Lecture Notes in
Physics
Edited by J. Ehlers, M0nchen, K. Hepp, Z~irich R. Kippenhahn, M0nchen, H. A. WeidenmLiller, H...
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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
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
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 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) (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 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 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 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,
--~-~ = 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
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__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 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 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
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 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 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 ) 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)=ro (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
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, .
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)(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, -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 = (f 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
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
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)l2II
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,)
= .
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
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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( ~ 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 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, 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 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 ]
= 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(°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 : 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(@)= -+
->
+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 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.
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415
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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" (-~
=
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< ... ÷ 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).
= ~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
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 < "'" )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 = 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. (*)
al0 ,
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~
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
As an illustration first let calculated as
(3.2.2)
n=l.
In this case (3.2.1.) is explicitly
m -i =~-l[ ] WF(X) ~ 2 [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
)
negative) frequency part, the expectation value
contains only the positive (reap.
(reap. )
is analytically prolongable to the upper (reap. lower) half Euclidean plane 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 /< ~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)
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)
= ~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 " 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
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 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') = /T n , _+ Wjj,(p,p') = -/T n , W+,T
Wjj,(p,p') = -.
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),
= ' 0 = = 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
+-~
.. ~(+) ~'2 .(-) " . ")°n> . . . . -I i< ~l
- i 8a ~ -2L < ~1 2 "'en >
i
n
....
i
(4.10.3)
YI
i
>
0
"""
"
n
.. _i
" n
_(+). (+)
-i
-in
Tn = .
-) ""~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
, 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
: >+-
:
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+
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 +
=
(37)
0
and the notation _+
(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
