int = -g E~Jl(X)Jll (x)

~F(X) ,¢(x)

are bosons satisfying the microcausality

[~(x),~(~)]=0 for (x-x')2<0.

473

(current- pseudocurrent interaction behween two species of fermions). censt=~is

re!~nted to ~4~e exponent of the monodromy

(3.6.5)

g = 2~.

The coupling

References There are an enormous number of literatures concerning the Ising model. [l]

B.M. McCoy and T.T. Wu,

See e.~

The two-dimensional Ising model, Harvard University

Press, 1973. Explicit formulas of the spin operators and correlation functions on the lattice are given from the viewpoint of the Clifford group in [2]

M. Sato, T. Miwa and M. Jimbo,

Holonomic quantum fields V, preprint RIMS 267,

Kyoto Univ. (1978). Some recent developments in this branch are quoted in the references of [2]. Deformation theory and closed expression of correlation functions were introduced by [3]

M. Sato, T. Miwa and M. Jimbo,

Proc. Japan Acad. 53A (1977), 147-152, 153-158,

183-185. This work is a generalization of the monumental papers [4]

E. Barouch, B.M. McCoy and T.T. Wu,

Phys. Rev. Lett. 31 (1973) 1409-1411,

T.T. Wu, B.M. McCoy, C.A. Tracy and E. Barouch,

Phys. Rev. BI3 (1976) 316-374.

in which the 2 point functions are obtained in terms of a Painlev~ transcendent of the third kind. [5]

Details along with some generalizations of [3] are found in

M. Sato, T. Miwa and M. Jimbo,

Publ. RIMS, Kyoto Univ. 15 (1979) 577-629.

,

Holonomic quantum fields 9,

to appear in Publ. RIMS, Kyoto Univ.

,

Supplement to Holonomic quantum fields 9,

15 (1980). preprint RIMS 304, Kyoto

Univ. (1979). Extension of the above Sheme to higher dimensions is briefly touched in [6]

M. Jimbo, T. Miwa, M. Sato and Y. MSri,

Holonomic quantum fields -- The unantic-

ipated link between deformation theory of differential equations and quantum fields --, preprint RIMS 305, Kyoto Univ. (1979) and the references cited therein.

474

§4.

1 dimensional

4.1

Free fermion

XY

Here

XY

H = - ~1

y

and

h

limit

operator

1 dimensional

(4.1.1)

model and its scaling

model

is defined by the Hamilton±an

~ ((l+y)oxox_1+(l-y)oYoY_~+2h@Z). m mt± m mr± m m~Z

are real parameters

x Om,

and

Z

oy m and

Om

are the following

spin

matrices.

x .... ° ~m

~.~.~

oYm "= . .

o

z

@ I1

....

@

m

i] @ li -i] @

111 ill ®

i

[oJ,ok,] m m

@

-i

o x = @~, o y = 0 2 m m m

These spin matrices

(4.1.3)

i~ 11 o I~ ~]° i~ ~1 o /~ ~1o... I~ ~1o-.-

and

3 = 2i ~ g., ~6 ,0 % , %=i JK~ mm m

oz = 03 m m

satisfy

the commutation

relation~

'

[oJ,o~]+ = 26jk m

where

gjk~

equation

(4.1.4)

is the anti-symmetric

tensor normalized

as

g123 = i.

The Heisenberg

reads as Sx = ~1 ((l_y)(_oz y I_oy ~oz)+2hoY), m m~± m-I m m by 1 z x x z x m = 2 ((l+y) (~m~m+l+Om_lOm)-2horn) ,

~Zm = 21 (_(1+7) ( jX m±± x_.+ m m-±'°Y)+(l-Y)m (°x°Y-"+°YoX-1))m mr± m m~±

where

denotes

the time derivative.

The main problems i)

Diagonalization

ii)

Calculation

we are concerned

of matrix

iii) Characterization

elements

of correlation

The key to the problem

(4.1.5)

Pm °

with are the following:

of the Hamiltonian.

~z ~Z OX ~m-l~

qm = ioY°Z m m-i

m-2

functions.

i) is to introduce

2

oZ

of spin matrices.

"'"

free fermion

operators.

475

They are fermions because they satisfy the anti-commutation

(4.1.6)

[pm,Pm,]+ = 2~mm,,

[qm,qm,]+ = -2~mm,,

The key point is that the anti-commutators

relations

[pm,qm,]+ = 0.

are not operators but they are constant.

By (4.1.6) we can define a symmetric inner product in the vector space generated by free fermion. Pm

and

We call this vector space the orthogonal space of free fermions. qm

are free because the Hamiltonian

1 H = ~

(4.1.7)

or equivalently

is quadratic in them:

~ ((l+y)qmPm+l-(l-y)pmqm+l-2hqmPm), mcZ

the Heisenberg equation for them is linear:

• i Pm = 2 ((l+y)qm-l+(l-y)qm+l-2hqm)'

(4.1.8)

• 1 qm = 2 (-(I+y)Pm+I-(I-y)Pm-I+2hPm)"

In order to diagonalize y

and

(4.1.9)

h.

(4.1.7) or (4.1.8) we should take account of parameters

There exist the following three different phases:

~i

:

h < -i,

y > 0,

~2

: -i < h < i,

y > 0,

~3

:

y > 0.

i < h,

We shall be concerned with the phase ~2" following Fourier transformation.

(4.1.10)

~#(-0) = ~1 ( a ( V ~

~(O)

Here we introduce

The diagonalization

Namely for

O ¢ IR/2~Z

is achieved by the

we set

~ e-i(m+l) Opm + V ~ - ~ - I ~ e-im8 qm ) , mEZ m~

-imO qm ,) . = ~1 ( a(~-~ ~ e -i(m+l) 8pm - 6a-(~ -I ~ e m4~ m~Z a(8), ~(@)

and

b(8)

(4.1.11)

a(e) = b(e)/b(-e),

(4.1.12)

~(e) = i/b(e)b(-e),

(4.1.13)

b(8) = i/~l-~+le i@) (l-~-lei@)

by

476

(4.1.14)

~± =

Note that in

~2

h±il-~h2-~ 2 l-y

if

h2+y 2 ! 1

hi~+~2-1 i-¥

if

h2+y 2 ~ i.

b(8)

is holomorphic in the unit disk

The commutation relations,

(4.1.15)

[~(e),¢(e')]+ = o,

(4.1.16)

H =

(4.1.17)

~#(9) = ~(8)~#(8),

''lei@l < i.

the Hamiltonian and the Heisenberg equation read as

[~*(e),~*(e')]+ = o ,

[~t(8),~(8')]+

~ 2~6(@-e'),

I ~de ~(e)¢*(e)~(e)

~(e) = -~(e)~(e). 4.2

Vacuum The vacuum state

ator

is defined to be annihilated by the annihilation oper-

Ivac>

~(e).

(4.2.1)

~(e) Ivac> = 0.

This is an eigenstate of the Hamiltonian with the zero eigenvalue. The n-particle state the creation operator

(4.2.2)

lel,...,en>

is defined to be created from the vacuum by

~t(e).

191,.-.,8n > = ~t(el)...~#(gn)IVac>.

From (4.1.15) and (4.1.16) we have n

(4.2.3)

Hlel,...,6n>

=

~ ~(e=)lel,... e >. j=l J ' n

A general state is a superposition of the above eigenstates.

(4.2.4) Here

f0

_

"'"

27

"'"

is a constant and

i~ f%(@l'''''@g) 1Ol'''''@n > " f%(el,.-.,@%)

is a function in

~

variables.

totality of such state is called the Fock space. The operation of

(4.2.5)

~*(e)

and

~(e)

on the Fock space is given by

~*(8) lel,-..,en>= le,el,...,en>,

The

477 n

(4.2.6)

~(0) IOl,--.,@n> =

A general operator

~

is of the form

r deI

r

(4.2.7)

°

~ (-)3-16(0-e~)lel, • .. ,ej_l,ej+l,- .-, On >" j=l J

dO r

rde I

de ,

x fR,,R,'(@l"'''@R,;@i '''''@#')~'l'(@l)'''~t(oR,)¢(@i)''''~(@~,')" Note that in the expression (4.2.7) annihilation operators are in the right while creation operator are in the left.

This is called the normal ordering.

The totality

of such operators is called the operator algebra. The dual Fock space is generated by the (dual) vacuum state

<el,...,0nl


defined by

(4.2.8)


(4.2.9)

<el,...,enl =
We normalize the constant multiple so that the inner product between Ivac>

is

(4.2.10)

= £.

The vacuum-to-vacuum matrix element of an operator expectation value of

<~>

(4.2.11)

If

~


i.

~

and is denoted by

0

is called the vacuum

< ~>.

= .

is normal ordered as (4.2.7)

<~>

is nothing but the constant

f0,0"

The

correlation function is by definition the vacuum expectation value of a product of spin operators.

For example the 2-point correlation function of

x

is m

. mI m2

The calculation of correlation functions i s c a r r i e d out in the following two steps.

The first step is to give a normal ordered expression for spin operators.

The second step is to apply the Wick's theorem to the product of normal ordered operators.

As we shall see in the following sections, the first step reduces to a Wiener-

Hopf factorization problem and the second step reduces to a Fredholm integral equation.

4.3

Spin operator Let us consider

the phase

~2 ~

o x with the boundary condition o x > i for m m This amounts to considering the following product.

m

----+ _oo

in

478

(4.3.1) -x o

°m-X = (pmq m_l)(p m_lqm_2)

is characterized

...

up to constant multiple by the following commutation relations.

m

-x

-x Pm 'dm

=

(4.3.2)

-x -pm,Om

OmPm'

~mqm '

m

-x -qm'Om

m+l

'

_> m

m' < m-i .

the inner automorphism

serves the orthogonal In general,

> --

m' ~ m

-x qm' ~m

=

-x

In other words,

m'

of the operator algebra induced by

-x om

pre-

space of free fermion.

this characteristic

property is equivalent

to the following charac-

teristic form of operator

(4.3.3)

where

:CWl-..wkeP/2: ,

c

is constant, Wl,-..,w k

ratic form of free fermions.

belong to the orthogonal

The symbol

:

in (4.3.3) one should expand the exponential

:

space and

p

is a qua d -

means the normal product, namely,

and rewrite it in the normal product

form. -x 0

has the following normal product form.

m

(4.3.4)

-x -x Pm/2 o = :e :, m m

(4.3.5)

Pm =

;;d0d0(*t(-°)~(e))(j&7~y ~ ~2-~

O') ei(m-l)(0+0 ')

(~(o)+~(o')

-~(o)+~(o')) L e(o')

In the next section we shall explain how to derive

4.4

Wiener-Hopf

(4.3.5) from (4.3.2).

factorization

As a basis of the orthogonal

space of free fermions we use

(4.4.1)

p(0) = ~ - I ( ~ # ( - 0 ) + ~ ( 0 ) ) ,

(4.4.2)

q(0) = a(~-e~(~t(-0)-~(0)).

The kernel

l_e-i(0+0'-i0)

dO' T(0,0')-~-

of the orthogonal transformation T -x d o (see (4.3.2)) is given by

space of free fermions induced by

dO (p(e)2i~ ,q(e)~-ff ):

in the orthogonal

479

(4.4.3)

T(0,0') =

~ eim(@-@') m>O

~ e im(0-0'). m<-i

We set I-T P = T

(4.4.4)

The operators If

X±

P

and

I-P

are projection operators with the following properties:

is the multiplication operator with the kernel

is holomorphic for

(4.4.5)

X±(0)6(0-e')d0'

where

Xi(e)

leli01 < i, then we have

PX+(I-P) = 0,

(I-P)X_P = 0.

The projection operator

E±

of the orthogonal space of free fermions to the

subspace of creation operators (resp. annihilation operators) is the multiplication by the following

(4.4.6)

E+(O) (resp. E (0)) in the above basis:

E±(@) = ~

ia(0)- I

1

We set

(4.4.7)

E = E+ - E .

Let

JR++(0,0')R+-(O, 0'))] be the kernel of the quadratic form P0 with respect IN-+(0, e')

to the above basis.

(4.4.8)

Noting that

[p(0),p(O')]+=2.2~(@÷6') ,[q(0),q(0')]+=-2.2~(0+0'), [p(O),q(0')]+ = O,

the formula

(4.4.9)

where

R--(0,0'

(

)

is rewritten as

W(I-P+EP) = -2P,

W is the integral

operator with kernel

W(@,@')

dO'

given by

"2Rq-~(

If we find

Xi(e), which is holomorphic and invertible for

(4.4.11)

X

= X+E,

a solution

W

for the integral equation is given by

le±i01
480 (4.4.12)

W = -2X,Ipx+.-

In the present case of (4.4.6) we can take b (O) -1 (4.4.13)

(4.4.14)

x+(O) =

b(e)]

'

X_(O)= [b(-O)b(-o)-l}" dO ( p ( e ) ~ , q(O)2~ )

Finally changing the basis from we obtain (4.3.5) with

4.5

m=0.

The general

m

to

t dO d@ (@ ( - O ) ~ , ~ ( O ) ~ )

case is obtained by the translation.

Scaling limit Thus far we have argued the lattice theory.

limit.

(4.5.1)

Now let us consider the continuum

By this we mean the following limit.

g --+ 0, m --+ ~, e ---+ 0

keeping

ms = a

and

6/~ = p

In order to obtain non trivial results we should take simultaneously to critical values of parameters

(4.5.2)

~ = i, h = I-UE

y

and

h.

fixed.

a certain limit

Here we consider the limit

(£ --~ 0, p: fixed).

We shall list up the scaling limit of several quantities below.

(4.5.3)

(4.5.4)

~b(@)

> b(p) =-

i

a(@) -----+ a(p) = ~F~$~p/~f~Z~-ip,

(4.5.5)

g-l~(e)

> ~(p) = / p 2 ~ 2 ,

(4.5.6)

~¢t (o)

, ~t (p),

(4.5.7)

~-~ (O) -----+ ~(p).

The anti-commutation

(4.5.8)

(4.5.9)

,

relation (4.1.15) is scaled to

[~t(p),~t(p,)]+ = 0, [~(p),~(p')]+ = 0, [~#(p),~(p')]+ = 2~(p-p') .

~-ipm

> p(a) = f ~

~-~-l(~t(-p)+@(p))eiap,

481

(4.5.10)

~-lqm

(4.5. ii) (4.5.12)

> q(a) = I ~

g-iH

>H = I ~

(2g)-i/8 ~x

~

(~*(-P)-~(P))eiap'

~(p)~t(p)~(p),

~ ~(a) = ..eP(a) /2

m

(4.5.13)

P(a) = ff d-e ( ~ fd-el 2~ (-p)~(p))2~

i

~(~7~Yp)~ ' ) [O~(p)-~(p')-~o(p)-~(p'i] _ieia(p+p ') [~#(-p')] × lw(p)-Mo(p') -~(p)+~(p' p+p'-iO (~(p') J " The commutation relation (4.3.2) is scaled to

p(x) y(a) (4.5.14)

x > a

~(a)p (x)

~(a)q(x)

(4.5.14)

(-p(x) ~(a)

x < a,

i q(x) ~(a)

x > a

[-q(x) ~(a)

x < a.

We note that (4.5.13) follows either from (4.3.5) by scaling or from (4.5.14) and (4.5.15) by a similar integral equation as (4.4.9).

We note also that

~(a)

is

identical with the (time zero),scaling limit of Ising spin below the critical temperature. 4.6

Operator product and Fredholm equation Now our main interest is in the operator product

correlation function

~(al)...

~(a n)

and the

< ~(al)... ~(an)>.

The quadratic kernel for

~(a)

was obtained by solving the integral equation

(4.4.9), which is a consequence of the commutation relation (4.5.14), (4.5.15). Let us assume, for a while, that

n=2

and

al
Then the product ~(a I) ~(a 2)

satisfy the commutation relation

(4.6.1)

i p(x)~(a I) ~(a 2)

x < aI

or

~-p(x)~(a I) ~(a2)

a I < x < a2,

i q(x) ~(al) ~(a 2)

x < aI

(-q(x) ~(al) ~(a 2)

a I < x < a 2.

x > a2

~(al) ~(a2)P(X) = or

x > a2

~(a 1) ~(a2)q(x) =

From this we can deduce an integral equation similar to (4.4.9). cannot expect a simple solution like (4.4.12).

But this time we

Instead the solution is characterized

by monodromy properties and by a system of linear differential equations.

To see

this we ~se a different integral equation from (4.4.9), which we shall explain in this section.

482

It is natural to enlarge the space of free fermions and the operator algebra as follows. We introduce copies 4*(J)(p) and 4 ~j)" "(p) (j=l,--.,2n) of creation and annihilation operators, respectively. (4.6.2)

We assume the following commutation relation

[4t(J)(p),4*(k)(p')]+ = O,

[4(J)(p),~(k)(p')]+ = O,

[4 (j) (p),4t(k)(p')]+ = ~2~(p-p')

if

i

%J_nok_n2~(p-p')

Here

%jk (I ~ j < k j n)

(4.6.3)

A =

0

~12

0

0

0

0

lj-kl = n

if n+l ~ j < k ~ 2n otherwise.

are arbitrary complex parameters and we set ~13 "'" %in ]" ~23 "'" %2n 0

"''

0

For a I < ... < an , we denote by yj the operator : e0(j)/2 : with the following quadratic form p(J).

(4.6.4)

p(J)= SI2d-2~~ '

(4.6.5)

R(J) (P'P') = ~

"4t(j+n)(_p,)] (4t(J+n)(-p)4(J+n)(p))R(J)(p,p')4(j+n)(p,)J . 1

We define (4.6.6)

~

2n × 2n matrix kernel

[~(P)-~(P') .~(P)+~(P')

-~(P)-~(P')I

-ieia'(p+P')3

-w(p)+~(p')J p+p'-i0

W(p,p , ) = -(Wjj,(p,p'))g,g,=~;j,j,=l,...,n Eg'

Wj~.,(p,p') = <4(J')(-p')4(J)(-p)~l..-~n>/T n , _+ Wjj,(p,p') = -<4 (3-, ) (-P~)~I''" ~n4* (J) (P)>/Tn ' jj,(p,p') = <4 (j) (-p ')~i'" •~n 4f (j') (p')>/T n , W+,T

Wjj,(p,p') = -<~l'''~n 4#(j)(p)~f(3 )(p,)>/T n , (4.6.7)

Tn = <~l'''~n >.

The quadratic form Pn for ~l---~n = Tn :epn/2: two particle matrix elements:

is determined by the above

by

483

(4.6.8)

pn= ~

IId~ d--~CdJ*(J+n)(-p)~(J+n)(p))I j W ~ . , ( p p ' ) W ~ j , ( p p ' ) / [ ~ t ( J ' + n ) ( - p ' ) ] -+ -~b(3'+n). ,

Now we shall .write down the integral equation which characterizes the kernel W(p,p').

We denote by

R "~"

d ' R(J)(p,p')2~ " We also

the operator with the kernel

0 0 denote by K(resp• tK)- the operator with the kernel (i 0 )~(p+p')dp' 0 3 (0 -~)~(P+P')dP')" U- We set 0

= IR1-RnJland

=

%12 X ..... %In K

_%12tK

0

-%intK Then the Operator

W

(resp•

.....%2nK

-imntK ... d

with the kernel

W(nr'~n ' "2~ ~

!

satisfy the following Fredholm

integral equation

(4.6,9)

(I-RA)W : R•

Moreover the correlation function

T

is nothing but the square root of the

n

Fredholm determinant. i

(4•6.10)

4.7

T

= (det(l-RA)) 2.

n

Operator expansion Before proceeding further we need the asymptotic formula at

products of ~(a)

and

~#(p), ~(p).

±~-1

[@*(-p)±*(p),~(a)] = 2ip z

where the operators (4.7.2)

~(±)(a)

for the

Making use of the explicit form (4.5.13) of

the kernel, we obtain the following operator expansion at (4•7.1)

p = ~

.

-

i

p = ~. '

-

e-lap(~(+)(a)- ;~a~+)(a)+'-'),

is defined by

~(~)(a) =:(~2d~--~p)-l(~T(-p)±~(p))eiaP)e0(a)/2:.

We note that

~(-)(a)

is identical with the (time zero) scaling limit of Ising

spin avobe the critical temperature• We also need the formula for the products of ~(±)(a) and ~t(p), ~(p): i . i (4• 7 . 3 ) [~(-p)±~(p),~±) (a) ]+ = ±2pi~e-laP(~(a) - ~ ~a ~ (a)+•••)'

n

484 [~t(-p)+~(p),~)(a)]+ = 2ip~2e-iap(~(+)(a)- ~i ~ ~(+)(a)+-.'),

(4.7.4)

where the operators ~(±)(a)

are defined by

(4.7.5) ~(-+)(a)=:(I2~/~c~l(+t(-p)±+(p))eiap)(I~--~l(~t(-p)±~(p))eiaP)eP(a)/2:. Let us explain how to use the above formulas to obtain the asymptotic expansions of W~:(p,p'). ~(J) (p)

The anti-com~utation relation (i'~,6.2)implies that

co~nute with

~k)

if j#k. Hence we have

~t(j)(p),

<~(J')(-p')~(J)(-p)~l-..~n >

= <~(J')(-P')~I'""~(j)(-p)~j'''~n>' 0 = <~(J')(-p')~l'''~n~(j)(-p)> : <~(J')(-P')~I •..~j~(J)(-p).--~n > and )(-p),

<~(J')(-P')~(J)(-P)~l'''~n > = <~(J')(-P')~I''"

1••

gel l) It is convenient to eleminate the p-dependence of Wjj,(p,p limit p -~ o% _

(4.7.6)

W

Thus we set

(I)EE'

-iap , +e -£' e j (Wjj,(p,-p')-gWjj,(p,-p')),

I+C j j, (p')=lim 2~~ i ~ ) p-~

::I ~(~p')<+(J')(p' , <

1~

...

~i

)~1. • -~)

t(j')

(O...

~j

• • • #n>/rn

~n~

(c'=+)

_

(P~Tn

(g'=-).

In the next section we study the monodromy property of the matrix (w(i)~'(

by taking the

W(1)(p) =

. .

jj' P))g,£'=~;j,j'=l,-'-,n"

4.8 Monodromy property The Fredholm integral equation can be solved by using a Neumann series. the limit of (4.7.6) we obtain the following series for (4.8.1)

Taking

W(1)(p).

W(1) £~' -ia. ,p jj,(p) = e 3 {~jj,(~l ~+~_IEE') co +Z

n

t dP 1

i(a. x e

x

31 a j 2 ) P l +

• I dp£

k.

...~.

x

..'+i(aj£-aj £+1 )p£

~-e'CjlJ2~ 1 ~I+EjlJ2J3~2.. ~£-i+~-J~_13%3£+I. . ~£ P-Pl+i0 Pl-P2 +i0 p£_l_P£+i0 (~le~O£-~_iggj £j £+i),

4~

where we set

~jk = kkj

(j > k),

kjj = O; ~ = ~(p), ~j = w(pj); gjk = i 1

aj >ak, =

gjk% = ~jkgk%

and

Jl

=

J'' J%+l

L

J"

=

a.J ak aj < ak matrix whose elements -

Further we denote by

W(2)(p) (resp. W(3)(p))

are given by (4.8.1) with W(1)(p)

p-pi+i0

(resp. W(2)(p))

lower h~if plane

~ 2 ).

(resp. W(2)(p))

replaced by

contour of

Pl

Hence

2nX2n

(resp. p-pl-igjlJ20).

is well-defined in the upper half plane Since

~(p) = p ~ + p 2

has a branch point

g. . ~ 0. JIJ2

the

p-pl-i0

integration for W(3)(p)

p = i~

W(3)(p)

in

~i

can be deformed into

Ci

(resp. in the

p = ±ip, W(1)(p)

(resp. p = -ip

is well-defined in the region

in ~ 2 ) . The according as

~3

/I

ip

/

i./

• /

~i

has branch points at

/

>

/

H/ W (k) (p)

(k=l,2,3)

The relations between i -i p--~+ ~ = 2~(p).

is easily seen from (4.8.1) using

the identity (4.8.2)

W(3)(p) = W(1)(p) [ 1 I+2A I'

(4.8.3)

We set (4.8.4)

,(k)Eg', , i ,,,(k)E+ , , jj,kP) = ~ kw jj,kP) + g'w(k)jj,(P)).g-

We note that

+-~<~l'''[~t(J)(-p)+~

(j)(p),~j,]'''~E)'''~n>/Tn

±~(~2~) <~l-..[~'l'(J)(-p)_+~(J)(p),~e)]+...~n>/Tn ¥

(4.8.7)

(j = j ' )

~ ( ~ ) <~ ..~!e)..[@t(J)(,p)_+~(J)(p),~j ]...~n>/Tn (j <j,) 2 tl" "j "

The relations (4.8.2) and (4.8.3) read as (4.8.6)

(j >j')

y(3)(p) = y(1)(p)

I+A -A

y(2)(p) = y(3)(p)

l+tA tA

-A ], I+A J tA ] • l+tA

J

486

The monodromy proper~ (see 2.1 and 2.2) of

y(k) (k=1,2,3)

is explicit if we

write them as (4.8.8)

where

y(k)(p) = ~(k)(p)o0(p)DeAoop

D =

.

and

A

=

l1 1wlial "1

",

-ia n _ial '

0''. 0

-ia Here

~(k)(p)

series

i s holomorphic and i n v e r t i b l e

in

~'k

n

and has the f o l l o w i n g i n f i n i t e

expression.

(4.8.9)

~(k) sg' jj,(p) = 6jj,6es,

oo

n

f dPl

+ %~i j2,-!.j%=lJ~2~l

..

[ dp9~ ~. " J2~i~

"''~..

jlJ2

ei(aJl-aj2)Pl+..-+i(aj -aj%+l)p%

J%J%+l

× 61g'-6-1c'gJlJ2~ ~ . ~l+gJlJ2J3~2 ... ~%-i+~J%_i j~jk+l~ (~I ~ -~ IE~j%j%+I )" p-p, +ig!k! 0 p %_i- p ~+i0 313 2 Pl-P2+I0

(k) = {I

where

~. .

JlJ2

~jlJ2

y(p) = y(3)(p) at

k=l k=2 k=3

has an i r r e g u l a r

singularity

at

p = oo and regular singularities

p = ±i~.

At

p = ~ we take sectors

(4.8.10)

: (k-2)~ < arg x < k~

~k

and denote by Yk(p)

Yk(p)

(k=1,2,3)

multipliers

Sk

the analytic continuation of

y(3)(p)

from the real axis.

have the same asymptotic expansion in each sectors. (k=l,2), which is defined so that [ l+tA

(4.8.11)

(k=1,2,3)

S1 =

tA

tA

)[ -i

l+tA

Yk÷l(p) = Yk(P)Sk, is given by tA ] - 1 ,

][ l+tA 1

The Stokes

tA

l+tA -A I+A

The exponents of formal monodromy at Let wise.

Let

T±

p = ~

the closed path starting from (y±Y)(p)

are p = 0

(i,...,i,0,''',0). and encircling

be the analytic continuation of

Y(p)

±iD

anticlock-

along the path

y±.

487 Then the monodromy matrices

M+

at

+i~, which is defined so that

(y_+Y)(p) =Y (p)M~,

is given by

(4.8.13)

14+ = S2 ,

M

: SI.

The exponents of formal monodromy at

4.9

i

_+iD are

I ,'-',~,0,''',0).

Deformation theor[ In this section we show that the monodromy property of

y(k)(p)

leads us to

linear and non-linear system of differential equations. First let us consider species

~dY(k)(p)-y(k)(p)-l. ~P k=1,2,3, we shall drop the index k.

The multi-valuedness of

Y(p)

cancels and

Since it does not depend on the ~Y(p).Y(p)-i

is single-valued

UlJ

holomorphic except for at

p = ~i~

(4.9.1)

p = ±i~

and

p = ~.

A+ +..p-i~

~py(p).y(p)-i =

A p_--i~+-'' (4.9.2)

The singular part of

~DY(p).Y(p) -I

is given by

A+ : Y(1) (i~)

p = i~

at

p = -i~,

~(i)(i~)-1

2.

i1

at

"! 2 0.

y(2)(_i~)-i" (4.9.3)

A, = Y(2)(-i~)i~ "'i

20.. "0 At

P = 00 _ ~ . _ y ( p ) . y ( p ) - _l op

(4.9.4)

is

of

the

form

~py(p).y(p)-i = Ao° + O(1).g

From (4.9.1) and (4.9.4) we conclude that

Y(p)

satisfy the following system of

linear ordinary differential equations. A+ (4.9.5)

~pY(p) = A(p)Y(p),

A(x) = ~

A_ + ~

+ Aoo.

For the later convenience we write down the relations between oo

coefficients

Yj (j=l,2,'--)

of the expansion of

A+, A .

Y(p) = l+j~iYj/p]

and the

_

at

p = 0%

488

(4.9.6)

[YI,A ] + D = A+ + A_,

(4.9.7)

[Y2 ' A ] + [A'YI]YI + [YI 'D] - Y1 = i~(A+-A_).

Now let us deform

al,-.-,a n

and

D

keeping

A

fixed.

Namely we deform

Y(p)

keeping ths Stokes multipliers, connection matrices and the exponents of formal monodromy fixed. (d

This isomonodromy property assures that

is the exterior differentiation with respect to

dY(p)'Y(p~ I is single-valued al,...,a n

and

~).

Then by a

similar argument as above we can derive the following system of linear total differential equations.

(4.9.8)

mY(p) = ~(p)Y(p),

!~_+-iA ip_~]_ _~ + dH.

~(p) = p d A + [ Y l , d A ] +

The compatibility condition between (4.9.5) and (4.9.8) is written as

(4.9.9)

dA(p) = ~ p (p) + [~(p),A(p)].

Comparing the singular parts at

p = ~iv

in (4.9.9) we obtain the following system

of non linear total differential equations for

A~.

(4.9.10)

dA+

=

(4.9.11)

dA

= [d(-iHA)+[YI,dA ]+A+dlog~,A_]

[d(i~A ) + [ Y I , d A ] + A d

We note that from (4.9.6)

4.10

[YI,dAj

logD,A+] ,

.

is solved in terms of

A+.

Correlation function In this section we shall give closed expressions for the correlation functions

of operators ~(a), The

YI

~(+)(a) and ~(-)(a).

term in the expansion of

Y(p) = i + ~ Y./pJ at p = ~ has the j=l J following operator expression as a consequence of (4.7.1) ~ (4.7.5) and (4.8.5).

-i-~-~-<9~8a 1 i~'''2 ~n >

.. ~(+) ~'2 .(-) " . ")°n> . . . . -I
.(-) ~2 .(+) "" e n > i< ~l

- i 8a ~ -2L < ~1 2 "'en >

i

n

....

i<~l~+)."~n.(-)>

(+) >

-n

....

n

>

489

(4.10.2)

-

YI + =

i

0

_i<~-)~2.

""

~(-) n

>

n

• <.(-),^(-)

~i

*2

%>

i<~-)~2" " "~n(-) >

(4.10.3)

YI

i<~l~-) " " "~(-)n>

h

n .

""

i
0

= +

_i<~l~-). ~(-)

0

h

"''~n >

>

0

"""

"

n

.. _i<~+)~2 . .~(+)>

" n

_(+). (+)

-i<~ i V 2

0

"''% >

.... i <~i~ +)''" ~0(+)n>

-i<~+)~2'''~(+)>n

(4.1o.4) £< = +

.<~o(-),o(+)

i<~-)~2.

(+)

n

• ..(+)_(-) -~<~1 ~ ""Z > - i ~~ 1 ~ r " % >

... i < ~

-i<~i(+)~2"" "~-)>

.... i~<~i~2""" ~n > n

First let us consider

-i <~i~ +)'" "~(-) >n

Tn = <~I-..~_>.

-) ""~n(+) >

Note that (4.5.8) and (4.5.13) are -i a + c a, ~ + c~, p ÷ cp,

scale invariant under the scale transformation ~t(p) ÷ / ~ % ( c p ) transformation (4.10.5) keeping (4.10.6)

and

~(p) ÷/~(cp).

~ ÷ c~, p + cp.

Hence

Tn

is invariant under the scale

Thus it is sufficient to consider

~ = d log Tn ~ fixed.

Then from the diagonals of (4.10.i) and (4.10.4) we obtain

i ~ E - ~ trace Y I d A

mod

d~.

Finally using the diagonal part of (4.9.7) we obtain (4.10.7)

~ E ~i trace((A+-A )I~dA+}[YI,dA ~ (A++A_))

mod

d~.

490

Correlation functions containing number of

~(~)

the number of

~(±)

is calculated as follows:

is two, they are given in (4.10.1) ~ (4.10.4) ~(±)

If the Tn . If

divided by

is greater than two, they are written as Pfaffians.

For

example

<,~(-).^(-)- (+)--(+)>-

(4.10.8)

'~i ~/2 %°3 ~F4

/T4

= c<~(-)~(-)<~o~

~(+)~(+)>

" "i --2 --3'4 "1--2--3 --4

<,.(-),. (+).. ><,^ _(-).^ _(+) >+<_(- ),. .^ _(+) ><,^_(-)_(+),. >.. -~fi vF2W3 u/4 Ufl~/2 v/3u/4

4.11

~ui v/2W'3%u4

2

~iv/2 ~f3 5v4 )/T4"

Two point function In this section we show that if

n=2

the deformation equations (4.9.10) and

(4.9.11) reduce to the Painlev6 equation of the fifth kind. First we note that if

n=2

SI=M-

and

S2=M+

split into two

2x2

blocks.

] (4.11.i)

S1 = M_ =

-I -i

21

-21 (4.11.2)

S 2 = M+ =

1

-i

21 ]. -i -21

Hence

Y(p)

(4.11.3)

i

i J

itself splits into (1,4) and (2,3) blocks.

Z(p) =

YII (P)

YI4 (P) leia2 p

Y41 (p)

Y44 (p)

and reconstruct the deformation theory in It is known (see [ 3]) for

2x2

matrices

(4.11.4)

A+

2x2

We set

matrix form.

that the deformation equations (4.9.10) and (4.9.11)

reduce to the following fifth Painlev~ equation

d2y = (i + 1 )(~)2 i d_x +_~21)2(~y~) dt 2 2y y-i - t dt t

+ ~ty + ~y(y+l) y-i '

by setting I 11+z (4"ii'5)

A+ = [ u-i z

u(12-11-z)] 12_ z

]

,

A

KI-li-z

v (-
v- 1 (KI-II-~]I-Z)

K2-12+z

=

-i t=2p(al-a2) and y=u v. Here (11,12) , (~I,N2) and (KI,K2) are the exponents of formal monodromy at p=ip, -ip and o% respectively. The parameters ~,~,y,~

491

is given by

(4.11.6)

i 2 e = ~(KI-%I-P 2) ,

In the present case

(4.11.7)

o~ :

i

~

,

i. % .2 8 = - 2[
~ Y = -I+%1-%2+PI-~2 '

%1 = ~2' %2 = 0, ~i = ½' ~2 = 0,
B :

-

i

~- ,

y

:

0

,

cS :

-

i = - 2"

Hence we have

i

~- .

We note that (4.11.4) is rewritten into the following first order system.

(4.11.8)

t ~

= -ty-2z(y-l) 2 -(y-l)(-91Y+V2+~3) ,

dE t ~

=

-yz(~l-z)-y l(92-z)(~3-z).

where

Vl =
~3 = K%-%1-~i"

In the present case

i

~I =

is given by

z

(4.11.9)

~t log T2 = ~ + ~t(-2z2-yz(21----z) + y

If we set

O = 2t d

-i

i

z(~z)).

log Y2, it satisfies the following second order non linear

equation.

(4.11. lO)

1 - ~do.(l (t d2----q~)2. = (O - t ~dff + 2 (dq. ~ ) 2.2 ) + 4..do. ( ~ ) 2 (~ ) ~ + ~dff ). dt 2

References This review is based on

[13 [23 [33

M. Sato, T. Miwa and M. Jimbo, Holonomic quantum fields V, RIMS preprint 267 (1978), to appear in Publ. RIMS, Kyoto Univ. M. Jimbo, T. Miwa and M. Sato, Studies on holonomic quantum fields XV, Proc. Japan Acad. 55 A (1979) 267-272. M. Jimbo, T. Miwa, Y. Mori and M. Sato, Density matrix of impenetrable bose gas and the fifth Painlev6 transcendent, RIMS preprint 303 (1979), to appear in Physica D.

In [2] the correlation function for the non-linear Schr~dinger equation with finite particle density at the infinite coupling is studied in detail. See also

[4]

H. G. Vaidya and C. A. Tracy, One particle reduced density matrix of impenetrable bosons in one dimension at zero temperature, J. Math. Phys. 20 (1979), 2291-2312.

2-'

RECENT RESULTS

FOR THE PLANAR ISING MODEL by D. B. ABRAHAM

Theoretical Chemistry Department University of Oxford I S o u t h Parks Road Oxford OXI 3TG The planar Ising ferromagnet theory of phase transitions

since the pioneering work of 0nsager

who showed that the phenomenon from Gibbsian

statistical

The burgeoning

has played a vital role in the

mechanics without

extra ad hoc assumptions.

of the study of critical phenomena,

largely a result of

the work of the King's

college

the scaling hypothesis

which can be subjected to rigorous

extensions

of 0nsager's

that the enhanced a length scale

[3].

to

tests using

It has long been realised

typical of the critical region occur on

~ , called the correlation

the temperature the limit as

group [2] led, amongst other things,

original work

fluctuations

[I],

of change of phase could be obtained

+ , Tc T @ Tc _

length, which diverges as

being the critical temperature.

~ ÷ ® , the lattice

In

can be replaced by a continuum;

the

models thus obtained turn out to be Euclidean quantum field theories, one of which,

obtained as

recent work,

four separate

functions theories.

T-) T c - , has symmetry breaking. groups

~,5,6,7]

In

have obtained the n-point

both for the Ising model and for its associated

field

Other advances have focused on boundary perturbations

restricted geometries,

leading to refinement

of ideas on phase separation

of scaling theory

and surface tension

in [~

and

[9].

In the following, the approach used by the author to investigate the above problems will be given.

It relies on the transfer matrix

and theory of spino~ @igebras. Definition

of the p l a n a [ Ising ferromagnet:

Consider a subset 1 ~ m ~ M }. variable ~(r) ~ r

a(~) = ±I. each

A

of Z 2

of the form

Assign to each point of r

A configuration

in A , denoted

{0};

~ on

A = {(n,m);

of Z 2

1 ~ n ~ N

a classical

spin

ANM is an assignment

of

the set of all possible

m

configurations energy

is the configuration

space denoted

{-I, I} NM.

An

,

493

E

({a})

= - [

J(i-j)

a(i) a(j)

li-j I=I -H ~ a(i)

-

i is associated couplings

with each configuration;

and the

H

and

of perimeter

points

of

given by the

canonical

H(i)

A .

where ture label

The statistical

({a})

Z , called the partition

system is supposed T

and

given

The Transfer

(1)

J(x)~Oare fields;

called ~ e ~ ¢ o ~ ~A

mechanics

is the set

of the model

is

kB

normalises

(2)

the measure;

the

with a heat bath at tempera-

is Boltzmann's

the boundary

constant.

fields

H(i),

The

ic~A

Matrix

The essential

idea behind the transfer matrix

the exponential

in (2) and to write

each column

A

of

each adjacent

correlation

in terms

is to factorise

the sum over spin variables

as a matrix product.

pair of columns

The n-point is defined

a(i)

= Z -1 exp - BE ({a})

function,

where

specifies

H(i)

measure

to be in equilibrium

8 = I/kBT

B(A)

the

are magnetic

probability

PA, B(A)

[ iE~A

The matrix

is called the transfer function

associated

< >

with

matrix.

for the probability

of the expectation

in

measure

(i)

by

n

g((r) n

I A, B(A))

--

Here

rj : (xj, yj)

in Cartesian

: < n a(rj)> 1 A,B(A)

form and the notation

(E)n : (r I . . . . . will be used. toroidal

It is a standard

boundary

conditions

G((r) n I ^, t)

= Tr

(S)

result

r n)

(4)

that, 'on a lattice

A

with

denoted

B = t, i.2.1) may be written x2 -x 1 V xl ax(yl ) V aX('y,2) . %

aX(yn_l ) V xn- Xn-laX(yn)

VN - xn J/I/Tr V N

(5)

where

v = v2½ v I v2½

(6)

with M

V2: exp K 2

~ 1

ox(j)

qx(j+l)

(7)

494

and

,

Vl= The parameters

(2 sinh 2K1)M/2ex p - K 1

M

[ oz(j) 1

(8)

in the above are K3•

=

8J

j '

j

=

I

,2

(9)

and exp - 2K The

operation

= tanh K

in (2.8) is an involution.

duality

transformation

for this

defined

as shown i n t a b l e

It is involved

model[~.].The s p i n o p e r a t o r s

in the

x(j)

are

1.

Table site in column

(10)

1 : Hilbert

1

2

space

. .

j

" " U~

°~1

j÷l .

M

~(j) f~Cj)

U

In the above ~

~J

is a 2-dimensional

complex

Hilbert

~

space ~ : x,y,z

and

ox U Notice

°0 ] oy:I 0

o

j

= -0"z that if we define

f+(j) then the

= (fx(j) f(j)

[ f(j), The basis

+ i fY(j)~2

are Fermi operators,

for each

f(k)]+ is

= 0, [ -1 )

that is

[ f(j), f(k)%]+ = ~]k and

1 J

1

495 Table 2 shows the relationship of the steps followed in solving the model. Table 2 : Orsanisation of solution Transfer matrix represented by operator

V

on

~M Jordan-Wigner Transformation

Reduction to Quadratic form in Fermion operators I Symmetry Reduction to canonical form

J

Spectral decomposition of V in (5); n-point function written in terms of matrix elements of ox(j) with ei~envectors of

V

t

Representation of matrix elements as generalised Pfaffian

Remarks:

1.

The Jordan-Wigner transformation is shown in line 3 of table 1.

2°

The symmetry used in step 2 is invariance of

V

under unitary

transformation by P : exp Since

p2 = 1

projectors

i 2 1

we can project P± = (1 ± P)/2. Io>

, f+(j)lo>

~M

(11) on to subspaces ~M(±)

by

Using the usual basis ,

(121

496

f C i where 2)1o> = 0 ~ j , we see that ~ M ( + ) | respVLM(-) | contains k 2 states with even (res~ odd) number of fermions.

3.

2~

3:S~mmetr ~ :

o~(j)

here we use the translation

o~(j_l)

~t =

j

2,.'"

=

defined by

M

(13)

a~(1) ~t : a~(M) The non-local character of the Jordan Wigner transformation that

T may well act on states of (12) in a curious way.

means

Nevertheless,

let us define Ft(k)

= M-½ ~ eik] ft(j) 1

(14)

Then we have the Lemma:

expikjM = (-1) r-1

If

j : 1,

r

(15)

then T with

FT(k 1)

FT(kr)lO> : exp(ik)

r k:lkj

F % (k 1)

F#(k r)

Io>

(mod 2~)

(16) ' (17)

1 Sub-remark8 :

I. When

r

is even we use antiperiodic wave numbers

kj

2. The proof of this is a special case of the implementation boundary conditions 4o

Step 3 :

Fermi operators

of¢~c[~c

in the Bethe ansatz.

Note that from (14) and (15) the

F(k)

are also

in the sense that

provided

IF(k1) , F(k2) ~+

=

0

(18)

[F(kl)

=

~(kl, k 2 )

(19)

=

1

, F'(k2) ~+

expi M(k I + k 2)

But F(k I) ,

2

F(k2)% ~

i(k 2 - k 1) M(e if

expiM(k I + k 2)

=

- 1 .

-I)

The latter rather subtle effect is

(20)

497

central 5.

to our development.

Step

3 continued

:

write V : V+ P+ Then,

for instance,

with

even

exp(isM)

(1~ and (19.

V2(8)

i

eos8

- 1

3

step

and

[F(8)tF(8)

sin 8 [F(-B)#F(8) ÷

V1(8) = exp-2K 1 Completing

=

VI+ (8)

V2+(S)½

(23)

Now

= exp2K 2 -

(22)

: V2+(8)½

V+(8) using

(21)

P_

V

: >
V+ where

H

+

*[

F(8)¢F(6)

just requires

+ F(-~)%F(-8)

-

-1]

F(8)F(-8)]

+ F(-g)*F(-8) a canonical

-1

]

(2g)

transformation

of the

type

(25)

G±(k) t = cos e (k) F(k) % - i sin e (k) F(-k) where

exp(ikH)

e2ie(k)

= [~-]½

ei~

= ±1

l ei~m e i~

the correct 2e(~)

:

branch of the square

0 ,

mod 2~ .

A = coth K 1

indexes

G(k) +

1 A.J)(e i

-!)

and

--i I

(26)

A-1)(e i~ - B)~ |

root being

The parameters

coth K 2

A

such that and B

B = coth K 1

are given by tanh K 2

(27)

Using the notation n

I(~)n>+-

:

1"[1G-+(~J)t I°>+-

(28)

with G_+(~) the results

Io>+

:

0

in table

(29)

3

can be derived

498

Table 3 : eigenvectors of V T>T c

T
I0>+

t ~

I(~)2n+i>-

10>+

I0>_

I(~)2n>+

I(')2~

decreasing eigenvalue

l(e)2n>+ n

Q

l(~)n>

= exp (- 7. Y(ej)) l(.)n > 1 n

[(~)n

cosh

>

y(~)

Remarks

on

= exp (i [ mj) 1

= eosh

table

2,

2K 1

eosh

(30)

l(~)n> 2K 2 -

(31) sinh

2K 1

sinh

2K 2

G((E) n It) =

ml,..mn_l =0

•

n-2

FX((el~)ml) 7

n-1

(32)

( 5 ) gives

I~I d(mi)ml '''d(mn-l)mn-i [ - I 1,, 1 (2~)mjmj!

.

FX((el~j) mj

(eimj+l)mj+ I) Fx( e l~n " l)mn_i

m~

exp

[-y(mk£)(X£+l-X £) + i(Y£+l-y£)mk£ sgn ~] ~=1

m

continued:

Step 4 : Taking the thermodynamic limit N,M-+- in

6.

cos

(33)

k=l

where the matrix elements are Fx [(m)ml(e)n] =

+< (~)mloX(1)l(m)n >_ +

(34)

499

Determinati£n of matrix elements Note that (20) and (26) imply that _~ G+(m)

:

[~

~ ( e i m ')

i

~

d~

I

~.

e

- I

+lJ G (.,)

G (-~')+ ~(e

im )

(35)

where (e Im )

:

(36)

exp - 2i8(m)

Using the result that <(mlnl

G+(e)lo> +

=

(37)

0

and the notation _<(m)nlO>+

(38)

F ((m) n)

:

we derive the recurrence •

n

(yF) (elm) n

=

[ 2

(39)

(-1) j h(ml,m j) FCAlj(m n )

where h( u,v

)

=

e L(u+v)

2

[

1

-1] (40)

e

i(u+v)

-

I

(eiv)

and (YF)(eim)n = ~----

I"

dm e i(~'ml)- i

F(e i~, eiW2, ... ei~n) Finally, for index sets AI (~)J

O ( e i~1)

(41)

I C J, =

(~)JII

(42)

50O

Thus we have to invert Y ; this can be don@ using the theory of Wiener-Hopf factorisation and semi-infinite T6plitz forms [ ~ To im determine FX((e )nl(elm)m), use (35) and note the relationship (from table I) ox(1) = f(1) + + f(1) (43) The results are collected together in table 4. Table 4 : Matrix elements for V T > T~: F((ei~)m{(ei~)m,2n)

in : ~ (-l)Jf>(~l,~j)F(Alj(ei~)ml(el~)m, 2n)

(44)

2 with

ei(el+mj) (g>(~l)g>(-ej)+c(j)g>(ej)g>(-ei))

f>(ml'mJ) =

(45)

ei(ml+eJ )- i

where E(j) : I (resp. -i) if m < j < 2n (resp.

I < j < m)

(46)

and g>(~) = [(ei~-A)(eim-B-l)]-½

(47)

boundary condition is F(¢) = (i - (sinh 2K 1 sinh 2K2)2)1/8/cosh K1 *

(48)

FX((ei~)m{(eiW)m,2n)

(49)

= 0

•

2n+l

FX((el~)m{(eiW)m,2n+l)

h>(e) =

=

[

•

(-l)Jh>(mj)F(Aj(ei~)ml(el~)m,2n+l

j=l

(5o&i

(B/A) ½ eie / [ ( e i~ - A - 1 ) ( e i~ - B~ ½

(50hi

T < Tc: FX((ei~)ml(eim)m,2n+l)

= 0 (51) 2n FX(( eim)m (ei~)m, 2n) : Z (-l)Jf<(~l,mj)FX(Alj(ei~)ml(el~)m, 2n (52) 2 with i(~l+~ j ) f<(~l,mj ) = e

(g<(ml)g<(-~j)+E(j)g<(~j)g<(-ml)

(53)

ei(ml+mj )- 1 g<(~) = [(ei~-A)/(eim-B)]½

(54)

Fx(¢) : m* : (I -(sinh 2K 1 sinh 2K2)-2) 1/8

(SS)

¢(j) is given by (45) aDove.

501 Graphical Representatio n By combining (33) and the results of table 4 a graphical representation of the n-point function can be obtained.

Consider

(n - I) disjoint subsets of vertices, denoted Vj , of the vertex set

V

of a graph ~

: {V, E}

by the integration variables that

Vj = ~ is also allowed. I : 2~

Wk~

exp

~

=

En_ I

~,

k = 1 . . . . mj

E

U

; notice

There is an associated vertex weight

Y(Wkl)(x£+ 1 -x£) + i(y~+ i - y~)

The allowed edge set E

The vertices of Vj are labelled ~jk e ~ , ~

sgn £

(56)

can be partitioned in the form (Ej U Ej ,j+1)

(57)

!

where the edges of

Ej connect vertices in

Ej,j+ I connect a vertex in

Vj to one in

vertices of an edge be m I and ~2"

Vj

whereas those of

Vj+ I.

Let the terminal

Then the edge weight is given by

e(~1,m 2) : f ( ~ m 2) from (45) and (53) of table 4, with ¢(2) = I (resp. -I) if e g E j , j + l ( r e s p. Ej) for any j The rules for drawing the graphs are as follows: T < To:

(mj + mj+ I) for j = I, .... n-2 and m I and mn_ 1 must be even; n must be even (for toroidal boundary conditions).

Allowed graphs are unions of disjoint cycles such that each and every vertex has valence 2.

With + boundary conditions any

n

is

allowed. T > Tc:

both (mj + mj+ I) for j = I,...n-2 and m I and mn_ 1 must be odd~ n must be even. Each matrix element produces

one termination with weight h>(m). Graphs are unions of disjoint chains and cycles compatible with these rules. Remarks

I.

The integrals over

are in general of principal part type ;

recalling the Poincar~ - Bertrand formula, the order must be specified;

given any chain of edges from

Ej,j+ 1

, the principle

part integrals are ordered from either end of the chain.

This has

important consequences for truncation. 2.

The scaling limit:

exp(-xy(m)

+ iym)

As we have seen, each vertex carries a weight

where

Take the formal limit

x

and

y

are relative displacements.

x,y ÷® , y(o) ÷ o

such that ~(o)x = s I ,

502

~(o)y then

: s 2 are p

fixed.

(-®)®)

in the

exp

- s I (I + p2)½

form

for p a r t i c l e

as p a r t i c l e s 3.

The

and the

+ i p s2

mass

produced

I. by

which

We

can

scattering

representation

n-point

in t h e

for T > T c is s t i l l

W u at el. function

transcendent notionally 5.

~0]

to the

dimensional

see

described

here

ground Bose

state

gas w i t h

is Euclidean

fermions

in the

theory

spins. to o b t a i n

an

functions

is to o b t a i n

and t h e r e b y

field

:

as in

a similar

to i n v e s t i g a t e

for T > T . c

The n a t u r e

of the

a mystery.

limit

kind.

et el.

p : ~I¥(o)

of the n - p o i n t

a remarkable

scaling

of the t h i r d by S a t o

at zero

obtained

in the

The m e t h o d

tions

field

of the

question

functions

singularity

weight

can be a p p l i e d

A challenging

by

appropriate

off the

compact

analyticity

4.

think

more

for t r u n c a t e d

point

vertex

theorem

[7 1 .

number ~

is the

cluster

, [6J and

form

limit

the w a v e

linked

apparently [51

Replace

This this may

X - Y infinite

representation

in t e r m s has

Volume

been

two

generalised

.

be a p p l i e d chain

of the

of a P a i n l e v ~

with

problem

6 function

trivial

and to the

modificaone

repulsion.

References I.

2.

3. 4. 5. 6. 7. 8. 9. ]0.

L. Onsager, Phys. Rev. 65, 117 (1944) ; B. Kaufman and L. Onsager, Phys. Rev. 76, 1244 (1949) ; T.D. Schultz, D.C. Mattis and E.H. Lieb, Rev.Mod.Phys. 36, 856 (]964). C. Domb, in Phase Transititions and Critical Phenomena Vol.3, ads. C. Domb and M.S. Green, M.E. Fisher, 1974 : Academic Press, London, New York. Lectures in Theoretical Physics VII C, ]964 : Boulder : University of Colorado Press. Rap. Progr. Phys. 30, 615 (1967). C.N. Yang, Phys. Rev. 85, 808, 1952. D.B. Abraham, J. Statist. Phys. 19, 349 (]978) and references therein. D.B. Abraham, Phys. Letts. 61A, 271 (1977) ; Commun. Math. Phys. 59, 17 (1978), 60, 181, 205 (1978). R.Z. Bariev, Phys. Letts. 64A, 169 (1977). B.M. McCoy, C.A. Tracy and T.----T.Wu, Phys. Rev. Letts. 38, 793 (1977). See work by Sato, Miwa and Jimbo reported in this volume. M.E. Fisher and H. Au Yang, preprint. D.B. Abraham and P. Reed, Phys. Rev. Letts. 33, 377 (1974) ; Commun. Math. Phys 49, 35 (1976) ; D.B. Abraham, to be published (I980). E~. Barouch, B.M. McCoy, and T.T. Wu, Phys. Rev. Letts. 3~I, 1409 (]973) ; T.T. Wu, B.M. McCoy, C.A. Tracy and E. Barouch, Phys. Rev. Bl3, 316 (1976) ; T.T. Wu, C.A. Tracy and B.M. McCoy, J. Math. Phys. 18, 1058 (1977).