Borel-Laplace Transform and Asymptotic Theory: Introduction to Resurgent Analysis [1 ed.] 084939435X, 9780849394355

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Bo rel-Laplace UW:

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3 06248

= oO & a Co) = S = £> 2 ) Library

001259177 L

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Borel-Laplace Transform and

Asymptotic Theory Introduction to Resurgent Analysis Boris Yu. Sternin -

Professor of the Moscow State University Doctor of Science in Physics and Mathematics Moscow,

Russia

Victor E. Shatalov Professor of the Moscow State University Doctor of Science in Physics and Mathematics Moscow, Russia

Boca Raton

CRC Press New York London

Tokyo

Library of Congress Cataloging-in-Publication Data

Catalog record is available from the Library of Congress. This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage or retrieval system, without prior permission in writing from the publisher. CRC Press, Inc.’s consent does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from CRC Press for such copying. Direct all inquiries to CRC Press, Inc., 2000 Corporate Blvd., N.W., Boca Raton, Florida 33431. © 1996 by CRC Press, Inc.

No claim to original U.S. Government works International Standard Book Number 0-8493-9435-X Printed in the United States of America 1 2 3 4567890 Printed on acid-free paper

UWEC Niclatyre Lil

PREFACE

The resurgent function theory introduced by J. Ecalle in the early 1980s is clearly one of the most interesting and beautiful theories in the field of mathematical analysis. The main aim of the resurgent function theory is to work out a resummation method for divergent power series and to apply this method to various mathematical problems. From this viewpoint, this theory had to attract the attention of specialists on asymptotic methods. Namely, it is well known that asymptotic series are, as a rule, divergent, and one could obtain new and interesting results in this field with the help of the above mentioned resummation method. Hence, there is nothing astonishing in the fact that some specialists in the asymptotic theory of differential equations turned their attention to the resurgent function theory. The matter is that the creation of new refined asymptotic methods is at present one of the most urgent tasks of the theory of differential equations. Actually, a powerful theory of global asymptotic expansions of solutions to differential equations was created in the middle of the 1960s (see [112, 66, 114] and others). This theory allows us to construct asymptotic expansions of solutions to differential equations up to an arbitrary power of a small parameter. After this, the problem of creating a theory of asymptotic expansions with exponential accuracy became one of the primary tasks of the asymptotic theory. The analysis of the problem shows that it is natural to construct the theory of asymptotics of this kind in a special function space, that is, in the space of the so-called resurgent functions. The exact definition of this notion will be given. Now we remark only that resurgent functions are ramifying analytic functions with endlessly continuable Borel transforms. Clearly, for effective investigation and usage of the notion of resurgent function, it is useful to have in hand the corresponding harmonic analysis; that is, one should be able to investigate spectral characteristics of the objects under consideration. The tool for investigating spectral properties in the resurgent function theory is exactly the above mentioned Borel-

Laplace transform (as well as generalizations of this transform to the spaces of analytic hyper- and microfunctions). In particular, this transform allows us to carry out the microlocalization procedure, that is, to “split” the Borel transform of the resurgent function in question into distinct microfunctions, each corresponding to its own component of asymptotic expansion. We remark that the classical WKB theory differs substantially from the analytic theory of asymptotic expansions with exponential accuracy. The matter is that the analytic continuation of asymptotic expansions (for example, of WKB type) does not in general coincide with the asymptotic expansion of the analytic continuation of the function in question. The latter can have jumps when encircling focal points (the Stokes phenomenon). Thus, in the investigation of asymptotic expansions of ramifying analytic functions, two types of monodromies arise: the real monodromy (that is, the monodromy of the considered resurgent function itself) and the formal monodromy (that is, the monodromy of asymptotic expansion of this function). The nondirect connection between these two types of monodromy is described by the connection homomorphism. This homomorphism allows one to construct the global picture of asymptotic expansion of the function in question using the above mentioned microlocal elements. The aim of this book is to get the reader acquainted with methods and ideas of the resurgent analysis; we do not suppose any preliminary knowledge in this field. As a consequence, the book goes as follows. We begin with the Introduction in which we consider simplest examples from the asymptotic theory of differential equations. By these examples, we try to show how main notions of resurgent analysis do arise and what effects do take place when constructing asymptotic expansions with exponential accuracy. Chapter I is aimed at the construction of the “spectral analysis” of the resurgent functions theory — the Borel-Laplace transform of ramifying analytic functions. We introduce also the notions of analytic hyperfunction and microfunction including the notion of generalized hyperfunction necessary in the resurgent function theory (as far as we know, this is the first appearance of this notion, at least in the explicit form). Then we investigate the Borel-Laplace transform on the corresponding function spaces. In Chapter II, we consider the resurgent analysis itself. We present the notion of resurgent function of several independent variables (which

was introduced by the authors in [153, 158, 161]) and investigate the main properties of this notion including the above mentioned connection homomorphism. This chapter contains a lot of examples illustrating the introduced notions. Finally, Chapter III contains some applications of the resurgent analysis to the asymptotic theory of differential equations. Clearly, it is far from realistic to review within the framework of a single chapter all the variety of applications of the resurgent analysis to mathematical problems. Thus, this chapter is aimed only at illustrations of the above described technique

to some (though very serious) mathematical problems. To conclude our short Preface, we remark that the resurgent functions theory is strongly connected with the theory of differential equations on complex-analytic manifolds developed by the authors (see [157]). A frag-

ment of this theory (the so-called 0/0s-transformation widely used in the resurgent functions theory of several variables) is presented in the Appendix to this book. We hope that this book will help mathematicians working in different fields of mathematics to get acquainted with a new and interesting branch of the asymptotic theory — with the resurgent functions theory. Acknowledgments. We are deeply grateful to Professor Frédéric Pham who attracted our attention to the resurgent functions theory. The discussions with him were always of extreme use for us. We are also very grateful to Professor Jean Ecalle for the discussions of fundamental questions in the resurgent functions theory. We are thankful to our friends and colleagues from the Laboratory of Jean-Alexander Dieudonné (Université de Nice-Sophia Antipolis), especially to Eric Delabaere, Francine and Mark Diener, and Herve Dillinger for a lot of interesting conversations which took place during our visits to the University in 1992 and 1994. Authors

Moscow, March 15, 1995

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Contents

INTRODUCTION. Resurgent Analysis in the Theory of Differential Equations 0.1 Singular Points of Ordinary Differential Equations ..... O.¥:1 Classification of singular points... .... ©... .. «> 0.1.2 The Borel-Laplace transform ............. UL d athe Ruler Cxalmpicee to . +4, a he ae oe ee: ke 0:2" Equations onlnfinite Cylinder... ..+... 5. eo tl. ee 0.2.1 Modification of the Borel-Laplace transform... ..

0.3

0.2.2 Asymptotics of functions of exponential growth . . . 0.2.3 Asymptotic expansions of solutions. ......... Semi-classical Approximations. ................ 0.3.1 WkKB-expansions. Elementary theory ........ 0.3.2 Exact WKB-approximation (quantum oscillator) . .

0.3.3 0.3.4

Asymptotics at infinity (the Airy equation) ..... Concluding remarks SY¢:P20aqa. (44k has 2

CHAPTER I. Borel-Laplace Transform 1.1 Entire Functions of Exponential Type ............ LLcl ae a MaGCUDILIONSs tm seob4n tener as rola bi ~ BEE 1:1;2The Borel-Laplace transform .. .0cecueieh) < k-s © > + Pee eK AI eS ae re re od te Ee ee ge 1.2 Hyperfunctions with Compact Support. ........... 1.21 = Mat GCUNIGIONS Ser ta fe ae 8a ee a. oye 2 1327 Nhe Borel-Laplace tYAnsionin cs 1 OES Bh ONE Fey0)OLR Ae Oe ks a 1.3 Hyperfunctions of Exponential Growth. ........... 32) ON OCNDISIONS SE .o. crete eet eee as a cas #:3:2° “The Borel-Laplace transiorm, «=. -. 1. The asymptotics of a solution to such an equation can again be represented as the sum of terms having the form shown in the last column of the table, but it differs from the previous one by the fact that the corresponding series are, as a rule, divergent. Our starting point will be the

4

INTRODUCTION.

Resurgent Analysis

consideration of this last (but not the least) case. For simplicity, we restrict our considerations by the case of a simple irregular singularity. The Euler example

After what was told in the previous subsection, the task of computing the asymptotic expansion of a solution to a differential equation

d Pr («.2*=) ¥=0

(0.1)

near its irregular singular point looks rather simple. Namely, to construct such a solution one has to search for the asymptotic solutions of the form yx el D5 an r*

(0.2)

k=0

for some complex numbers w and a,x, k = 0,1,2,....

It can be expected

that the numbers w and a, are determined from the equation (0.1) by some computation procedure and expansion (0.2) will give an asymptotic solution to equation (0.1) near the origin. Unfortunately, the reality is much more complicated than such a simple and clear scheme.

To find out the reasons due to which expression (0.2) cannot be an asymptotic expansion for solutions to equation (0.1) (at least, globally), we consider the simplest example of an equation of the type (0.1) which is

often used in the literature on resurgent functions for illustrations — the

Euler example”. Let us consider the equation

—2’y +y=2

(0.3)

on the complex plane C with the coordinate zx. First of all we shall try to carry out the simplest scheme described above, that is, to search a solution

to equation (0.1) in the form (0.2). To begin with, we shall search for a particular solution to this equation in the form of (formal) power series y2 Sere

(0.4)

k=0

Substituting the latter expression into equation (0.3) we obtain co

co

= wD (k —1) Op—12* + = ape* = x. k=2

k=0

?On the contrary to the above considerations, we consider here a nonhomogeneous differential equation. It is not essential since all solutions to equations (0.3) are at the same time solutions to the following homogeneous equation

—z3y" + (2 —2”)y—y=0.

Section 1. Singular Points

5

Hence, for series (0.4) to be a (formal) solution to equation (0.3), the coefficients a, have to satisfy the following recurrent system

a

= 0, a4 =1, og = (K—1)an-1,

k > 2.

Evidently, the solution of the latter system is a, = (k-1)!

and we obtain the following formal solution to equation (0.3):

yz > (K- 1) ic".

(0.5)

k=1

Certainly, the obtained formula does not determine a solution to equation

(0.3) since the series on the right in (0.5) diverge. But, perhaps (0.5) gives an asymptotic expansion of a solution to equation (0.3). Let us try to investigate this question. It is not hard to construct (with the help of variation of parameters method) the formula for the general solution to equation (0.3): z

1

y(x) = Ce-'/* — e-/* [ela

(0.6)

with an arbitrary constant C. To obtain the asymptotics of the second term on the right in the latter formula, we shall distinguish the two following cases. Case 1. Let us suppose that Rez > 0. In this case the exponent e~1/ decreases as x — O and the asymptotic expansion of the second term on

the right in (0.6) can be obtained simply with the help of integration by parts:

| —el/z /el/¥ 1dy y x

K

K

>> (k= Vick - e/* 5° (k - 1)! k=1

k=1 z

ere

Wileq i= /yK-lel/¥dy.

(0.7)

1

The second sum on the right in the obtained expression is evidently an exponentially decreasing function in the half-plane Rez > 0. The last

term has in this half-plane the order O (x*) which can be easily derived with the help of the integration contour drawn on Figure 0.1. As a result, we obtain the following estimate K

y(z) — >> (kK -1)!2* 0.

for any integer K. Clearly, this estimate is uniform in any sector

1

T

If ={-S +e 0. The constant C may certainly depend on e. Thus, we have obtained the following result: Series (0.5) are asymptotic series for any solution to equation (0.3) in

the sector I;. Case 2. Let us now consider the half-plane Rez < 0. Here the situation

is quite different. Namely, in this region both terms in the right-hand part of (0.6) are of equal order as x — 0. Besides, the above method cannot be directly applied to computation of the asymptotic expansion because

the second term on the right in (0.6) has exponential growth. Therefore, before calculating the asymptotic expansion we reduce (0.6) to the form y(z) = Cie"* -e

1

=28

3° dy,

=z

7(z) where the contour 7(z) is drawn on Figure 0.2. Now, quite similar to the considerations of Case 1, we obtain

K y(z) = Cye7= + ey. (k= 1)i k=1

0 (ee)

(0.8)

Section 1. Singular Points

t

Figure 0.2. Integration contour for Rez < 0.

uniformly in the sector iw

I=

us

ee

3a

ee

oe

:

Thus, the following result is valid: In the sector I; the principal term of the asymptotic expansion of func-

tion (0.6) equals to Cye~= with some constant C, depending on the considered solution. Besides, series (0.5) are asymptotic expansion for the difference

ay]

y(xz) — Cye™=. Let us summarize our considerations.

The solution y(z) to equation (0.3) is a ramifying analytic function with the asymptotic expansion of the form (0.5) in the sector J; and the asymptotic expansion of the form C) e~= in the sector I; (see Figure 0.3). What concerns the sectors J! and J;’ which are neighborhoods of the positive and negative parts of the imaginary axis, respectively, here both terms of asymptotics

Cye~= and )*(k—1)!2* k=0

have equal orders and no one of them is negligible.

8

INTRODUCTION.

Resurgent Analysis

Figure 0.3. Domains in the plane x.

We remark that (due to formula (0.8)) in the domain J> we have managed, except for the dominant component C, e-= of the asymptotic expanco

sion, to write down its recessive component

)> (k —1)!z*. It gives rise to k=0

the guess that in the sector J+ we also have the recessive component equal

to Cye~= and that in the regions J! and J!’ takes place just the change of leadership between these two components (the recessive one becomes dominant and vice versa). Such phenomenon is known in the theory of ordinary differential equations as the Stokes phenomenon. However, for such a treatment asymptotic expansions of the form

to be possible we must interpret the

co

y(x) ~ S* (k -1)!2* + Cye7* k=0

in the regions, where the second term on the right decreases exponentially. The latter, however, is not a simple task. Certainly, one can easily formulate in the precise terms that the first series on the right in the latter formula is the asymptotic expansion of the solution up to any power of z. However, to make the second term “visible,” one should extract the first term from y(z) and write down something of the kind

y(z) - = (k —1)!2* ~ Cye"1/?. k=0

(0.9)

Section 1. Singular Points

9

However, the latter relation is meaningless because the series in its left-hand part diverge and, hence, do not determine any function at all. To overcome this difficulty, one has to resummate the series on the left in (0.9), that is, to find (with more or less regular procedure) a function for which the series plays the role of the asymptotic expansion. If we denote by o the operator of resummation , then we can replace (0.9) by the relation

/= y(x) -o@ bs(k- via Ol e-1 k=0

which already makes sense. The mentioned resummation procedure can also help one in constructing exact solutions to differential equations, provided that this procedure is regular enough. Namely, suppose that we had constructed a resummation procedure o satisfying the following three conditions: 1°. The operator o determines a homomorphism of (some subalgebra of) algebra of formal power series in z to the algebra of analytic functions. 2°. The operator o takes convergent power series into their sums. 3°. The operator o commutes with the differentiation.

In this case, the function o[y] will be a solution to the differential equation in question for any formal power series 7 being a formal solution to this equation. In particular, we shall see below that the resummation of

power series (0.5) leads to the exact solution to the Euler equation (0.3). Thus, for investigation of asymptotic expansions of solutions to differential equations in the complex domain one needs to work out a theory of resummation of divergent power series taking into account the Stokes phenomenon. To analyze how such a theory can be constructed, we shall present another method of solving equations of the type (0.3) based on the so-called Borel-Laplace transform.

0.1.2

The Borel-Laplace transform

Here we recall some facts from the theory of the Borel-Laplace transform. Below we shall present the detailed theory of this transform; here we restrict ourselves to the formulation of main definitions and theorems in the simplest form we need at the moment. Let € € Cg. Consider a function F(€) which is holomorphic in the sector 0, < arg E < 02

which has not more than exponential growth in |€| in this sector:

|F(€)| < Cee*l, 6. +e 0. To provide the analytic continuation of the function y+(zx) into the left half-plane Rex < 0, we must rotate the direction of the integration contour to the angle x. The result, however, will be quite different for rotation of this contour in different directions. Namely, if the rotation is performed in the positive direction, then the analytic continuation of the function yt(z) will be given by the integral coe’*™

Atile*

[y* (z)]

=

at

Defers (=

es

0 (where by At we denoted the operator of analytic continuation to the left half-plane along any path coming above the origin). Inversely, if we rotate the contour in the negative direction, we obtain coe’*™

Aq [yt (x)] = -2mie/* + / ete 0 with the similar notation. The appearance of the first term on the right in the latter formula is due to the fact that, during the rotation, the integration contour intersects the pole € = 1 of the integrand giving rise for a term which is just a residue of the integrand at this point. Thus, we see that the Stokes phenomenon can lead to the appearance of multivalued solutions obtained by the Borel-Laplace method. Let us now try to understand how the above considerations lead to the method of resummation of the divergent series. To do this, we remark that

the solution (1 — €)~* to equation (0.14) can be obtained directly from the series (0.5) without applying the Borel transform to the equation itself. Namely, since

Ce" = bie

one can define the formal Laplace transform Lf of a (formal) power series in € by the formula

Ly

SS at!SSS kage? k=0

(0.16)

k=0

This transform is defined on the space of formal power series. The inverse transform is the formal Borel transform given by co

co

k

By Sat we ye as : k=0

(0.17)

k=0

From (0.16) and (0.17) one can see that the formal Borel transform accelerates the convergence of power series. Thus, if we apply this transform to

Section 1. Singular Points

13

(divergent everywhere) series (0.5), we obtain the series

By S Katt = ye k=0

(0.18)

k=0

which is convergent in the unit circle centered at the origin. The sum of the latter series is exactly the solution Y(£) to equation (0.14) obtained earlier by the application of the (not formal) Borel transform to the Euler equation:

So =¥O= k=0

% g

Thus, we can consider solution (0.15) to the Euler equation as a result of resummation of series (0.5) which is a formal solution to this equation. In this treatment the resummation procedure is a composition of the following operations:

1. Formal Borel transform. Application of this operation to (divergent) series (0.5) gives series (0.18) which converge in the neighborhood of the origin and, hence, determine a germ of a holomorphic function at € = 0. Here we use the acceleration property of the formal Borel transform. 2. Analytic continuation of the obtained germ Y(€) of a holomorphic function up to the function which is defined everywhere except for some discrete set of singularities. In the considered example the only singularity of the obtained continuation is the point € = 1, where the continuation has a polar singularity; as we shall see below the latter fact is quite occasional. 3. Laplace transform (not formal). The application of this transform to the constructed analytic continuation gives us (in the considered case) an exact solution to the Euler equation. Thus, if we denote by o the operator of resummation of divergent power series, we obtain

y(xz) =o bsnat =LoAoBy; k=0

‘ katt k=0

for the solution to the Euler equation. We remark that, in our case, the resummation procedure applied to the formal solution to a differential equation has led to the exact solution of this equation. Certainly, it happens not for every equation, because, as we shall see later, the theory of Borel resummation deals with asymptotics of analytic functions up to arbitrarily fast decreasing exponentials rather than with functions themselves. However, as it was mentioned above, if we want to obtain solutions to differential equations up to exponentially decreasing functions starting from formal solutions to these equations, the resummation procedure must satisfy rather strong requirements. In particular, it must be a homomorphism of algebra of divergent power series (or, possibly

14

INTRODUCTION.

Resurgent Analysis

of some its subalgebra) to the algebra of analytic functions of exponential growth given on some sectors in the complex plane centered at the origin. Later on, this homomorphism must commute with the differentiation operator. Finally, it is evident that the resummation procedure applied to convergent series must give the usual sum of this series as a result. As we shall see below, the described Borel resummation method satisfies these requirements.

Let us examine the requirements on the power series needed for the above program listed in points 1—3 above to be applicable to this series.

First of all, one can easily describe the set of power series which become convergent in a neighborhood of the origin after application of the formal Borel transform. For this it is necessary and sufficient that the coefficients ax of the series (see (0.17)) satisfy the estimates

laz| < CR*k!

(0.19)

with some positive constants C and R. The set of (formal) power series > a,2* with coefficients subject to (0.19) is known as the Gevrey class Gy of order 1. So, the first requirement needed for some power series to be summable is that it belongs to the class G,. Later on, if }> axz* € G;, then the formal Borel transform of this series determines a germ of a holomorphic function F(€) near the origin. The second requirement is endless continuability of this germ, that is, continuability up to (ramifying in general) analytic function having a discrete set

of singularities on its Riemannian surface®. Finally, to apply tained by the analytic has the power growth can be omitted if we to rapidly decreasing

the Laplace transform to the analytic function obcontinuation, we must require that this continuation at infinity. As we shall see below, this requirement construct the asymptotic theory of resummation up functions of exponential type.

We remark that, as we have shown, the Borel resummation allows one also to investigate the Stokes phenomenon. The above introduced scheme of resummation

method

can be illustrated with

the help of the following diagram.

>The precise formulation of the notion of endless continuability the reader can find in Chapter I.

Section 1. Singular Points

15

Power series

Formal

Formal

convergent

power

power

near origin

series in x

series in

Analytic

Analytic

functions

functions

in x

in €

Usual sum

The functions obtained with the help of the described resummation procedure we shall call resurgent functions. On the above diagram, a resurgent function corresponds to such formal power series in z whose Borel transform admits an analytic continuation up to an endlessly-continuable function of the variable € (see lower right corner of the diagram). Let us present some concluding remarks. First of all we shall note that the Laplace transform £ takes the convolution of the two functions in €

[F* aye)

€

4hF(€ — n)G(n) dn 0

into the product of the Laplace transform of factors. Therefore, the natural

structure in the space of functions f(x) (as well as in the space of formal power series in x) is the structure of algebra with respect to the usual

multiplication, and the natural structure in the space of functions F(€) is the structure of algebra with respect to the convolution. In particular, it means that the Laplace transform should be determined on the space of generalized functions (and even on the space of hyperfunctions) — for example, the unit element with respect to the convolution is the Dirac delta function. The introducing of the generalized functions allows us also to include into consideration power series with constant term (note that the above considered series (0.5) begins with the first power of z).

16

INTRODUCTION.

0.2

Resurgent Analysis

Equations on Infinite Cylinder

Here we illustrate the appearance of the main notions of the resurgent functions theory on the example of partial differential equation on the infinite cylinder, that is, on a direct product R x 2, where R is the real line and Q is some smooth compact manifold without boundary. The equations of

the kind were considered earlier by Agmon-Nirenberg [1], Sternin [145], [144] and others, who obtained asymptotic expansions for solutions to such equations with exponential weight at infinity. Since one of the main tools of investigation of this section is the BorelLaplace transform (see Subsection 0.1.2 above), we shall consider equations in special function classes adapted for application of this transform. As a result we obtain asymptotic expansions of a solution not only for real values of t € R but also in sectors of the complex plane C; containing the real axis.

It is worth noticing that terms of an asymptotic expansion of solution at infinity can be originated both by the equation itself and by analogous terms of asymptotic expansion of the right-hand part of the considered equation. We shall include in our considerations all these types of asymptotic expansions.

In order to simplify our presentation, we shall carry out all considerations on the example of the Laplace equation

on the circular cylinder

Ou

Ou

Ot?

= Ax?

at

t

f(a, t)

i

hea

C = R} x Si (so that the manifold 2 mentioned

above is acircle S!). Thus, we can suppose that all the functions considered below are periodic functions in the variable z with the period 27.

0.2.1

Modification of the Borel-Laplace transform

Here we shall describe the modification of the Borel-Laplace transform introduced in Subsection 0.1.2 in order to include into consideration not only positive but also negative direction on the real line R}. We begin with the description of the class of functions used in the investigations below.

Let a, and a_ be two real numbers such that a_ > ay. By Fa,,a_(S*) we denote the class of functions f(z,t) on C satisfying the following conditions:

Section 2.Equations on Infinite Cylinder

17

1. The function

f+ (x,t) = {ee

:a

extends to be an analytic function in the sector

S; aioe Gangetic} of the complex plane C;. Also, the function

0,

Se

t>0,

{f(z,t), t +00.

To conclude this subsection, we shall try to write down the representation of the function f (x,t) € Fa,,a_ (C) in the general case in the form of superposition of exponentials e“* over some subsets in the complex plane

C;. (We remark that formula (0.24) gives us an example of such a superposition using a discrete set of such exponentials.) To do this, we remark that, due to inequality (0.22) one can extend the function F(z,£) up toa

smooth function F* (x,€) in the domain 2_ (a_,e) which will satisfy the same estimate in this domain as € — oo. Applying the Stokes theorem to the integral (0.23), we obtain a representation

f(et)= /eff du(€), é,

(0.29)

where dy (€) is a (complex-valued) measure. The support of the measure dp will show which exponentials are included in the (given) representation of the function f(z,t). The asymptotics of this kind in cases when the exceptional set is not discrete are called continuous (see, for example, [135]). However, one must take into account that the choice of the measure dy is not unique: one can add to this measure any measure orthogonal to the space of entire functions (this measure has necessarily a compact support) without changing the function f (z,t) in the representation (0.29). So, the question arises what is the minimal support of the measure dy admissible in the representations of the type (0.29) of the given function f. It occurs, however, that there exists no measure with minimal support for given f except for the simplest case of functions with meromorphic Laplace image. Even in this simplest case, to obtain the representations with minimal supports, one has to replace the notion of measure in (0.29) by the notion of distribution (generalized function). Then formula (0.24) can be written down in the form (0.29) with the “measure” dy(€) having the “density” co

m,—1

see)

m()= > So (-1)"?4-5"(€ - &(z))k=

0.

To rewrite expansion (0.25), (0.26) in the form (0.29) with (in some sense minimal) support one requires the notion of microfunctions instead of distributions. We postpone the discussion of this notion until Chapter I. To conclude this subsection, we only mention that the support of the measure dy in the case of analytic functionals (that is, when the function F(z,€) can be continued up to the holomorphic function outside some compact K) can be chosen arbitrarily close to this compact. Similar, in

26

INTRODUCTION.

Resurgent Analysis

the case of hyperfunctions, this support is an arbitrary neighborhood of the union of K with some ray emanating from K in the direction of the negative part of the real axis.

0.2.3

Asymptotic expansions of solutions

In this subsection we investigate the asymptotic behavior of solutions to equation (0.20) as t > oo provided that the asymptotic behavior of the right-hand part of this equation is known. 1. First of all, the following result can be easily obtained with the help of the above introduced Borel-Laplace transform.

Theorem 0.1 [f the interval (a4,a_) of the real aris € does not contain integer points, then equation (0.20) has the unique solution from Fa, ,a_ (C) for any right-hand part from the same function space. We shall give here only a brief sketch of the proof. First of all, we apply the Laplace transform in t to equation (0.20). We obtain the equation

fal + €U = F(z, €) Ox?

i

,

(0.30) :

for the Laplace image U(z,€) of the function u(z,t). Evidently, equation

(0.30) is equivalent to (0.20). With the help of the Green function for an ordinary differential equation, the (unique) solution to this equation can be written down in the form z+2nr

cos(y —x — m)€& U(ag= |SBR a re,6a. nk

(0.31)

=z

Since, due to the assumption of the theorem, the domain 2. (a;, a_,€) does not contain zeros of the denominator on the right in the latter formula,

function (0.31) is holomorphic in this domain. Later on, it is easy to see that the kernel of representation (0.31) satisfies the estimate cos(y-—x—m)€

2 C

2€ sin n£

= I

in the domain 2 (a;, a_, €), so that the function (0.31) satisfies the estimate of the type (0.22) provided that the right-hand part F is subject to an estimate of the same type. This proves the theorem. Now let us investigate the asymptotic expansion of the solution u(z, t)

to equation (0.20) as Re t - +00 (for definiteness) in the case when the asymptotics of the right-hand part of this equation is known. We consider

Section 2.Equations on Infinite Cylinder

at

a rather simple, but quite representative example of the right-hand part

having the form

f(z,t)= {0, Here € (x),

Pe

e

(0.32)

(zx) are smooth 27-periodic functions in z.

First of all, let us compute the Laplace transform F (z,€) of this function. We suppose that there exist numbers a;,a— such that

f (x,t) € Fa, ja_ (C)

(0.33)

and that the interval (a;,a_) does not contain integer points. We remark that inclusion (0.33) will be valid if and only if the singularities of the function Fy (x, €) given by (0.21) lie outside the domain 24 (a4,¢) for all values of x. Since the integral (0.21) for the function f (z,t) given by (0.32) equals co

7, _ F—€(a)’ y (x) Fy (x,€) Se fe—té e ,t&(x) dt

0 this means that the set of values of the function €(z) is contained as a whole in Q_(a_,¢). This set is a closed curve in the complex plane C, which possibly has singularities at stationary points of the function €(z).

Evidently, the function F_ (z,€) vanishes identically for function (0.32) and, hence, we have Er, 2) ae Nee Zye) eee (ze) =

g(x) Gi e(z)a

Substituting the latter expression into formula (0.31) we obtain r+2nr

U(.8)

= sees

/ cos€(y

—x —7)

ely)

E— €(y)

dy

(0.34)

for the solution to equation (0.30). Let us investigate the singularities of the obtained function U (z, €). From formula (0.34) it follows that the function U (z,€) is a meromorphic function of the variable € for each given z if € belongs to the complement of the compact domain K bounded by the curve = €(z) mentioned above. It is evident since for such values of € the integrand on the right in (0.34) is an everywhere smooth function in z and, hence, all singularities of U (x, €) are determined by zeros of the denominator 2€ sin 7€. Moreover, the poles of this function located exactly at integer values of the variable € are not included into the domain Q(a;,a_,€).

28

INTRODUCTION.

Resurgent Analysis

Later on, to obtain the asymptotic expansion for the function u(z, t), as it was explained in the previous subsection, one has to deform to the left the integration contour in the Borel transform

u(t) = 5 ie€U (a,€) dé. ay

As a result, we obtain the following formula for the asymptotic expansion of the solution:

u(x,t) = > f (a6) de+ 1

> Res fe€U(x,6)].

t€

ae

té

(0:35)

k ;:

a=)

(that is, a sequence of functions f; (€) satisfying Ff;(€) = fj-1 (€)). To obtain the asymptotic expansion by smoothness when the point £ approaches one of the stationary values of €(z) (say, to &), one can deform the integration contour [(z) in (0.37) in the fashion shown on Figure 0.10 and compute the residue of the integrand at point 7 = €. The result is U(z,&) ~

2mi__

v(x ()) cos€ (x (€) — x — 7)

2€ sin 1€

€' (x (€))

up to a function holomorphic near € = €,. Thus, in a neighborhood of this point, the function U (z,€) has a finite-sheeted ramification and can be expanded into the Puiseux series

(¢-&)/" ge)== 2 b; Chapa a

O(7.6)

j=

0.39 (0.39)

— Ft

with some smooth b,’s. Here, I is the Euler gamma function and m is the order of zero of the function € (x) — €; at the corresponding stationary point.

3) The third (and the last) step in the investigation of the first term of asymptotic expansion (0.35) is to obtain an explicit asymptotic expansion

for the corresponding term of the solution u(z,t) using expansions (0.38) and (0.39) of its Laplace image. To do this we, as usual, deform the integration contour in the Borel transform of U (z,€) to the left and obtain the following asymptotic representation of the first term in (0.35)

=; [eu(@,Od

Qni

a |U (a,0146+af ut é) dé,

7’

where the contours [9,T'),..., are shown on Figure 0.11. The contour To corresponds to the ramification point € = € (x) and the contoursT;, 7 = 1,...,.N correspond to the points £,,...,€, respectively. These integrals can be rewritten in the form 1

| oa 7!

e§U (z,€) dé

ners

=

te

sale var U (x, €) dé Yo

32

INTRODUCTION.

Resurgent Analysis

Figure 0.11. Decomposition of the contour. N

as [efvaru (e,6) dE,

j=l

201

where varU (z,€) is the variation of the function U (z,€) along the path surrounding the corresponding point of singularity. Thus, we see that this term is exactly the resummation of the (divergent) series 1 a

t€ |e U (2, €) df

~ =

té(z = waC Gy

7!

Gane

—(j+1 G+1)

jg=1

N

+ Soe k=1

co

.

So oj (x)t~Grt) j=l—m

(see Subsection 0.1.3). The coefficients a; (x) and 6; (x) are given by formulas (0.38) and (0.39). Thus, for entire functions € (x) and y(zx), the asymptotics of the solution to equation (0.20) is given in the form of a resurgent function and, hence, we again come to the necessity of constructing of the corresponding theory. Certainly, all the effects which arise in this theory, such as the presence of focal points and the Stokes phenomenon, are present in the investigation of the asymptotic expansions of solutions to elliptic equations on an infinite cylinder.

Section 2. Semi-classical Approximations

0.3

33

Semi-classical Approximations

In this section we shall show how the notion of resurgent functions appears in the theory of asymptotic expansions of solutions to differential equations with respect to the small parameter. For such a situation, the notion of resurgent function occurs to be strongly connected with the so-called WKB method of construction of asymptotic expansions of solutions to a spe-

cial class of differential equations which are usually called 1/h-differential

equations [112], [114]. Therefore, in the first subsection of this section, we shall briefly recall the main notions concerned to WKB-expansions, and in the subsequent parts of this section, we shall show what progress can be achieved in this area with the help of the notion of resurgent functions. Later on, as we shall show in this section, the theory of WKB-approximations of solutions to 1/h-differential equations is strongly connected with quite another asymptotical problem — the problem of constructing asymptotic expansions to the same differential equations for large values of independent variables (and for fixed values of a parameter). Below we shall try to clarify this connection on the example of an ordinary differential equation. Since the aim of this introduction is just to illustrate the main notions of the resurgent analysis on concrete examples rather then to present all the definitions and theorems of the resurgent functions theory, we shall carry out almost all our considerations on the example of Schrédinger equation h2

What is more, in all subsections below, we consider only the one-dimensi-

onal equation, that is, z € R!.

0.3.1

WkKB-expansions.

Elementary theory

In this subsection, we shall briefly recall the known facts from the theory of WKB-approximation to make our presentation self-contained. The reader familiar with the notion of semiclassical approximation in the quantum mechanics can omit this subsection or just look it through in order to be familiar with the notation used in the sequel. So, let us consider the one-dimensional Schrédinger equation h? d?u

34

INTRODUCTION.

Resurgent Analysis

where V(x) is some entire functon of the variable z. Let us search for the

solution to this equation in the form!° u = er S)a(z,h),

(0.41)

where a(z,h) is a formal series in powers of h:

a(2,h) = Sa; (2) (—ihy’.

(0.42)

j=0

We remark that the asymptotic expansion (0.41) is known as a WKBexpansion. The function $(z) is usually called an action and the series (0.42) is called the amplitude function (or simply amplitude) of this expansion.

Substituting expansion (0.41) into equation (0.40), we obtain

Fee ah

+

(aya

idS d

fdr ae

Bu= {|e (=) +t)

aT

oe da

-

2

Te

:

et) 0. Then the contour [ involved into representation (0.54) must be chosen as it is shown on Figure 0.14.

40

INTRODUCTION.

Resurgent Analysis

Figure 0.14. Initial choice of the contour.

2. Now we shall use the analytic continuation process to construct the solution in all regions of the real axis R}. Certainly, since we consider only nonsingular WKB-expansions (that is, expansions outside focal points), to

continue the function from the region z < —V/2E to the regions -/2E < z < V2E and x > V2E, one must use the path lying in the complex plane C! which does not go along the real axis. We shall perform the analytic continuation along the path | in C} shown on Figure 0.15. On this figure, A; and Ap» are the focal points for the considered namely z = +/2E.

equation,

It is quite simple to perform the analytic continuation of the constructed solution along the part of the path / which goes from —oo to the point

(a) along the real axis. However, when the variable x moves along the circular arc connecting the points (a) and (b), the situation becomes not so simple. The matter is that, while the point x moves along this arc, the corresponding singularity points will rotate to the angle 37/2 in the positive direction (the direction of this rotation is shown by arrows on Figure 0.14). During this rotation, the lower point of singularity comes to the upper half plane, intersects the contour of integration and extracts from this contour

(generally speaking) two other integration contours I’ and I’ (see Figure

0.16).

At this point, one should stop and think a little about the situation. The matter is that it is absolutely unclear how many contours will be extracted from the contour [ by the point of singularity intersecting this

Section 2. Semi-classical Se eeApproximations a

LS

Figure 0.15. Path of analytic continuation.

mE

2nE

3nE

Figure 0.16. Extracting additional contours.

41

42

INTRODUCTION.

Resurgent Analysis

contour and, hence, the asymptotic structure of the continued solution in

the region —/2E < x < V2E will be, in turn, unclear. The reason is that both integration contour I and singularity points S (zx) of the integrand are located not on the complex plane C, but on the Riemannian surface R of

the function U(z,s) involved into representation (0.54). In particular, two vertical rays which are parts of the contour [' can lie on different sheets of R. Thus, it is possible that the singular point S; (zx) will extract from the contour I’ only one additional contour I” (if this point lies on one of the sheets of the Riemannian surface R used by the contour I) or will extract from T no additional contour at all (if this point of singularity passes over some other sheet of R). Certainly, the number of contours extracted from I depend on the structure of the Riemannian surface FR of

the function U(z,s) over some domain containing both S(z) and 5S; (2). However, up to the moment, we have no information about the structure

of R except for the information given by formula (0.55). Unfortunately, this information concerns only the local structure of ® near each of the points S (x) and 5S; (x) and gives no information about the behavior of the function U(z,s) in the domain containing both these points (semi-global information). And the question which must be solved now is: from what reasons can such semi-global information be obtained? The method of obtaining this information proposed by the resurgent functions theory is as follows: First of all, we know that solutions to equa-

tion (0.51) determined on the whole R} are real-analytic functions and, hence, can be analytically continued as holomorphic functions (in particular, univalued) in some neighborhood of focal points. Hence, one can try to obtain the required information from the condition of univaluedness of

analytic continuation of the function (0.54) to a neighborhood of the point A, which we encircle by the path | at the moment.

To do this, one must

perform the analytic continuation of function (0.54) along a closed path encircling this point, to take into account all additional contours which can appear during such continuation due to intersecting of the integration contour by singular points of the function U(z,s) and equate the result of the continuation to the initia! function. The conditions obtained with the help of such a procedure are known in the resurgent functions theory as resurgent equations, and they really do allow us to obtain the needed

information about the structure of the Riemannian surface R. The algorithm of writing down these equations will be considered in detail later (see Sections 2.4 and 2.5 of Chapter II); here we shall mention only, that the structure of 2 obtained by this method in our particular case is such as it is shown on Figure 0.17. From this figure, it can be easily seen that only one contour I” will be extracted from I when this latter is intersected by the point S; (x) of singularity. Now we can describe the result of the analytic continuation of the function (0.54) into the region -V2E < x < V2E along the circular part

Section 2. Lae Semi-classical pelt haa nce aa a

Approximations i

432

ne

Se 4 Se

Figure 0.17. Riemannian surface of solutions.

(a) — (b) of the path J. This result is i

ieee) =

A

1

3

| ctu es) ds + J, | cfu @s) ds,

vi

vid

where contours [I and I” are shown on Figure 0.18. As we see, the asymptotic expansion of the considered solution in this region is represented as

a sum of two formal series of the type (0.52) (unlike the case of the region xz < —-V2E, where the asymptotics reduces to one formal series). One can also notice that performing analytic continuation along the path encircling the focal point from above instead of the path /, we obtain the same result since we have constructed the function which is univalued néar the focal points.

:

The further process of the analytic continuation is quite similar to that considered above. The result is

TNCet)

—

1

a

[ efu Gs) ds

vi +

: [ctu

vh J

?

3) ds+—

[ ctu (a s) ds,

Vh J,

d

?

(0.56) ~

where the contours I, I’ and I” are shown on Figure 0.19. Hence, we obtain that the asymptotics of the continued solution contains, in general,

44

INTRODUCTION.

-32E

-2xE

-xzE

REDE,

Resurgent Analysis

Skt

Figure 0.18. Integration contours in the middle region.

three formal series of the type (0.52) two of which are of exponential growth as x — +00 (and as h > +0 as well). 2. Now we shall show how the described procedure can be used for computation of the spectrum of the considered operator. First of all, it is almost evident (and can be proved rigorously) that the asymptotics of the spectrum is given by the set of values E such that there exists an approximate solution to equation (0.51) which exponentially decreases both for x — —oo and for zt> +00. It is evident that the only way to obtain the solution of such kind is to require that the first two terms on the right in (0.56) are cancelled out. So, we obtain a relation

fetus) s= fetus) ds r

(0.57)

I’

which is essentially an “equation” for determining approximate eigenvalues E. Let us try to solve this “equation.”

Since the contours included in (0.57) differ from each other by shift 27, in order to understand for what values of EF relation (0.57) can take place, it is necessary to examine the periodicity properties of the function

U (x,s) with respect to the variable s. Due to formulas (0.49) and (0.55)

Section 2. Semi-classical Approximations pen r ai irda Lid r r

NE

2nE

45 ©J

3x2E

Figure 0.19. Terminal integration contours.

above, we have U(z,s)

= (Z>e0

or

;

To begin with, we shall investigate the periodicity properties of the func-

tions y; (x) included in the latter formula. The following affirmation takes place:

Proposition 0.1 plex curve

The functions y; (x) are univalued functions on the com-

L={(z,p):

p? +27 =2E).

Proof of this proposition is based on the investigation of the properties of solutions y; (x) to transport equations (0.50). These equations can be written in the form

V2E — =F

= 0,

ry,pF qulvionty= Wi /apm : 0.58 -V2E- PTee(54-1), (058) dx

46

INTRODUCTION.

Resurgent Analysis

ED Ce ee

RE,

Figure 0.20. Characteristic set as Riemannian surface.

From these equations, the needed affirmation can be obtained by straightforward computations with the help of induction on 7. More precisely, the curve L determines a two-sheeted covering over the complex plane C,. The structure of this covering is shown on Figure 0.20. Hence, to prove the univaluedness of the functions y; (x), it is sufficient to verify that the integrals

of the right-hand sides of equations (0.58) (divided by /2E — x?) over the contour 7 shown on Figure 0.20 equals zero. corresponding computations.

We leave to the reader the oO

Remark 0.1 This proposition describes the periodicity properties of the function U(z,s) since bypassing the curve y leads to the change of the action by the additional term 27L. Now we are able to compute the approximate spectrum of the Schré-

dinger operator (0.51). Let us rewrite the left-hand part of relation (0.57) in the form

etu (z,s) ds = (OE is

(z,s +2mE) ds

rv

On the other hand,

ae

ie oo

SoS”

Section 2. Semi-classical

Approximations

47

where the prime denotes that we use values of all functions obtained by the analytic continuation along the path y drawn on Figure 0.20. Due

to Proposition 0.1, the value of y; (x) will not be changed under such a continuation. The value of ,/dt/dz will obviously change its sign, and the value of S (x) is changed by the term 27E. Thus, we obtain that

U (z,s + 2nxE) = -U (2,8) and, hence, the integrals in different parts of relation (0.57) differ by the factor-exp (2itE/h). Equating this factor to unity, we obtain the approximate eigenvalues of the Schrédinger operator corresponding to equation

(0.51)

E=h (i+ 5) which coincide with the exact spectrum for the considered Schrédinger operator as well as with the spectrum obtained with the help of BohrZommerfeld quantization conditions.

3. To conclude this subsection we note that, in order to justify the above considerations, one has to prove the resurgent character of formal

solution (0.52) obtained with the help of the WKB-method.

This is one

of the most difficult problems in the resurgent functions theory in its applications to partial differential equations). One of the ways to solve this problem is to reduce equation (0.51) to the for the function U (z,s) involved into the representation (0.54). substituting this representation for solution into equation (0.51) easily obtain the following equation

10°U

(=

(at least possible equation Namely, one can

)0?U

err Aon vee reels for the function U in the complex space Chas s)> This equation can be investigated with the help of the theory of differential equations on complex

manifolds worked out by the authors (see [157]). The other way of investigating this problem proposed by E. Delabaere, H. Dillinger, and F. Pham

is based on the usage of resurgent equations (see {39]).

0.3.3

Asymptotics at infinity (the Airy equation)

In this subsection, we shall consider the computation of asymptotic expansions of solutions to differential equations at infinity (that is, for large values of independent variable). We shall see that exact semi-classical asymptotics with respect to a small parameter (the coequational or quantum resurgence) are closely related to asymptotics at infinity (equational

resurgence)!4. As above, we shall carry out our considerations mainly in 14The terminology is due to J. Ecalle.

48

INTRODUCTION.

Resurgent Analysis

the simplest case of ordinary differential equations though the method of computing asymptotic expansions obtained in this subsection will work in the case of partial differential equations as well (see Section 3.2 of Chap-

ter III). First of all, let us present some simple reasons leading to the form of asymptotic expansions of solutions to differential equations at infinity. In the previous subsection we have obtained the (formal) asymptotic expansion of the solution to differential equation (0.40) which has the form (0.41),

(0.42).

Then, by resummating the series (0.42) involved in this asymp-

totic expansion we have come to the exact asymptotic approximation of

the form (0.54), which uses the summation procedure based on the BorelLaplace transform of analytic functions (see Subsection 0.1.2). We claim that expressions (0.41), (0.42) (for fixed value of h and special choice of arbitrary constants in solutions to transport equations) determine asymp-

totic series with respect to x — oo and, hence, the representation (0.54) will give us the exact WKB-approximation at infinity if we fix the value of h (in what follows we shall put h = 1). Let us illustrate this observation on the simplest example. Let us consider the Airy equation with the small parameter h:

=h?—— =u = 0: To begin with, let us compute

(0.59)

a WKB-approximation for solutions to this

equation:

TH Ee) ee ek S(z) = (-ih)? @; (x).

(0.60)

j=0 1. The Hamilton-Jacobi equation for the considered case becomes

ds Na

ent (see formula (0.44) above) and one can easily find solutions to this equation

2 3/2 : S (2) = S2(2)i— $32

(0.61)

(We remark that another choice of an arbitrary constant in the solution of the Hamilton-Jacobi equation will lead only to the multiplication of the constructed solution by a constant factor and, thus, the choice of an arbitrary constant in the latter formula is not of importance.) To be definite,

we shall carry out all computations below for the + sign in (0.61) (and omit the corresponding subscript).

Section 2. Semi-classical

Approximations

49

2. The system of transport equations (0.48) for the action given by

(0.61) is 4200 + ag = 0,

40S

+a; = —2ye

Sip, ee eS

The solution to the first transport equation is ag (x) = Cunt, Choosing the constant C equal to unity (which is simply the normalization of the

solution), we are led to the formula

ao (x) = 27/4,

(0.62)

Now for the function a; (z), one has arts) = Cz

a

Ded oer ae

(We omit the trivial computations leading to the latter formula.) However, if we want to construct a solution which is asymptotical not only as h > 0 but also as x — oo, we must choose the constant C in the latter expression

to be equal to zero since only for such choice of C the function a, (z) will have less order as x — oo. (The solution for which each function a;41 (z) has less order in x than a; (x) we shall call a well-normalized one.) Then we obtain

Se a (x) = 752 vee

(0.63)

Similar computations show that the coefficient a2 (x) for the well-normalized solution is +4 pus

and so on. The analysis of the relations (0.62)—(0.64) leads to the assumption that, for the well-normalized solution, the coefficients a; (x) involved into asymptotic expansion (0.60) have the form a; (z) =

Gyr

tee

with some constants 7. This assumption can be easily verified by induction and the following recurrent formulas for coefficients c; can be obtained simultaneously: Coral

07

cj =

Th483(G7

5) Cj-1;

et es

(0.65)

3. Thus, we have obtained the asymptotic series in powers of h which is well-normalized in z:

u(z,h)

= exp ae ApS (—th) cja 74-9972,

(0.66)

50

INTRODUCTION.

Resurgent Analysis

where the constants c; are computed with the help of recurrent system

(0.65).

From (0.66) one can see also that asymptotic series in h are at

the same time asymptotic in x. From recurrent system, one can easily see

that the series on the right in (0.66) diverge and, hence, the resummation procedure is needed for obtaining the exact semi-classical solution. Due to formulas (0.52), (0.54), and (0.55), we obtain for the exact semi-classical approximation of the solution

=

u(z,h)

e

ae

ne:

2% @ Jexp {Fx} nm,(-ih) cja—1/4-33/2 j=0

=

‘

1

€ e eayr mfelic

ie

Z,S) enue)

, 0.67 ) (0.67

d as,

where 3\J-1/2

U

tion

Sa (sely See(2,3)s so e +2) EGa

op

ee

ails 33/2

(0.68)

4. Now in order to investigate the behavior of the solution to the equaa u

(0.69)

aE a

at infinity, one must simply pose h = 1 in the relation (0.67), thus obtaining the following formula for the asymptotic expansion of solution to (0.69) as x 00

a(z)= feues) ds, 8

(0.70)

where the function U (z,s) is given by (0.68). Up to the moment, we have obtained only formal asymptotic expansion. Below we shall show that this asymptotic expansion (at least in the considered example) is also a real asymptotic expansion of solutions to (0.69). (In fact, we shall show that this formula gives the exact solutions

to this equation for some concrete choice of the function U (z, s) and, what is more, every solution of this equation has the form (0.70).) 5. Let us search for solutions to equation (0.59) for h = 1 in the form of the Laplace integral

u(2) = i)efFU(€)dé. Substituting the latter relation into equation (0.59), we obtain the equation for the function U (€)

idaho

‘pum

U-=0:

Section 2. Semi-classical le a Approximations

51

h =

ae

Figure 0.21. Admissible contours.

The solution to this equation is exp (—i€3/3) and we obtain solutions to equation (0.59) in the form we

feos) dé.

(0.71)

Y

Up to the moment, the solutions (0.71) are formal since we must choose contour 7 in the latter formula in such a way that the integral on the right converges. It is easy to see that the integrand of the considered integral rapidly decreases if the contour y comes to infinity in a direction lying in

one of three sectors (which is not hatched on Figure 0.21) 7 0 and M such that

IF (é)| < CeM¥l.

(1.1)

Certainly, the constants C > 0 and M can depend on the concrete element F from the space E).

We shall consider the space FE as an algebra with respect to convolution.

We recall that the convolution of the two functions F' (€) and G (€) is defined by the formula g

FxG(Q)= [ FE=n)G(n) dn

(1.2)

0 The following affirmation takes place: Proposition 1.1 For any two elements FG € Ey the convolution F'*G belongs to the same space. The stated affirmation shows that the space E, forms an algebra with

respect to the convolution. We leave to the reader the easy proof of this proposition which goes by the direct estimate of the right-hand part of integral (1.2). The definition of the Laplace transform is as follows:

Definition 1.1 Let F'(€) be an element of the space FE, (Cg). Then the function coe?

f()=clry

/ e€F (E) dé

(1.3)

0

is called the Laplace transform of the function F'(€). Here the integration is carried out along the ray in the complex plane C¢ with origin at £ = 0 which is inclined to the positive direction of the real axis by the angle 6. However, the above definition needs some verification. First of all, one should describe a convergence domain of the integral on the right in (1.3) (in principle, it is possible that an integral does not determine a function

Section 1. Entire Functions

59

for all values of independent variable x). Later on, if we are interested in the construction of the inverse transform (this is really so), then we must describe the image of the transform L. This description must be precise to have an opportunity of defining the inverse transform. Finally (though it is not directly concerned with the definition, but we have used the word “transform”) it is worth doing to define the inverse transform using the description of the image mentioned above. Let us proceed with the realization of the described program:

Proposition 1.2 If the function F (€) satisfies estimate (1.1) then the integral on the right in (1.3) converges for all x with |x| > M. More ezactly, for any fixed 0 integral (1.3) converges in the half-plane Re (ex) > M and all functions fg (x) defined by integrals of the type (1.3) coincide on the intersections of the corresponding half-planes. Proof. Denote

:

Seen

;

erg

e

Then, with the help of estimate (1.1) the integrand in (1.3) can be estimated as follows le"

F (€)| < CEM? e-prcos(e+8)

=

Ceh(M-rcos(et9))

(1.4)

Hence, integral (1.3) converges for M — rcos(y+ 6) < 0, that is, in the half-plane mentioned in the proposition (see Figure 1.1). The coincidence

of the integrals fp (x) on intersections of their domains of definition easily follows from the integral Cauchy formula. This completes the proof. Oo From the proved proposition, it follows that the function f(z) is a holomorphic (and, in particular, univalued) in a neighborhood of infinity.

But, what is more, the estimate (1.4) allows us to obtain the estimate

aks

|z|

(1.5)

with some positive constant C, in each sector of the complex plane with vertex at the origin with opening less than 7. Since the whole neighborhood of the infinity can be covered with the finite number of such sectors, one can see that estimate (1.5) is valid in the whole neighborhood of infinity. Thus,

the image f(z) of the function F € EF, can be expanded into convergent

power series in z~! without zero term:

f(z) = Drage.

(16)

n=0

We denote by 02, the space of germs of holomorphic functions at infinity representable in the form (1.6). Thus, we obtain the following corollary of Proposition 1.2:

60

CHAPTER I. Borel-Laplace Transform

ia’

bad\ >

Figure 1.1. Convergence domain.

Corollary 1.1 The Laplace transform L given by (1.3) determines a map-

ping L:

Ey(Ce) > O%.

This gives us a needed description of the image of the Laplace transform. Now we are able to define the inverse transform, which is traditionally called the Borel transform.

Definition 1.2 For any element f (z) € O%, the function

FQ) =BU# def =1 [efne a) de =

=>

(1.7) ily

Cr

is called a Borel transform of the function F (€). Here, Cr is a circle of the radius R centered at the origin which is contained as a whole in the domain of regularity of the function f(z). The following affirmation is quite evident: Proposition 1.3 The Borel transform B given by (1.7) determines a map-

ping B:

O8% —

Ex (Ce).

Section 1. Entire Functions

61

In the next subsection, we shall see that the above defined Laplace and Borel transforms are inverse to each other and that these transforms are algebra homomorphisms. (The space 02, is considered with the algebra structure induced by the usual multiplication of functions.) However, even at the moment we can see why the Borel-Laplace transform considered on the space of entire functions is not sufficient for needs of asymptotic theory (and this, in particular, is the reason for considering the corresponding theory On more wide spaces of hyperfunctions). Actually, with the help of the above defined Laplace transform we can represent only convergent power series in z~! at infinity! This is insufficient even for the simplest Euler example considered in the introduction. Besides, the space O®, contains only power series without zero term, so that both 02, and EF, are algebras without unity. Thus, the generalization of the introduced Borel-Laplace transform is necessary. However, we postpone this generalization until the next section and now will proceed with investigating the main properties of the Borel-Laplace transform on the space of entire functions of exponential type.

1.1.2

The Borel-Laplace Transform

Let us first prove that the two transforms defined in the previous subsection are inverse to each other. The following affirmation takes place:

Theorem

1.1

Transforms (1.3) and (1.7) are mutually inverse.

We present two versions of the proof of this theorem. The first version is based on the inversion theorem for the classical Laplace transform. However, it is possible to give a proof of this statement using the methods of the function theory. This proof will be of interest for us since it is strongly connected with asymptotic expansions of the function f (x) at infinity.

Proof of Theorem 1.1 (first version). Without loss of generality, we can assume that @ = 0 (in the opposite case we can perform the variable change

€' = e~%€). Then formula (1.3) determines a usual Laplace transform of the function F' (€) considered as the function of real variable € given on the positive real half-axis R;. As it is known, the inverse formula is given by a+ico

FO) = 5 A e€ f (x) dz g—i00

with the integration contour being the vertical straight line Rez = o (see

Figure 1.2). Since the function f (z) is holomorphic outside the disk |z| < R and tends to zero as |z| — oo uniformly with respect to arg z, one can close the contour with the circular arc Cs for some R> R (using the Jordan

62

CHAPTER I. Borel-Laplace Transform

Figure 1.2. Closure of the integration contour.

lemma).

Thus, we obtain that transform (1.7) is inverse for (1.3). The

proof is complete.

G

To present the second version of the proof, we first introduce the notion of convergence into the spaces Ey and oF involved in the definition of the

Laplace and Borel transforms, correspondingly. Namely, we shall say that the sequence {F, (€), mn=1,2,...} tends to zero in the space E, if and only if:

1) there exists a constant M such that |F, (€)| < C, exp(& |€|) for all values of n with one and the same constant MV;

2) lim sup |F;, (€)| exp (—M |€|) = 0. Similar, the sequence {f, (xz), n =1,2,...} tends to zero in the space

O°, if and only if: 1) there exists a constant M such that f, (x) is holomorphic for |x| > M for all values of n;

2) lim sup |f,z(x)| =0. no? |z|>M —n It is quite evident that the set of powers {x~",

n= 1,2,...} is dense

in the space 02, with respect to the above introduced convergence (since every function f (xz) regular in a neighborhood of infinity can be expanded into power series in x~! which converge uniformly in a sufficiently small neighborhood of the infinity). The similar property for the space EF, is stated in the following assertion.

Section 1. Entire Functions Lemma

63

1.1 The set of powers {€", n = 0,1,...} is dense in the space EF.

Proof. Evidently, each function F'(€) from E, being an entire function can be expanded into the Taylor series

Ee)

(1.8)

ae dn n=0

We claim that this series converges to the function F'(€) in the sense of the space E;. To prove this, we first note that the coefficients a, of series (1.8) satisfy the estimate

Peers

(1.9)

with some constants C > 0 and M . Actually, we have

IF (1 dle Mi esl =f en Soe Rr

n,MR due to estimate (1.1). Now, minimizing the expression R~"e™** with respect to R € R,, we come to the estimate

M\” n

lan| < Cn! (=) e" < Ci V2rn (=)

Tee

VEN n

ee

-

(=) e" R}. In other words, the following affirmation takes place: Proposition

1.10

The following sequence

0 > O(B(R))

> O (% (R,e)) 20 (5(R,e))

(1.26)

1s exact.

Proof. The only nontrivial affirmation in the statement of the proposition is that the mapping var is an epimorphism. Let us verify this fact. To

do this, we consider a locally finite covering of the ray {argé = 6, |€| > R} consisting of intervals R; < |€| < R;. Let G(€) be a function holomorphic

in So (R,e€). Define the functions F; (€) by the relations Rie?

1

G

F(Q= 55

a dn. R;e*4

Sectiona 3. Hyperfunctions of Exponential Growth eee eee aa ie abla eh la

79

Figure 1.3. Covering (R,e).

These functions being given by the Cauchy-type integrals are defined and holomorphic everywhere except for the segment on the ray arg€ = 6 given

by R; < |€| < Rj and have a jump on this segment equal to G(€). It is also evident that (due to the analyticity of the function G in the sector

So (R,€)) these functions can be extended domains

to analytic functions in the

Q;= {€ : 0-€ M.

Then

(1.33)

Actually, the integrand of the last integral on the right in (1.29) with

y (€) = e~*§ can be estimated as follows: |var F (€) e=*s] < CeMe-prcos(8+41)

= Ce7P(r cos(8+81)—M)

and the corresponding integral converges in the domain (1.33). The domain in the z-plane determined by inequality (1.33) is a half-plane bounded by the straight line | orthogonal to the ray arg x = —6@ and intersecting this ray at point |x| = M (see Figure 1.6 a)). Later on, since inequality (1.32) is valid for directions 6’ sufficiently close to 6 (with constants C and M possibly depending on @’), the domain of definition of the expression

LiF] = (Ren)

(1.34)

is a union of the above described half-planes. Evidently, this union contains the domain

Op tre l= {zx : larg « + 6|
R’}

(1.35)

shown on Figure 1.6 b) for sufficiently large R’ and sufficiently small e’. Definition 1.9 Expression (1.34) is called the Laplace transform of the hyperfunction F of exponential growth.

As we have shown above, the function £[F] is defined in the domain (1.35). It is easy to show that this function has exponential growth in this domain. We denote by EF (Q6 (R’, e’))

Section Exponential Neca abe 3.eaHyperfunctions > Scena oaofLc Coe ye Growth lll

85

Figure 1.6. Definition domain for the Laplace transform.

the space of functions of exponential growth in 0 (R’,«’). Now we formulate the theorem on properties of the introduced Laplace transform.

Theorem

1.7 Formula (1.34) determines an algebra homomorphism

Lo: He(Ce) > Ey (No (R’,€’)). Besides, the commutation formulas

S ) = £[-€F] (2) dx clr Ve

6 Ba (x) = 2£[F](z) are valid.

Proof. We shall prove only the first part of the theorem, leaving to the reader the easy verification of the commutation formulas. Thus, we need to prove the fact that the Laplace transform of the convolution of two hyperfunctions equals the product of the Laplace transforms of the factors.

Let F and G be two hyperfunctions and F'(€) and G(€) be their representatives. Then, since the variation of the function F' * G(€) equals the

86

CHAPTER I. Borel-Laplace Transform

variation of the function H, (€) given by integral (1.30), we have

L([F *G] =

lim

emis { [Fenvee=n) dn > dé. if

a

Changing the integration order in the latter integral we come to the formula

C[F*G]

=

lim (|e-*"F (n) i

=

/e*(&-G(é —n) dé}

dn

r

L[F|LIG]

as desired. This completes the proof.

Oo

Now we shall define the Borel transform on the space FE, (% (R’,e’)) and prove that this transform is inverse to the above defined Laplace trans-

form. Let f (x) be a function from Ey (Q (R’,e’)). In particular, it means that

If (z)| < CeMltl for cE 9(R,e’). Denoting, as above, € = pe” , x = re, we come to the estimate

lif crete lee

LM tecosl ten),

Hence, the function f (x) e** is exponentially decreasing in the intersection of the domain 0 (R’, e’) with the half-plane

Lo = {M + pcos(@ +61) < 0}. We define the Borel transform of the function f (x) by

Bie defY=1 a

e** f (x) dz,

(1.36)

YA

where the contour 74 (which is shown on Figure 1.7) has its origin at some (arbitrary) point A € 0 (R’,e’) and goes to infinity along a direction of exponential decrease of the function f (x) e7*. It is evident that the righthand parts of (1.36) corresponding to different points A, A’ differ from each other by an entire function in € and, hence, determine one and the same element from H, (C¢), that is, the definition of the Borel transform is correct. Later on, it can be easily seen that the function defined by (1.36) is determined for all values of € with arg € € (9—€,27+6+ 6) if |€| is

sufficiently large. Moreover, the result of analytic continuation of B [f] (€) from arg € = @ to arg € = 2m + @ is given by the expression of the type

Section 3. Hyperfunctions of Exponential Growth

87

EO

LE SWNN Meee

S \

A

Figure 1.7. Computation of the variation.

(1.36) with y4 replaced by 7/, (see Figure 1.7). Hence, we have var B[f](€)

=

zhu

ef (x) dx — hs

211

211

eri

ef (x) dr

(1.37)

bz “f(a) da,

1

=z

y

where the contour 7 is such as it is drawn on Figure 1.7. In particular, the latter formula shows that the function varB [f] (€) is an exponentially increasing one. The following invertibility theorem is valid: Theorem

1.8

The Laplace and Borel transforms determined by formulas

(1.34) and (1.36), respectively, are inverse to each other. Proof. First of all, we note that, without loss of generality, we can carry out the proof for 9 = 0. Let us consider first the composition Bo L. If

F (€) is the function representing an element F € H,(C¢), then, due to the above definitions, we have the following formula for the composition

BoL[F):

88

CHAPTER

F’'(é) &

BoL{F]

(1.38)

Se

=

I. Borel-Laplace Transform

1

GS

coe’

i

eaghe (n) dn + / e*"var F(n) dn > e§dé

A

Cr

Rei€

(here, € is a sufficiently small in modulus and the integration in the outer integral on the right in (1.38) is carried out along the ray shown in Figure 1.8 a)). It is evident that, when integrating over €, one can use one and the same integration contours in the inner integrals in (1.38) and we can change the order of integration in this formula: ~ 1

PQ) = sf

coe

Prin}

fo

271

:

2

[ e@ ae} dn

Cr

1

i@

A

coe’?

e

/ var F' (7)

(eee

Re*é

dn.

A

Evidently, the inner integrals on the right in the latter formula converge for sufficiently large values of |€|. Under this assumption, we obtain

F=fF@ Cr

e4

(€—n)

Rs

dn,

(1.39)

where the integral is understood as the limit of integrals over the contours I'rR (obtained by truncation of the contour I'z at |€| = R’; see Figure 1.8

b)) as R! > +00. Deforming the integration contour in (1.39), we replace the integral on the right in this formula by the integral over the contour Ir, with |£| < Ry plus the residue of the integrand at point 7 = € (see Figure 1.8 b)):

FQ =F@+ [|FQ) peas, dn. eAlé-7)

Tr,

The integral term on the right in the latter formula is an entire function in the variable € and, hence, the function F’ (€) determines the same hyperfunction as the function F'(€). This proves the first part of the theorem.

Section 3. Hyperfunctions of Exponential pole eee RA is Is i Growth lthi

rrr rrr

89

Figure 1.8. Integration contours in the Laplace and Borel transforms.

Let

us

now

consider

the

composition

Lo

B.

For

any

function

f (x) € Ex (Qo (R’,€’)) we have co

coBif= fei) d+ [evar BI z

Cr

= +h. (1.40)

R

Let us consider the first term J; on the right in the latter formula. We have

Ve /eae2071 t ekf(y)dy de, Cr

(1.41)

7(€)

where 7(€) is a contour lying in the domain of exponential decrease (in the variable y) of the function e¥. If € moves along the contour Cr, then the domain of exponential decrease of the function e% will rotate counterclockwise and, hence, the system of contours ¥y (€) for different £ € Cr will look like as it is drawn on Figure 1.9 a). Hence, integral (1.41) can be written down as the integral over the two-dimensional surface Up which is projected on the y-plane into the system of contours 7 (€), where € changes along the contour Cr:

himige iel¥-2)EFf(y) dé Ady. 1

Nt

=r

90

CHAPTER

I. Borel-Laplace Transform

Figure 1.9. Contours included in the composition Lo B.

Evidently, we can choose the system of contours 7 (€) in such a way that the point y = z is located outside the projection of Lr to the y-plane. Then we have 7

=a fe{ Sra} 1

-

e(y-z)é

5

= 5 [ SStww. |

Rr

el(y—Zé

s

042) an

OZR

From the definition of the surface Ug, it is clear that its boundary consists

of two contours 7 (Re) and —y (Re?"*) (where € = R) plus the part of this boundary lying over the point A, where € € Cr. The form in the integrand on the right in formula (1.42) vanishes on the latter part of the boundary (since y = A = const there) and we obtain

1 =

e(¥-z)é

ae

7

f (y) dy,

(1.43)

71

where 7 is a contour drawn on Figure 1.9 b). Let us consider now the second term Jp on the right in (1.40). For computing this term we note that, due to formula (1.37), the variation of

the function B [f] (€) equals to

varBUA 6) = => |etFW) dy,

Section 3. Hyperfunctions of Exponential Growth

91

where ‘2 is a contour drawn on Figure 1.9 b). Since, during the integration over the ray [R, +00] in Jp the contour 72 remains unchanged, we obtain 1

co

co

=) ext 2 aa fe R

[etre ay) d= [pes fag} dy. 72

72

R

For convergence of the inner integral, we must assume that the real part of zx exceeds the real part of y on the whole integration contour 72 (see Figure 1.9 b)). Under such an assumption, we have

0

a

——

f (y) dy.

(1.44)

Substituting expressions (1.43) and (1.44) into (1.40), we obtain for the considered composition )R

cope pasteles =,Tt 9

[relics f (y) dy. y-2 bleep

Using the residue formula, we obtain the required equality Co B[f] = f. This completes the proof of the theorem.

1.3.3

Oo

Generalized Hyperfunctions

In this subsection, we introduce the notion of generalized hyperfunctions needed to extend the theory of the Borel-Laplace transform to the space of functions of exponential growth given in sectors of x-plane of arbitrary small opening. The matter is that the Laplace transform of hyperfunctions

acts from H? (Cg) to the space E; (Q (R,e)) of functions of exponential growth given on the sector

M (Rye) = {2 : larg 2+6|< ate, Iz| > R} with opening more than 7 (see formula (1.35) above). We want to extend the definition of the Borel-Laplace transform in such a way that the Borel transform will act on the space FE, (Sg (R,€)), where Sg (R, €) is a sector in the z-plane with the arbitrary opening € > 0 given by

So (R,e) = {x : larg x —-6| R}. To do this we introduce the space of generalized hyperfunctions of exponential growth as a quotient space

GH? (Cy) & By (ae (R, :))/Ex (Ce),

(1.45)

92

CHAPTER

I. Borel-Laplace Transform

~(1) where Q, (R,€) is a sector in the €-plane: rAGly

Q

(Rye) = {61 04 Se < ame al

with the opening a +2e. We remark that the space H? (Cg) of hyperfunctions is defined as the factor similar to (1.45) but in sector with opening

more than 27. We remark that the fact that in (1.45) we use the space ~(1) Ei (Q% (R, .))of entire functions of exponential type instead of the whole space of holomorphic functions in this domain used in the definition of the space H? (Cg) of hyperfunctions is not essential.Actually, due to results on the Borel-Laplace transform of hyperfunctions below, in every equivalence class defined by any hyperfunction there exists a representative of not more than exponential growth. This is the reason of the name generalized hyperfunctions for the elements of quotient space (1.45). Similar to the case of hyperfunctions of exponential growth, we can

define the convolution of generalized hyperfunctions (see the concluding part of Subsection 1.3.1). We remark that, due to the results on the BorelLaplace transform obtained below, each equivalence class in (1.45) contains a function F'(€) decreasing at infinity as |€|~*. Hence, the corresponding integral

[Fmee-n dn y converges for such representatives F'(£) and G(€). The choice of the contour is similar to that for hyperfunctions given in the previous subsection. The details are left to the reader. Now we define the Laplace transform on the space GH? (Cg) of generalized hyperfunctions by the formula

f(x) =c[F)® /e-*€F (€)dé, def

4/

where F (€) is a representative of a generalized hyperfunction F in the space Ey (% (R,e)) and the contour 7 is defined similar to the corresponding contour shown on Figure 1.7. One can easily check that the latter formula correctly defines a function from EF; (S@(R,¢€)) for any given generalized hyperfunction F’. Thus, we have defined a mapping

L: GH’ (Ce) > Ei (So (R,e))-

(1.46)

Similar to the case of hyperfunctions of exponential growth, it can be shown that the Laplace transform C (1.46) is an algebra homomorphism between the convolution algebra GH? (Cg) of generalized hyperfunctions

Section 3. Hyperfunctions of Exponential Growth

93

and the multiplicative algebra E, (So (R,e€)) of functions of exponential type.

We define also the Borel transform on the space E; (S@(R,e)) by the formula 0)

coe?

FQ) =Bi = = | ef (a)de, def

1

t

(1.47)

A

where the integration is performed over the contour with the origin in an arbitrary point A of the domain Sg(R,e) and going to infinity along some ray arg x = @’ in this domain. It is evident that the functions F (€) corresponding to different choices of the initial point A of the integration contour differ from each other by an entire function of exponential type and, hence, this formula correctly defines a mapping

B : E,(So(R,e)) > GH? (Cz). Theorem

1.9

(1.48)

The mappings (1.48) and (1.50) are inverse to each other.

Besides, the commutation formulas

FLIF\(e) = L{-€F\(2), dF c|F| (s)ee

2C(F|(2)

(1.49)

are valid.

Proof of this theorem is based on the fact that (up to changing sign of the independent variable x and the factor 1/277) the Laplace and Borel transforms determined in this subsection coincide with Borel and Laplace transforms on the space of hyperfunctions of exponential growth examined in the previous subsection if these latter transforms are written down in terms of variations of the corresponding hyperfunction. One must take into account also that the Laplace transforms of hyperfunctions with com-

pact support (which form the kernel of the variation mapping) are entire functions of exponential type. corresponding proof.

We leave to the reader the details of the Oo

The action of the Borel-Laplace transform in spaces of hyperfunctions is illustrated by Table 1.1.

1.3.4

Examples

In this subsection, we present several simple examples of hyperfunctions and their Laplace transforms.

94

CHAPTER I. Borel-Laplace Transform

He (Ce) = Oo0/O (Ce)

#9 (Ce) = Er (fo (Rye) /Es (Ce) |Es (M0 (R’,e’)) GH? = Ey (9§) (Rye) /Ex (Ce) |Ei (So (R’,e")) Table 1.1. Borel-Laplace transform 1.The Heaviside function Y (€) is a hyperfunction with the following

representative! 1

Evidently, this function determines a hyperfunction of exponential growth and, hence, the generalized hyperfunction. The Laplace transform of this hyperfunction equals

LY = x fev*tingag=fen*tag=-. r

0

2. As it is usual in the theory of the classical Laplace transform, in order to embed the space of entire functions of exponential type into the space of hyperfunctions, one should multiply any entire function by the Heaviside function Y (€). In particular, computation of the Laplace transforms for polynomials can be carried out with the help of the formula

f Sy | =a

(1.50)

which can be obtained from the preceding formula with the help of the

commutation formula (1.49). 1 We remark that the variation of the function Y(£) equals unity. Thus, this definition coincides with the usual definition of the Heaviside functions which equals unity on the positive part of the real axis.

Section 6. Microfunctions

95

Thus, we see that the correspondence the embedding

E; (Cg)

F (€) +» F'(€)Y (€) determines

GH" (Cg).

One should be careful using such an embedding since it does not commute with the differentiation. Actually, we have

d

FT

Sais oa = YRS

Gt

eee

up to an entire function of exponential type. In other words,

d

dé {Y (€) F (£)} = Y (€) F(€) + F (0) oo (€)The latter formula explains the appearance of the term —F (0) in commutation formulas for “classical” Borel-Laplace transform of Subsection 13122. 3. For any noninteger o the function |

(€) a

(€ — £)” (e2™ #2 ~ 1)T(o+

1)

evidently determines a hyperfunction with the only singular point € = £. For this hyperfunction we have

CIF. (©|=—

1.4

Microfunctions

In this section, we introduce the notion of an endlessly continuable hyperfunction. This notion is of importance for applications of the Borel-Laplace transform theory to the investigation of asymptotic behavior of analytic functions due to the following reasons. As we have seen in the introduction, asymptotic expansion of an analytic function of exponential growth is determined by singularities of the Borel transform of the considered function. So, to consider the class functions with so-called discrete asymptotics , one must consider the subclass of the class of hyperfunctions consisting of functions having only the discrete set of singularities (at least on Riemannian surfaces of these functions). However, the definition of the notion of hyperfunction assumes that the corresponding analytic functions (which represent the considered hyperfunctions) are defined only for sufficiently large values of their argument

96

CHAPTER I. Borel-Laplace Transform

€. Thus, to investigate the class of analytic functions with discrete asymptotics one should consider a class of hyperfunctions whose representatives extend to functions having only discrete sets of singularities. This leads us to the notion of endlessly continuable hyperfunction (certainly, this notion requires more rigorous definition which will be given below).

Later on, from the examples considered in the introduction, one can see that each summand in the asymptotic expansion of an analytic function is determined by one of the singular points of its Borel transform. Thus, we must give the procedure of decomposition of a hyperfunction into, so to say, a sum of the (micro)local elements, each corresponding to exactly one point of singularity of the considered hyperfunction. To give the exact formulation of such a “microlocalization procedure” we introduce the notion of a microfunction (it occurs that microfunctions are exactly the above mentioned local elements). Below, we introduce this notion and prove the corresponding decomposition theorem.

1.4.1

Endlessly continuable hyperfunctions

Before investigating the notion of endlessly continuable hyperfunction, we introduce the auxiliary notion of extendible hyperfunction which will be of use below for the description of the decomposition procedure.

Definition 1.10 A generalized hyperfunction F € GH’ (C¢) is called to be an eztendible (generalized?) hyperfunction with support at point & if there exists a representative F'(€) € Ey {Q

(R,2)) of this hyperfunction

such that this representative extends up to an analytic function on the covering

My (60,8) ={E: EF bo, 0 -e < arg (E-&) —1 and some C>0.

98

CHAPTER

I. Borel-Laplace Transform

Figure 1.11. Appearance of a microfunction.

We denote the space of regular extendible hyperfunctions with the sup-

port at & by Cie

Proposition 1.11

(Cg). The following affirmation takes place:

The function f (x) € FE, (Se (R,e€)) ts a Laplace trans-

form of some regular hyperfunction with the support & = 0 if and only if this function satisfies the estimate

If (2)| < [2

-—a-l1

with some a> —1 for large values of |x|. Proof. Without loss of generality we can assume that 6 = 0. Let us now proceed with the “only if” part of the assertion of the Proposition. Let F be a regular hyperfunction. Then its Laplace transform f (xz) can

be written down in the form

f(z)

=

1

ea

| e*F(€) d€= | e“*F(E) dé + ie-"*F (€) d€+ iie *§F(£) dé

(1.53)

for some positive value of a, where the contours ,, 74, and y_ are shown on Figure 1.11. It is quite evident that the two last integrals on the right

Section 6. Microfunctions

99

in (1.53) are estimated as follows:

/eR (é) dé| 0. The function f (x) is defined uniquely up to a rapidly decreasing function.

Proof. Let a > 1 be a number such that the function e~°*" is a decreasing function at infinity in the considered sector for any positive constant b. Consider the series

me

(1.57)

ec

AO) n=1

(

)

with some (undefined at the moment) positive constants b,. Due to the estimate

\l1-—e77| f@)=-Liflo"+ The first sum on the right in the latter relation is evidently a rapidly decreasing function. The estimate of the second sum can be carried out as follows:

fale) (1-e hs") n=N+1

106

CHAPTER I. Borel-Laplace Transform

Ns

ye evs iG

ya Cnbalzla a n=N+2

Cabu (——2— n=N+2

= Cres

Vel

The latter estimate proves the lemma.

Oo

The result of this lemma can be reformulated as follows.

Let us intro-

duce the topology on the quotient space

Eq(S2)/E7™ (Se) with

{E,“(S.)/E, (Se), M € R} as the fundamental system of neighborhoods of zero. affirmation is the reformulation of Lemma 1.2.

Then the following

For any given functions fr(x) € E,”™(S_) the series Ds, fn (z) n=l

converges in the quotient space E} (S,) /E;© (Se). Now we can define the Laplace transform of an extendible microfunc-

tion. Let F € MQ .,, and let F (€) be a representative of F. Consider the integrals

fn(a) = fe“(6)de-+ fe-*Evar F(6) dé, Cy

r

where the integration contours are chosen similar to the formula (1.54). The differences f,41(z) — fn(z) evidently satisfy the requirements of Lemma 1.2 and, hence, the sequence {f, (xz), n = 1,2,...} converges in the space

E? (S.) /Ey, ~ (S_-). We introduce the Laplace transform of a microfunction F by the formula

CIF) = tim fa (x) € ER (Se) (Ey (S-).As above, the inverse (Borel) transform

B : EY (Se) /Ey™ (Se)

7 Moject

is given by formula (1.47). Namely, the following theorem is valid.

Section 6. Microfunctions

Theorem

107

1.10 Let F € MQ ..,.

Then the function Bo L[F] is an ez-

tendible hyperfunction determining the microfunction F under the mapping GH) (C¢)

=e WA veee:

In particular, isomorphism (1.55) can be obtained as a consequence of this theorem.

Proof of the Theorem.

Let us show that the function BoL|F]

can be

analytically continued into the intersection of Np (€) with the half-plane Rez < n for any given integer n. Due to the definition of the function £[F] we have Tr

L (F] a [tre

d£ + [erharF

GCF

d£+h(z),

T

where h (zx) is an element of the space E; ”(S,). Thus, the Borel transform of the function h (x) is holomorphic in the half-plane Rez < n and we must only prove that so is the Borel transform of the function

fer dee [em varr (@ ag are-*€F (€)dé, r

iD

where I’, is a contour shown on Figure 1.12, can be analytically continued

to the intersection Qe (e)N {Rex < n} for any n (of course, in our case 6 =0 and a=7n). Denote

FQ

2 B| [etre del = fe 1/e*9F (n) dnp de A

n

n

for some fixed positive value of A. Changing the order of integration, we come to the formula co

Fu) =a [4[eM “, de} Fn) an. 1

Be

A

Calculating the inner integral on the right in the latter formula we have

F(é)

=

=
0, there exists a number N such that N

f (z)- D> fala) € Ey 4(S(R,e)).

(1.58)

n=0

The reader can easily see that the convergence introduced by this definition is none more than the convergence with respect to the above introduced

topology in Ej/E,~. The following affirmation is a direct consequence of Lemma 1.2. Proposition 1.12

Under the above assumptions series

fal) SEF] n=0

(1.59)

n=0

converges in Ey (So (R,€)) /E; © (So (R,€)). Proof. Let us consider the functions

f(z) =”

i e~**F, (€) dé

(1.60)

rv) over all values of n such that the corresponding point €,, lies in the halfplane Re€ < N. The contours r™) included in the latter formula are obtained from those in the Laplace transform by truncating at Reé = N. (See Figure 1.13; for example, contours I’ (N) and Eoce shown on this Figure

are included into the sum sn in contrast to the contour BOs One can easily.check that the differences f(Y+1) (x) — f(%) (z) satisfy conditions of Lemma

1.2.

Hence, sequence (1.60) converges in the space

E, (So (R,€)) /E,* (So (R, €)). Now it is not hard to check that the limit function f (x) = Nim, f) (z) satisfy (1.58). This completes the proof of the proposition.

Oo

Now let f(x) € E, (So(R,€)) be a function such that its Borel transform F (€) = B[f («)] is an extendible hyperfunction in the direction 6 = 0. Then the function F'(€) determines a sequence of extendible microfunctions F,, Fo,... corresponding to singularity points €), 2,...of the function F (€) ett) which are visible from the domain Q, (R,e) (see Figure 1.14). The following decomposition theorem is valid:

110

CHAPTER

I. Borel-Laplace Transform

Figure 1.13. The sum of microfunctions.

Theorem

1.11

Under the above assumptions, the decomposition formula

f(z)=)>_ Ci]

(1.61)

n=0

is valid. Inversely, if Fi, Fo,... is a sequence of extendible microfunctions supported in the set &1,&2,... contained in some angle of opening m — 2€

bisected by the real axis, then the Borel transform of the function f (x) given by (1.61) ts an extendible hyperfunction determining microfunctions Fy, Fo,... in the above described way. Proof. If f (x) = £[F] for some extendible hyperfunction F’, then we have

f(e)= /enF(6)dé, Oy

where the contour y is shown on Figure 1.14. Deforming this contour as it is shown on this figure, we obtain decomposition (1.61) similar to the proof of Proposition 1.12. The proof of the inverse part of the theorem goes quite similar to the proof of Theorem 1.10 and we leave it to the reader. Oo To conclude this section, we remark only that if all the microfunctions F\, Fo,... are endlessly continuable, so is the hyperfunction F = B[f).

Section 6. Microfunctions

111

Q,(R, €)

DCW Figure 1.14. Decomposition.

1.4.4

Examples

In this subsection, we present an example of the application of the decomposition theorem to some resurgent function. We shall try to show how the process of the decomposition goes for various structures of the Riemannian surface of the hyperfunction in question and to illustrate the appearance of the Stokes phenomenon from the viewpoint of the decomposition theorem.

Let us consider a resurgent function f(x,y), where x € C is a resurgent variable and y € C is a complex parameter. Suppose that the Riemannian

surface of the Borel transform F(£,y) of the considered function has the form drawn on Figure 1.15 a) (such situation takes place, for example, in the investigation of the Airy function; see the next chapter). The mentioned Riemannian surface consists of three sheets A, B, and C with the two cuts emanated from the two points € = S;(y) and € = S2(y) along the direction of positive real axis (one should take into account that the complex plane C¢ on Figure 1.15 is turned to the angle 7/2 clockwise). The manner in which these three sheets are connected with one another is shown schematically in the bottom of Figure 1.15 a). Suppose also that the initial domain of definition of the hyperfunction F'\(€,y), shown on the figure, is located at the sheet A of the considered Riemannian surface. It is not hard to see that the deformation of the contour 7 included in

112

CHAPTER

I. Borel-Laplace Transform

Figure 1.15. An example for the decomposition theorem.

the resurgent representation

f(z,y) = /e-*F(E, y) dé

(1.62)

7

in the direction of the positive real axis leads us to the representation of the considered resurgent function as the Laplace transform of the microfunction

F,(€,y) determined by the hyperfunction F(€, y) at the point 5S; (y) (that is, in the form of integral (1.62) taken over the contour I’); see Figure 1.15 a)). This is valid due to the fact that the singular point S2(y) of the function F(€,y) lies on sheets B and C of the considered Riemannian surface and, hence, this point is not visible from the initial domain of definition of the function F(£,y). So, the decomposition of the resurgent function f(z,y) for values of the parameter y such that the Riemannian surface has the form shown on Figure 1.15 a) is

f(z,y) =L[Fi(€,y)],

(1.63)

where F\(€, y) is the microfunction determined by the singularity of F(€, y) at the singular point 5)(y). Now suppose that the value of the parameter y changes in such a way

that the singular points 5,(y) and S2(y) move in directions shown on Figure 1.15 by the thin arrows, so that the resulting position of these singular points occurs to be such as it is shown on Figure 1.15 b). Then the same

Section 6. Microfunctions

113

procedure of the deformation of the integrating contour 7 along the direction of the positive real axis leads to the completely different result. Now both points S;(y) and S2(y) are visible from the initial domain of definition of F(€,y) and, hence, the decomposition of the function f(z,y) for values of the parameter y such that the Riemannian surface has the form shown

on Figure 1.15 b) is

f(z,y) = £[Fi(f,y)] + £[Fa(é,y)] ,

(1.64)

where Fi (€,y) is defined as above and F>(€,y) is the hyperfunction determined by the function F(€,y) at its singular point S2(y). Comparing the representations (1.63) and (1.64) of one and the same resurgent function we see that the decomposition of a resurgent function can change by jump when the parameter intersects a Stokes line (clearly, the Stokes lines for the considered example is formed by values of y such that the singular points Si(y) and S2(y) have one and the same real part). This is exactly the Stokes phenomenon. Here we restrict ourselves by this short explanation postponing more detailed investigation of the Stokes phenomenon until the next chapter,

where the tools for investigation of this phenomenon (such as the connection homomorphism and the alient derivative) will be worked out.

ais

Charter t ah —

a

R}.

As above, we denote by EF; (KR) the space of functions f(z) holomorphic in the set Kr and such that, in any proper conical subset K’ of the set K, the estimate -

If (z)| < CeMl! is valid with some constants C and M possibly dependent on K’. The elements of the space E; (KR) will be referred as functions of exponential growth of order 1 in Kp. To begin with, we extend the notion of the Borel-Laplace transform to functions of several independent variables.

For any function f(z) of exponential growth of order 1 in Kr, we define a function

folI— fi lties.-:-, ea)

(2.5)

of one complex variable 7 € C which is defined for any fixed value of x in some sector

So(R,€) = {n : larg nl R},

(2.6)

where e€ > 0 can depend on the choice of x. Evidently, the function f, (7) is a function of exponential growth in 7 of order 1 in sector (2.6):

fe (n) € Ex (So (R,€)). Due to the results of the previous chapter, the Borel transform F'(s,z) of the function f, (7) in the variable 7 is defined as a generalized hyperfunction in the direction 6 = 0 by the formula

F(s,2) = B[fe(m)) © 5 fe" fe(n) dn

(2.7)

A

(we denoted by s the variable dual to 7). This function is defined in the domain ri A) (Re)

=

4 3 +5

3 ce

Rt

120

CHAPTER

Proposition 2.1 tion of order —1:

II. Resurgent Analysis

The hyperfunction F'(s,x) 1s a homogeneous hyperfunc-

F (As, b isneh fg Nahe) es Nak (aus eoanyent he

Proof. Due to formulas (2.5) and (2.7) we obtain Fi(syz)= aife f (nz)}:. saz ) dn. 1

s

Tn

A

Substituting As, Az!,..., Av” instead of

s,z!,...,2” into the latter relation,

we obtain 1

F (As, Ax) = —— sae (Ang gree Anz”) dn. 271

A

Now, using the variable change 7 = A~!m, we can reduce the expression for F'(As, Az) to the form 1

loo}

F(ds, An) = 15 [emf (ma!,...sma") dm = AF (6,2) AA

since the hyperfunction F'(s,r) does not depend on the choice of initial point A in formula (2.7). This completes the proof. Oo Definition 2.1 The homogeneous generalized hyperfunction (2.7) is called the Borel transform of the function f (zx).

defined by

Now we shall write down the expression for the inverse transform for B (the Laplace transform). We define this transform on the space of generalized hyperfunctions by the formula

LIF (s,2x)| we per (s,x) ds, r

(2.8)

where the contour I’ was defined in Subsection 1.3.1 of Chapter I (see Figure 1.4 there). Theorem 2.1 The transforms L and B introduced by formulas (2.8) and (2.7), correspondingly, are mutually inverse. Proof. The result of this theorem can be easily carried out with the help of

the inversion theorem for Borel-Laplace transform for generalized hyperfunctions of one variable. To be definite, let us prove that the composition

Section 2. Resurgent Functions

fa

£oB is an identity mapping. Due to the inversion theorem for generalized hyperfunctions, we have

f (nz) = fetres

ds.

iE

Putting in the latter formula 7 = 1 we obtain the relation

dja 6 ee exer

ds

{8

which proves the identity Co B = id. The fact that Bo L = id can be proved in a similar way. Oo Now we shall investigate the commutation relations between the above introduced transform and the operations of differentiation and multiplication by an independent variable. Clearly, these results are the direct subsequence of the commutation formulas obtained for the one-dimensional Borel-Laplace transform in Chapter I, but there are essential differences in the form of commutation formulas in the multidimensional case compared with those in one dimension. The following statement is valid:

Theorem

2.2 The following commutation formulas

Zein = e[(2) Se] 010Bl) eG sr are valid.

Proof. Let F be a Borel transform of some function f (x) € EF, (Kr). Then, due to formula (2.7) we obtain co

F(s,ie)

wa = [ens (nade nz”) dn. 1

(2.9)

Tt

A Differentiating the latter formula with respect to x’, we have

Fe (ani=

ical

ah [erage RE

wet an

AT

“

of

mire (+7,Repaytiaaiealtina ey = 555].

122

CHAPTER II. Resurgent Analysis

Applying the operators (0/0s)~' and CL to both sides of the obtained formula (note that the operator (0/0s)* is well-defined on the space of hyperfunctions) we come to the first of the desired commutation formulas. To obtain the second commutation formula, we multiply relation (2.9)

by 27. As a result, we have co

:

1

A

2/F (s,x) = oat f ert natn)

dn

A =—

271

feet (nz?) f (a Ae nx) dj

(=)

B [2 f] :

A

Applying the operators 0/0s and CL to both sides of the obtained formula, we arrive at the desired relation. This completes the proof. Oo

Now we are able to give the definition of the notion of a resurgent function in several variables.

Definition 2.2 The function f (x) € E; (Kp) is called to be a resurgent

function of the variables

z

=

(z',...,z")

if its Borel

transform

F'(s,x) = B[f (x)] is an endlessly continuable hyperfunction in s for any fixed value of x = (z!,...,2"). denoted by R.

The space of resurgent functions will be

We remark that, similar to the one-dimensional case, it can be proved that the set of resurgent functions is an algebra with respect to the usual multiplication of functions. To verify this assertion, it suffices to prove that the convolution (in the variable s) of endlessly-continuable function is, in term, an endlessly-continuable function.

Now we shall discuss the decomposition theorem for resurgent functions in several variables. However, it will be convenient for us to extend the definition of the Laplace transform of an extendible microfunction F'(s, x) to the case when the integration ray in this definition (see Subsection 1.4.2 of Chapter I) can meet some its singularity except for the one at the origin of this ray. In this case, there exist several possible definitions of the Laplace transform corresponding to integration contours encircling each of the additional singularity points from above or from below. The corresponding

Laplace transforms will be denoted by L., [F (s,z)], where w (the index) is an infinite tuple of pluses and minuses, e.g. w = (+,—-,—-,+,+,...), and each sign denotes the direction of encircling the corresponding point of singularity (+ for encircling from above and — for encircling from below; we suppose that singularity points lying on the integration ray are enumerated in the order this ray meets them).

Section 2. Resurgent Functions

123

Remark 2.1 Due to definitions given in this subsection, the direction of the integration ray can be chosen in any case parallel to the direction of the positive real axis. This means that in our formulas we can always choose 6 = 0. Then one and the same integration ray can meet points of singularity of the function F'(s,z) for one value of variables z and do not meet such singularities for another its value. The reader can verify that the decomposition theorem stated in Subsection 1.4.3 of Chapter I remains valid with this generalization of the Laplace

transform both in the case when the integration ray comes through some singular point of the function F'(s,z) and in the case when this ray is free of singular points. Thus, the following assertion is valid: Theorem 2.3 Let f(x) be a resurgent function. Then, for any fired value of the variables x and for any choice of the index w, the following decomposition

f (2) ==

-£51F;(s;2)]

(2.10)

Sj exu

takes place.

Here X., is the set of singular points of the function F (s,x) = B[f (z)| visible from the set Qo) (R’,€') along rays of direction @ = 0 and encircling singularities of F (s,x) from above for the + sign and from below for the — sign; Fj(s,x) are the corresponding microfunctions. Inversely, if

{F; (s,z), 7 =1,2,...} is a set of homogeneous microfunctions such that their supports 6,2... form a set which is contained in some angle of opening less than m bisected by the direction of positive real axis, then the right-hand part in (2.10) determines a resurgent function of the variables Eis We emphasize that the decompositions (2.10) corresponding to different indices w do not coincide with each other. This fact will be a topic of our consideration in one of the subsequent subsections of this section. In essence, Theorem 2.3 establishes the isomorphisms ® MM. Zont

a

R

(2.11)

sEC

corresponding to different directions of encircling of singular points. Here ®,ccMa,cont is the subspace of the direct sum ®secMs,cont such that

F (s,z) € ®feoMs,cont iff F(s,x) =

> F; (s,x), where F; (s,x) are micro-

functions whose supports belong to one and the same angle of opening less than 7 bisected by the direction of positive real axis (such an angle is shown on Figure 1.13 above). Isomorphism (2.11) is the inverse for the decomposition. This homomorphism is one of the main notions of the resurgent function theory. As we shall see below, the elements from Oya Viscount are elementary terms of asymptotics of the resurgent function when we

124

CHAPTER II. Resurgent Analysis

examine asymptotic expansions of resurgent functions, and the elements of each space Ms cont correspond to divergent series if we construct the resummation procedure for divergent series.

2.2.2

The connection homomorphism and the Stokes phenomenon

In this subsection, we shall examine how the decomposition (2.10) changes when we change the values of the variables x. There are two completely different cases of the behavior of the decomposition due to the variation of the independent variables. First of all, we shall carry out our considerations near points z € C,

such that all values S; (x) of the analytic (ramifying) function S(z) de scribing the set

p=

{(s,2) sisi 5 (x))

of singularities of the function F'(s,z) do not coincide with one another. Evidently, under this assumption, all values S; (xz) determine regular analytic functions in a neighborhood of each considered point z € Cy. Suppose first that, for some value Zo of the variable z, there exists no

one pair (S; (rz), Sz (x)), 7 # k, of singularity points of the function F (s, x) visible from its original domain of definition such that one of these points lay on the ray emanated from the another in the direction of the positive real axis, that is

Im; (2) = 1m op (x).

(2.12)

Then it is easy to see that the components’ C,, [F; (s,xz)] of decomposition (2.10) analytically depend on the variables z in some conical neighborhood of the point zo. In particular, the number of these components will be one and the same for all points z from the mentioned neighborhood. Quite another situation takes place when equality (2.12) holds for at least one pair of indices 7 # k (see Figure 2.1); in the situation shown on this

figure, that is, when Re S; (x) < Re S, (x), we shall say that the point S; (x) illuminates the point S;, (x) (along the direction of the positive real axis). In this case, the decomposition given by the inverse to the homomorphism to a, differs from that given by the inverse to the homomorphism q,)

corresponding to the different index w’ #4 w. To illustrate this fact, let us suppose for definiteness that the illuminated point lies at the upper side of the cut going from the point S; (z) along the direction of the positive real axis. Then, as it is shown on Figure

2.1 a), the point S, (z) is visible from AM) (R’,c') along this direction if 1In this case the subscript w in the expression L,, of the Laplace transform is inessential since any ray of integration used in the definition of the Laplace transform is free of singular points of the integrand.

Section 2. Resurgent Functions

125

Figure 2.1. Stokes phenomenon (revisited).

the cut used for the decomposition encircles singular point of the considered function from below. In this case, the element Lo) (f) contains the component corresponding to the singular point S; (zx). On the opposite, if we use cuts encircling singular points of F'(s,xz) from above (this means that we use the homomorphism |ire for the decomposition) then the point

S (xz) will not be visible from Ho) (R’,e’) and, hence, the microfunction corresponding to this singular point will not be included into the decom-

position Ly) (f). So, in the considered case, the element LO£4) (®)

(where & = Loy (f) © BlecMs,cont is the decomposition of the function f (z) from above) differs from the element F' by the microfunction F,(2) determined by thesingularity of the function F'(s,z) at the point S; (zx):

££) (#) - 8 = (koe; 7 id)(©) = Fs, (2): There exist two different ways of taking into account the above de scribed phenomenon. The first of these ways is based on the fact that, in the definition of the Laplace transform L, one can use different directions sufficiently close to the direction of the positive real axis. In this approach in a neighborhood of each point z, one has different representations of the function in question corresponding to different directions and the problem is to work out the tools for comparison of these different representations. It is almost evident that, for such a comparison, one needs only two definitions of the Laplace transform corresponding to the cases when all points

126

CHAPTER II. Resurgent Analysis

of singularity are encircled from the above and when all these points are encircled from the below. It is also clear that, in this case, the Stokes lines (that is, the lines on which the decomposition changes by jump) must be treated as some rays (directions) in the complex plane s. The second way is to fix the direction of the integration ray in the definition of the Laplace transform of microfunctions (the most appropriate choice of this direction is the direction of the positive real axis). In this approach, the Stokes surfaces are some surfaces in the space of the resurgent variables x (and the parameter variables if they present in the function in question). Then the problem is to compare decompositions of one and the same resurgent function corresponding to the two sides of the Stokes surface. Since the singularity points can approach the integration contour from different sides when the point x tends to the Stokes surface, it is clear that, for this approach, one needs generalization of the Laplace transform in which encircling of different singular points from different sides is allowed. Below we use the second approach to this problem since it is more convenient for the investigation of the ramification of the considered resurgent function and the notion of the Stokes surface for such an approach is more clear and classical. The above considerations motivate the following definitions:

Definition 2.3 For given hyperfunction F = F'(s,x), the set F of points xz € C,, such that

S; (z) = Sx (xz) for some j

Fk

is called a set of focal points of this hyperfunctions. In other words, the set

of focal points is exactly the ramification set of the function S(z).

Definition 2.4 The set of points z € C, \ F such that relation (2.12) holds for some pair of singular points S; (x), S, (x) of the function F (s, x) is called a Stokes surface of this function. In other words, the Stokes surface consists of those points x € C,,\F for which one of the singular points of the

function F'(s,x) illuminates the other. The connected domains bounded by the Stokes surface in C, \ F are called Stokes domains. Thus, the above considerations show that the components of the de composition a—! (f) of the resurgent function f depend analytically on the variables x inside each of the Stokes domains and can change by jump on the Stokes surfaces (which are boundaries of the Stokes domains). One can also see that the Stokes surfaces are smooth submanifolds of the real codimension 1 at all their points except for the set of focal points and the set of points x such that one of the singularity point illuminates more than one point of singularity. Of course, it can happen that the set of the Stokes

surfaces for some resurgent function f (x) is dense in the space of resurgent

Section 2. Resurgent Functions

12%

variables x. In what follows, we suppose that it is not so, that is, that the Stokes surface is a stratified set in the z-space of real codimension 1.

Let us consider now some point zo on the Stokes surface of some resurgent function f (xz) which is the point of regularity of its higher-dimensional strata. This means that the Stokes surface is a regular real submanifold in the space C? of the real codimension 1. Assume that some unit normal vector is chosen and fixed on the Stokes surface at the point zo. Then we can distinguish a point lying “above” and “below” the Stokes surface in a neighborhood of the point zo. Since the point zo is a point from the Stokes surface, there exists a nonempty set {93 (xo) eae (zo) siete YY

of singularity points of the function F'(s,r) = B[f (z)] lying on the integration contour [° Sj)(zo) emanated from the singularity point

S5q(20)5:550(20), F.S7x (xo) fork. = 1,2}... (We suppose that the points S;, (xo) are enumerated in the order in which they appear on the ray emanated from S;, (zo) in the direction of the positive real axis.) We denote by x5, the set of points from

195, (Lodo ATO) bert which approach the integration contour from above when the point x tends to the point zo from the positive side of the Stokes surface and by U5, the

similar set defined when the point xz tends to zo from the negative side of the Stokes surface. Denote now by wz the index wy

=(+,+,-,4,.-.),

where the — sign stands at all places corresponding to the points from x5,

and by w_ the index of the same form with — signs at all places corresponding to the points from Us, . Clearly, the decomposition using the Laplace transforms of the type C,, changes analytically when the point z moves from the Stokes surface in the positive direction and the decomposition using the Laplace transforms of the type C.— changes analytically when the point z moves from the Stokes surface in the negative direction. The above considerations motivate the following definition: Definition 2.5 The homomorphism dete US

=1 |Ome Loy

u >

@® M6s,cont sEC

we shall call a connection homomorphism.

u oe M s,cont sEC

128

CHAPTER II. Resurgent Analysis

The above discussion shows that the connection homomorphism describes the change of the decomposition when the value of the independent variable z comes through the Stokes surface in the direction opposite the direction of the positive normal (that is, from its positive to its negative side). In the sequel we shall use also the mapping def ,-] 6 = [nes Loy

: —id

f :

sEC

Ws

cont

ae

u we) M sEC

scant

which measures the jump of the decomposition when intersecting the Stokes surface. Evidently, the homomorphism 6 vanishes at points which do not belong to the Stokes surface.

2.2.3

Asymptotic expansions

In the next subsection, we shall investigate the asymptotic behavior of a resurgent function at infinity and the closely related procedure of resummating divergent power series. This investigation requires some knowledge of the theory of asymptotic expansions in the complex domain (having some features which are different from the corresponding real theory). This subsection presents a brief description of those notions and facts from the theory of asymptotic expansions in the complex domain which will be of use for us in the sequel. All the theory presented in this subsection concerns the case of analytic functions of one complex variable xz. For simplicity, we consider here asymptotic expansions at the origin; the asymptotic expansions at infinity can be obtained by the simple variable change y = z~!.

1. Let 2 be a sector in the complex plane C with its vertex at the origin:

ats

eee< argiz’ SF (to) (= 20) + 55f(w=) N £49 (W) dy. weet

n=0

1

x0

132

CHAPTER II. Resurgent Analysis

Let zo — 0 in the latter formula. We obtain then = je = Dae

i | ew

1

N

FOND(N41 (y) dy

n=0

re

1

Sy aysx nao

[a me) ft

n=0

0

ez) at.

The function

1

yn4i (zt) = = if(1 —t)* fN+) (tx) dt 0

is, evidently, bounded in 1’, so that the series )>° 5 anz™ give the asymptotic expansion of the function f (x) at the origin. The remaining affiirmations of the proposition are evident. Oo Let us turn our minds to the investigation of mapping (2.16). The kernel of this mapping consists of functions holomorphic in the sector 2 such that they tend to zero as x — 0 faster then any power |z|" uniformly in any 2’ < (2. The space of such functions we denote by P~ (QQ). The less trivial fact is that this mapping is an epimorphism. This fact, in essence, is exactly the affirmation of the following lemma which is known in literature as the Borel lemma. Lemma

2.1

The sequence

0 + P-(9) = G(2)8 Cf] + 0 1s exact.

Proof. Clearly, it suffices to prove only that the mapping As is an epimorphism. Without loss of generality, we can assume that the considered sector is bisected by the positive real axis, that is Q={xz : -%

< arg x < %,

|x| < R}.

Consider the function

b be ee ane! exp{ =} = exp { pe

}

(2.19)

(here z = re’). If b and @ are real and positive, we have the estimate

b b xp {=} = exp {00s 50}< exp {co}

(2.20)

Section 2. Resurgent Functions

133

in 2 with some positive constant c provided that the value of G is chosen

in such a way that G69 < 1/2. Estimate (2.20) shows that function (2.19) decreases in the sector 1 faster than any power of |x| and, hence, belongs

to the kernel of mapping (2.16). Let or. 9Qnx” be an arbitrary formal series. Consider the series

f(z) diva Lene ot ¢ (iat e vee ) with some positive constants b, which will be chosen later.

(2.21) Due to the

inequality |1 — e?| < C |z| valid for —7/2 +e < arg z < 1/2 —€

for any

€ > 0 with a constant C possibly dependent on e, we have the estimate for

the general term of series (2.21)

anes (1cent) alba a |z|

This estimate shows that there exists a choice of the constants b, such that series (2.21) converges and, hence, formula (2.21) determines a function holomorphic in the sector 2. The easy proof of the fact that this function

has the series )>~°,anz” as its asymptotic expansion is left to the reader. The proof is complete.

Oo

Remark 2.2 We note that this affirmation is not valid in the deleted neighborhood of the origin. Namely, in this case, the function f(z) admitting a power asymptotic expansion is a holomorphic function in the deleted neighborhood of the origin which is bounded at z = 0. Hence, the theorem on removable singularity shows that this function extends up to the function holomorphic near x = 0 and its asymptotic series converges. Thus, in this case, the image of the mapping As is a set of convergent power series. Remark 2.3 The analysis of proofs of the above affirmations allows one to suppose that the proved statements are not valid in the real case, that is, in the case when the opening of the considered sector equals zero. This is really so and the reader can construct the corresponding counterexamples by himself or herself.

2. Since our main goal in this book is the investigation of functions of exponential growth, we must consider the mapping As not on the whole space G(Q) of functions with power asymptotic expansions at the origin

but on the subspace of this space consisting of functions f (x) of not more than exponential growth:

If (z)| < CeM/ll, In such an investigation, the following statement known as the FragmenLindel6f principle is often of use:

134

CHAPTER II. Resurgent Analysis

Theorem 2.4 (the Fragmen-Lindeléf principle). A (the mazimum principle). Let a function f (x) be holomorphic in the sector of the opening less than x, continuous on the boundary of this sector (except for, possibly, the origin) and have in this sector not more than exponential growth. Then the following estimate

sup [f (2)| $ sup |f(2)| rE

(2.22)

zEon

as valid. B (the quasianalyticity principle). If a function f (x) exponentially decreases in the sector Q with the opening more than x then f(x) = 0 in gy Proof. 1) Let 2 be a sector of opening less than a bisected by the positive real axis: se

ER

Eo se Las BG Ie] < R}

with some positive e and R. Consider function (2.19) with some positive

value of b such that 3 (1/2 —e) < 1/2. Then, as above, estimate (2.20) is valid with some positive constant c. Choose the value of 3 subject to the inequality L +oo. The uniqueness theorem for holomorphic functions completes the proof.

oO

3. Let us introduce a subspace in the space of formal power series such that the formal Borel transform of this series determines an analytic microfunction at € = 0. As we shall see below, these are power series which can be asymptotic expansions at the origin of resurgent functions (if the latter do have a power asymptotic expansion at this point). The formal Borel transform is given by the formula co

By

n| > anZt n=0

def

1

CS 9 Sree!

201

a

=

ce

2 + Ing 2 ant ay

(2.23)

due to the definition of the formal Borel transform given in Subsection 1.1.2 below. One has to take into account that here we consider asymptotic expansions at the origin instead of examination of such expansions at infinity performed in Chapter I and that we use here microfunctions instead of their variations which allows us to take into consideration formal power series with a zero term. We see that, for (2.23) to define an analytic microfunction, we must require that the series in the right in this formula converge in some neighborhood of the origin and, hence, the coefficients a, of the series

Get n=0

(2.24)

136

CHAPTER

II. Resurgent Analysis

Figure 2.3. Illustration to the proof of Theorem 2.4.

must satisfy the inequalities

lan]

< CM™n!

(2.25)

with some positive constants C and M.

Definition 2.7 The series (2.24) is of type Gevrey 1 if its coefficients satisfy estimate (2.25). The space of formal power series of type Gevrey 1 will

be denoted by G; [[z]]. Now we introduce the subclass G, (92) of the class G (Q) of holomorphic functions with power asymptotics at the origin which consists of functions

f (x) such that the estimate 1

sup = fo (z)| E;°(2) > Gi(9) F G [x] > 0 for any sector 2 of opening less than 7.

QO

140 Remark

CHAPTER II. Resurgent Analysis 2.4 If the reader is familiar with the sheaf theory, he or she had

noticed that the spaces E;°?()

and

G,()

determine sheaves on the

unit circle S' = {|r| = 1} in the complex plane C,. If we denote these sheaves by €; ° and G,, correspondingly, then the above affirmation about the homomorphism As; can be formulated as follows: The sequence of sheaves

OS

= Gr ee Gia

0,

where G, [z]] is a constant sheaf on S', is exact.

Later on, if l is an arc

on the circle S', then the mapping Asi

: T'(l,91) —

G, [[z]]

is a monomorphism for |l| > x and an epimorphism if |l| < 7.

2.2.4

Resurgent functions with simple singularities

The most simple way to come to the notion of resurgent function with simple singularities is to consider the Borel resummation procedure for divergent power series. To simplify our presentation, we shall begin with the consideration of the one-dimensional case postponing the consideration of the general case to the end of this subsection. Here we again consider asymptotic expansions of analytic functions at infinity, so when applying the affirmation of the previous subsection, one must perform the variable

change rz > x7!. 1. Let us, first of all, formulate in the explicit form what is the resummation procedure. If

hontay

|

(2.32)

=0

is a formal power series, then to resummate this series means to find a function f (x) for which this series will be an asymptotic expansion. Of course, it is desirable that the introduced resummation procedure possesses some natural properties; for example, the result of resummation of the sum of the two power series must be equal to the sum of results of resummation of each of these series, and the similar affirmation must take place for the product of formal series. In other words, the resummation operator o

must be in some sense inverse to the algebra homomorphism As of taking asymptotic expansion examined in the previous subsection.

As it follows from the results of the previous subsection, the Borel resummation procedure based on the Borel-Laplace transform is applicable, not to the algebra G of all formal series, but only to its subalgebra

Section 2. Resurgent Functions Gi te

of power

141

series of Gevrey

1 type.

Later on, the results of

Propositions 2.5 and 2.6 show that only the investigation of asymptotic expansions in sectors (22 of the opening less than a makes sense (only in this case the corresponding homomorphism As; is an epimorphism). This fact is very closely related to the above considered Stokes phenomenon since from these propositions follow that, as a rule, asymptotic expansions of one and the same function f (x) will differ from each other in different sectors of the complex plane. We remark here that, in essence, the resummation procedure for formal

power series of the form (2.32) from the space G; [[z~"]] (in a sector 2 of the opening less than 7) up to exponentially decreasing term was constructed during the proof of Proposition 2.5. Namely, we have shown there

that, for any power series (2.32) from G, [[r~"]], the Laplace transform from the formal Borel transform of this series gives the function f (xz) for which the asymptotic expansion in the sector 2 at the origin coincides with the initial power series. Thus, the mapping

o' =LoBy

: Gi [[z~"]] > Gi (Q) /EZ° (Q)

is an isomorphism which determines the resummation procedure up to exponentially decreasing functions. However, if we want to construct the resummation procedure up to

rapidly decreasing terms, then even the class G, [[z~*]] occurs to be too large. Actually, the natural idea to construct the resummation procedure up to rapidly decreasing terms is to continue analytically the formal Borel transform F (€) = By »Sona n=O)

and then take the Laplace transform of the extendible microfunction which is determined up to rapidly decreasing terms (see Subsection 1.4.2 of Chapter I). Thus, we must consider the subclass of the algebra G; [[ost] consisting of those power series for which its formal Borel transform will be an endlessly continuable function. Thus, we have come to the following definition:

Definition 2.8 Formal series (2.32) is called to be a resurgent power series if its formal Borel transform is an endlessly continuable function. In this case, the composition £ o By determines

C= 6 0Be 3s Gy Weal

a homomorphism

— G,;(Q) /E;® (Q)

(2.33)

which acts to the space of resurgent functions of one variable z. However, the image of homomorphism (2.33) does not coincide with the space of all resurgent functions. The matter is that the singularity of the

142

CHAPTER II. Resurgent Analysis

microfunction =

By Isone n=0

corresponding to a power series are of a rather special type

RAEP+mey ons“] FCG) ar

(2.34)

(see formula (2.30) above). On the opposite side, if a microfunction F' (£) € M@ will have singularities of another type than (2.34), its Laplace transform will not have an asymptotic expansion of power type at the origin in the corresponding sector of the complex plane C,. For example, if the singularity of a microfunction F'(€) is

yeLonge g sai) Tr

F(é)=€

with some noninteger o, then the asymptotic expansion of the corresponding function f (x) will contain non integer powers of x:

f@yic SD ascot n=0

Certainly, a microfunction can have singularities of much more general type at the origin. The class of resurgent functions admitting the power asymptotic expansions is described in the following definition:

Definition 2.9 A resurgent function f (z) is called to be a resurgent function with simple singularities if its Borel transform (which is an endlessly continuable function due to the definition of a resurgent function) has singularities of the type

F()= 95 [pete tn

6) Dan EE

(2.35)

n=0

at each its singular point £. Combining the results on the decomposition obtained in Subsection 1.4.3 of Chapter I and in Subsection 2.2.1 of this chapter with the results on asymptotic expansions obtained in the previous subsection, we come to the following affirmation:

Section 2. Resurgent Functions

143

Theorem 2.5 If f(x) is a resurgent function with simple singularities, then, in any sector 2 of a sufficiently small opening, it has the asymptotic ezpansion of the form

Poe

3 e fee Soalk) e—”, k

n=0

where € are the points of singularity of the function F (€) = B[f (z)] included into the decomposition along the chosen direction @ and all the series on the right are resummated with the help of the Laplace transform L., described in Subsection 2.2.2 above.

2. Let us proceed now with the investigation of the multidimensional

case (so that from now on x denotes a point z = (z',...,2”) of the ndimensional Cartesian complex space C?). First of all, we note that the generalization of the notion of a power series is not quite clear in the multidimensional case. That is why we shall begin our considerations directly from the notion of a resurgent function with simple singularities and investigation of the form of asymptotic expansions of such functions at infinity. The definition of a resurgent function with simple singularities in the multidimensional case is quite similar to that given above for one dimension.

Definition 2.10 A resurgent function f(z) is called to be a resurgent function with simple singularities if its Borel transform (which is an endlessly continuable function due to the definition of a resurgent function) have singularities of the type

F(s,r) = el ee

pa.) +In(s — S(z)) SaAn+i (2) Cree n=0

at each ere point s = S(z). We note that, due to the homogeneity properties of the function F'(s, x), the function S (z) (the action) describing the singularity set of the function

F (s,z) is a homogeneous function of order 1 as well as the functions a, (x) (amplitude functions) are homogeneous functions of order —n. Then, quite similar to Theorem 2.5 above, we obtain the following result: Theorem 2.6 If f(z) is a resurgent function with simple singularities, then for any ray | = {Az,X € Ry} there exists a conical neighborhood K, of the ray | such that f(x) has the asymptotic expansion of the form

f(z) = paar n=0

a*) (x)

(2.36)

144

CHAPTER

II. Resurgent Analysis

in Ki, where S, (x) are branches of the function S(x) corresponding to those points of singularity of the function F (€) = B[f (x)] which are included into the decomposition for given value of z, ass) (x) are the corresponding branches of the amplitude functions a, (x) and all the series on the right are resummated with the help of the Laplace transform Ly described in Subsection 2.2.2 above.

Thus, the natural object replacing power series in the multidimensional case is the series of homogeneous functions with decreasing order of homogeneity of the type >, as) (x). The theory of resummating of such series goes quite similar to that in the one-dimensional case. 3. To conclude this subsection, we shall discuss one feature of the developed theory concerning asymptotics of resurgent functions at infinity which shows the necessity of generalization of the introduced notions. Since this feature is quite clear even for one-dimensional case, we shall restrict ourselves to the consideration of this rather simple case. Let us first consider an example. The function

f (x) = e*tV® is an element from EF, ({Q), where 2 is a right half-plane Rex > 0. Let us show that this function is a resurgent function of the variable zr. We have co

F (£) = B[f(z)| = = /eft -2tVF do 0

which converges for all values of € such that Re€ < 1. Performing the variable change y = r(1 — €) and deforming the integration contour again to the real axis, we obtain

FO

=

aang

| or

(H+ Jul V1=€) dy.

The latter formula shows that the function F (€) is an endlessly continuable function with the only singularity point € = 1. Therefore, the asymptotic expansion of this function must be written down in the form

f(z) =e“a(z), where the “amplitude function” a (x) equals to ev? and, hence, possesses no reasonable asymptotic expansion at infinity. Thus, the considered function is not in any case a resurgent function with simple singularities in the sense of the above definitions.

Section 2. Resurgent Functions

145

From the other hand, it is absolutely clear that this function must be represented in the form

f(z) =e" Sa(z), where the action S(x) equals z + \/z and the amplitude function a(z) equals 1. However, we cannot obtain such a simple asymptotics from our

above constructions! The reason for such phenomenon is that our definitions allow only homogeneous functions S (xz) as actions in asymptotic expansions of the considered functions. If we refuse from this (rather strong) restriction, then we can write down the representation for the considered function in the form

1

e “ds

f=

r

[|

of resurgent functions with simple singularities. So, to include into consideration functions of the considered type, we must examine representations of the type

f(xi)=— [eres 271

ip

with nonhomogeneous hyperfunctions F'(s,z). the topic of the following Subsection.

2.2.5

ds Such examination will be

Generalizations of the notion of a resurgent function. Resurgent representation

In this subsection, we present the two generalizations of the introduced notion of a resurgent function. The first of these generalizations concerns the extension of the notion of resurgent function to the spaces of analytic functions which have exponential growth with an arbitrary order k (unlike the above presentation, where k = 1). The second concerns the notion of resurgent representation; the preliminary discussion of this notion was done in the end of the previous subsection. 1. A lot of problems to which the resurgent analysis is applicable require the investigation of the spaces of functions which have the exponential growth at infinity with some order k > 1. (As an example of such problems, we mention here the problem of investigating the behavior of solutions to partial differential equations at infinity which shall be considered in the next chapter.) To extend our definitions to this case we shall introduce the following modification of the above introduced notions.

Let k be some positive integer and let f(z) € O(Kr) be an analytic function in the set Kr (defined in the beginning of this section) with exponential growth of order k at infinity. This means that the function f (z)

146

CHAPTER II. Resurgent Analysis

satisfies the estimate

If (x)| < Ce™'

k

in Kr with some positive constants C and M. The space of analytic functions in Kr with exponential growth of order k at infinity we denote

by Ex (Kr). Consider the function 1—k

1

1

n

fe Misnitat (néat,....nFz ) of one complex variable 7 € C. Evidently, this function is a function with

exponential growth of order 1 in some sector So (R’,e€’) bisected by the direction of the positive real axis:

fz (n) € E (So (R’,e')). Thus, the Borel transform B [f (x)] eB [fz (n)] of this function is defined and we denote 1

rs

re

1

1

F(s,z) = Blfz (n)] = 5 pert s (ae nt x”) dn. A

Similar to the case k = 1, one can check that this function is homogeneous of order —1 with respect to the following action of the group C, of nonvanishing complex numbers:

Fe(APS At) = Ne UG,a)

(2.37)

and that the inverse (Laplace) transform is given by the formula

f (2 =O LF (9:3) |= fevr (s,z) ds.

(2.38)

7

The interesting fact is that form (2.38) of the Laplace transform for the arbitrary value of k is just the same as for exponential functions of order 1; all the difference is in the homogeneity properties (2.37) of the hyperfunction

Fs; 2). All the theory developed above for the case k = 1 is valid for arbitrary positive integer k; the verification of this fact is left to the reader. We present here only the commutation formulas for transform (2.38) with the operators of differentiation and multiplying by an independent variable since these formulas differ from that given above for the case k = 1. The following statement is valid:

Section 2. Resurgent Functions

147

Proposition 2.7 For the Laplace transform in space of functions of exponential growth of order k the following commutation formulas

a

—L(F)

Oxi

=

F]

BN

=

c|(2)

aa,

Deiat

Fs[»(2) F

ti L[F| hold.

The statement of this proposition needs some comments concerning the definition of the operator (0/0s)*. To define this operator, we introduce the family of Borel transforms parameterized by the parameter 2 € R by the formula

BF f(a) = B[ fo )], fo) =a sf(att... nto”). The function F'(s,z) = B*[f (zx)] is a homogeneous function of degree —1-—-£

with respect to the above introduced action of the group C,.

Evidently, in this notation B[f (z)] = B° [f (z)]. The interesting fact is that, for all these transforms, the inverse is given by one and the same

formula (2.38) with functions of different order of homogeneity under the integral sign. We denote these inverse transforms by L°. The operator

(0/0s)* is now defined as

(0/ds)* & Bet o LA. The introduced family of Borel-Laplace transforms allows one to write down the version of commutation formulas which will not depend on the order k of the considered functions of exponential type. Such commutation formulas are more natural since the form of the Laplace transform does not depend on k and, hence, the commutation formulas should not also depend on this order. The following statement describes these commutation formulas:

Proposition 2.8

The following commutation formulas

a Dai

p

(F]

eds

oO pRB at Ad ofON Ain Se a (=) maa

2iLP(F] = £9+ [riFl hold. We leave to the reader the easy proofs of Propositions 2.7 and 2.8. If one takes into account that the upper indices in the Laplace transform indicate only the orders of homogeneity (and, hence, can be omitted

148

CHAPTER II. Resurgent Analysis

without any loss of information), it will be clear that the latter commutation formulas do not really depend on the value of k. This is especially important in view of the next generalization in which the function F'(s,z) will not be a homogeneous function at all. To conclude the considerations of resurgent representations with homogeneous functions F'(s,z), we shall mention that, in some cases, it is useful to consider more general action of the group C, of nonvanishing complex numbers than the above considered one. Namely, if we use functions F'(s,x) homogeneous of some order with respect to the following action of the group C,: (aa

enya)

=

(SSRs

aA?)

then we can include into consideration analytic functions with different orders of exponential growth in different directions.

2. The next generalization that we shall consider here is the integral representation

f (2)=L[F (s,2)] reveom (s,x) ds,

(2.39)

where each contour I’; encircles one of the points of singularity of the function F'(s,z) and goes to infinity along the direction of the positive real axis. The sum in (2.39) is taken over some subset of the set Ur of singular points of the function F'(s,z); this subset is referred below as the support

of the function f(z). In the introduced representation, we do not require any homogeneity properties of the function F'(s,z).

We emphasize that representation (2.39) of the function f(z) is not unique.

Actually, even in the case when we use homogeneous functions

F'(s,x) in (2.39) we can obtain different representations of one and the same function f (xz) composing the Borel transforms B* for different values of @ with the decomposition procedure described in Subsection 1.4.3 of Chapter I. The usage of arbitrary (nonhomogeneous, in general) functions F'(s,x) gives an additional degree of freedom and, as a result, the repre-

sentation given by (2.39) with arbitrary functions F'(s,z) is not unique at all. The mentioned additional degrees of freedom allow one to overcome difficulties of the type described in the end of the previous subsection. However, everything must be paid, and the obtained additional degrees of freedom which allow to represent asymptotic expansions of WKB type

with inhomogeneous action as a resurgent (in this new sense) function with simple singularities supply us also with some additional difficulties. Actually, if we require no properties of the functions F'(s,z) involved into the considered representation (2.39), then the convergence of integrals, validity of the decomposition procedure etc. occur to be under a question sign. In other words, we must introduce some other requirements on the function F'(s,z) instead of its homogeneity, for all affirmations on resurgent

Section 3. Legendre Uniformization

149

functions in classical sense to remain valid for the introduced generalization. Such requirements can be introduced in a lot of different ways and depend on the problem to which the resurgent analysis is applied at the

moment.

Here we shall mention one class of functions F'(s,z) for which

main features of the resurgent functions theory (except for the invertibility

theorem, of course) remain valid and which will be used in the next chapter when investigating asymptotic properties at infinity of solutions to partial differential equations with polynomial coefficients. Namely, if we consider the class of functions F'(s,z) which are restrictions to the plane ¢ = 1 of functions F'\(s,z,¢), ¢ € C, homogeneous of certain order a with respect to the variables (s,z,¢) in the following sense:

F (\*3,A2, XC) = A°F(8, 2,0) (such functions F'(s,z) we shall call asymptotically homogeneous), then all the above affirmations hold. To conclude this subsection, we must discuss a question of commutation formulas with differentiation and multiplying by an independent variable for representation (2.39). The matter is that, due to the ambiguity of the considered representation, one can write down different commutation formulas. For example, the commutation formula fa)

OQ

ail Wg Cia] ee Peace) is valid for this representation as well as the formula

—1

qe [F(s,x)]) =L (3)

oF (s,2)

included into Proposition 2.8. We claim that the commutation formulas of the latter form are the most suitable ones for application to partial differential equations since these formulas give the correct quantization of

the obtained differential equation for the function F'(s,z).

However, we

postpone the discussion on this question until the next chapter, where we shall consider different applications of the resurgent analysis.

2.3

Investigation Near Focal Points. Legendre Uniformization

In this and the next sections, we investigate the structure of resurgent functions near their focal points. This consideration includes two particular questions.

150

CHAPTER

II. Resurgent Analysis

First of them is the uniform representation of a resurgent function f (x) in a neighborhood of some of its focal points. Such a representation is possible in the case when the set /- of singularities of the corresponding

hyperfunction F'(€) is described by an action S (zx):

re

(652) 28 = (2)

(2.40)

such that the corresponding Lagrangian manifold

OAS Ls

=

{@2)

Say tia

3

= | CCT

OCr»

in the phase space C? © C,,, admits the continuation up to a nonsingular Lagrangian manifold over a neighborhood of the considered focal point. Here we obtain a natural generalization of the notion of a resurgent function with simple singularities which is invariant with respect to solutions of partial differential equations. The second question is the investigation of resurgent functions which are univalued in a neighborhood of the given focal point (such a situation often arises in the asymptotic investigation of solutions to differential equations). It occurs that, for the considered resurgent function to be univalued in a vicinity of some focal point, this function must satisfy some additional conditions which can be written down in the form of the so-called resurgent equations. For writing down these equations, the notion of alient derivative of the resurgent function at points of Stokes surface occurs to be useful and we introduce and investigate this notion in the following presentation. Besides, we present the algorithm of writing down resurgent equations using the known resurgent structure of the considered function, that is, the structure of singularities Lp of the corresponding hyperfunction F (s,z). This question will be investigated in the next section.

So now we shall investigate the representations of the form

f(a) by [evra

ds

(2.41)

ig, near a focal point zo € F of the function f(z). We recall that the set of focal points of the function f (z) is defined as a set of ramification of the action S (zx) describing the set of singularities of the function F'(s,x) (see formula (2.40) above). We emphasize that representation (2.41) is not, in general, valid at focal points since at these points an integration contour can be “pinched” between singularity points of the function F'(s,z). So, there arises a problem of writing down a representation uniform in a neighborhood of focal points. Later on, for simplicity, we shall consider here only regular hyperfunctions F'(s,zx), that is, such hyperfunctions which satisfy the estimate

|F (s,x)| < C|s - S(x)|"

(2.42)

Section 3. Legendre Uniformization

151

with some constants C > 0 and g > —1 in a neighborhood of each point of regularity of the action S(z). The space of functions F'(s,z) satisfying estimate (2.42) with the given value of g we denote by A, (=) (we consider the set of functions with one and the same singularity set D = Dp).

Since the validity of inequality (2.42) does not depend on the choice of the representative F'(s,x) of a hyperfunction F’, we can consider A, (2) as a subspace of the space of (generalized) endlessly continuable hyperfunctions. Since, for any regular hyperfunction F’,, its variation varF is an integrable function at the point s = S(z), representation (2.41) for such functions can be rewritten in the form

oe > fe“E(s.2) ds,

(2.43)

where, for brevity, we denoted var F'(s,z) by F'(s,z). To obtain a representation of the function f(z) near its focal point we shall use the so-called 0/0s-transform of ramifying analytic functions introduced by the authors in [150], [157]. Here we briefly recall the main statements and definitions of the theory of 0/0s-transform (see Appendix).

Let F(s,z) be a function from the space A,(z). 0/0s-transform of this function is defined by the integral a

Fe

i

n/2

Fi(s}p) = 2/2? (R(s,z)| = (=)

a

Then

n/2

(=)

s F(s —zp,z) dz, 3,p

where zp = z'p, +... +2"pn and h(s,p) is a ramifying homology class which is determined as follows:

Consider the plane L,,, in the complex space C"*+! with coordinates

(,x) defined by the equation Dee (s2)

s= s— ap.

Suppose that the plane L,,.,, has the quadratic contact with the set & at

some point (50,20). Then for any (s,p) sufficiently close to (so, p09) such

that L,,, is not tangent to ¥, the intersection L,,, 1 & is homeomorphic to a complex quadrics and we define h(s,p) as the vanishing class of this quadrics:

BAS) Calla

Lain, die p (hs)

The class h(s,p) is then extended to other values of (s,p) with the help of the Thom theorem. The inverse transform is given by the formula

re [Ronl£ (=) (FZ) f Re+enn a e

he

4

n/2

a

n/2

'

h(s,z)

152

CHAPTER II. Resurgent Analysis

where the class h(s,x) is defined similar to the above described class

h(s,p). Now let F'(s,xz) be, as above, an endlessly continuable homogeneous hyperfunction with singularities on the set

u= {(s,r) > s=S(z)}, where S(z) is an (in general ramifying) function in z. Require in addition that the homogeneous (with respect to the variables x) Lagrangian manifold

Ls={ (a)

> p=

OS

(x

=)

determined by the function S(z) outside the set F¥ of focal points of the hyperfunction F'(s,xz) can be continued up to a regular Lagrangian manifold over a neighborhood of a point x9 € F. Then (see, for example,

[114]) there exists a set of indices J C {1,2,...n} such that the equations of the Lagrangian manifold Ls in a neighborhood of the point zo can be represented in the form

OS;(z',p7)

+

DF = b= {tan pons elaast)

OS; (x! , pz

7) SSS oP T

with some regular function S; (x’,p7) homogeneous of order 1 with respect to the variables x’. Here J is the complement of J in the set {1,2,...n}, and we use the notation gi = (att ee ath) fOPSHtp

etc.

ess thee

We remark that the function S; (z’,p7) is then a solution to the

equation

dS; = prdz! — 2"dpy| written down in the local coordinates (z’,p7) on the manifold Ls. Such

function is called a (singular) action in the canonical chart (x',p;) of the manifold Ls. The main idea of the representation of the considered resurgent function

f (z) in a neighborhood of some of its focal points is to represent the corresponding hyperfunction F'(s,z) in the form of the inverse 0/0s-transform of a function Fy (s¥r" PT) which has simple singularities in the sense of Definition 2.10:

F, (s, 2", p7) =In (s-S(z',pz))

plea ebies

>> k=0

(2.44)

Section 3. Legendre Uniformization

153

(we omit the residual term since we consider here only regular hyperfunctions). Thus, we consider the following representation of the function F'(s,z) near its focal point zo: 0/Os

Ne

Bs ater[Fi (s, 2’, p7)|

(2.45)

I

- \ [7/2 =

ete (=)

a

|7|/2

weet (=)

v FY (s+2 ifPyz I Pr) dpr.h(s,z)

So we arrive at the following definition:

Definition 2.11 A resurgent function is called a resurgent function with simple singularities in a neighborhood of its focal point Zo if its Borel trans-

form F'(s,xz) can be represented in the form (2.45) with a (homogeneous) function Fy (s, 2", pz) which has singularities of the type (2.44) near its

singularity set Dy; = {s = S; (x’,p7)}. The following two statements show the correctness of the given definition: Proposition 2.9 If the function Fy (s, x! ,p7) is homogeneous with respect to the variables (s\2.) of degree |7| /2+k+1, then the function F'(s,z)

given by (2.45) is a homogeneous function of degree k with respect to (s, x). Proof. The only nontrivial point in the proof of this proposition is the verification of the fact that the ramified class h(s,z) is invariant under the action of the group C,. This fact, however, is an easy consequence of

homogeneity of the function S; (z’,p7) together with the above definition of the ramifying class h since the intersection

fee

{Pr

st+2'pr= Sy (x', pz)}

is C,-invariant. Here the equation of the plane lee is

. Lge {(3,p7) : 3=s+27pr} and

% = {5 = 5S; (2',p7)}

is the equation of the singularity set of the function Fy (s, x! . P7)- This completes the proof of the proposition. Oo

Proposition 2.10 The singularity set of integral (2.45) is given by the equation s = S(zx). If the function Fy (s, x", p7) has singularities of the type (2.44) then the function F (s,x) given by (2.45) has simple singularities in a neighborhood of each its nonfocal point.

154

CHAPTER II. Resurgent Analysis The proof of this proposition can be given with the help of the stationary

phase method for integrals of the above type carried out in the book [157] and we leave this proof to the reader. Now we can write down the integral representation of a resurgent func-

tion f (z) near its focal points.

To do this we note that, if the function

F'(s,x) is given by (2.45), then its variation F(s,z) has the form of an integral

: = i Fy (spy, © Pr) dp;,

i\MIZzs7a\lle (=) F(s,2) = (=)

(2.46)

hi(s,z)

where the class hy (s, x) can be constructed from the class h(s,z) as the sum of two copies of this latter class with opposite orientation lying on different sheets of the Riemannian surface of the function Fy (s+ apr, zi, Pr). The

class h; (s,x) can be considered as a homology class of the complement to the singularity set of the function F; in jing encircling this singularity set

as it is shown on Figure 2.4. Substituting (2.46) in representation (2.43) we obtain the above mentioned representation of the function f (x) near its focal points as a sum of the following terms:

JeUa)

i \M2 (5) pg \lt ei

di(=) bot

VW

( i F 1(3+2

mene py,x

,pr)\dapy p ds

hi(s,z)

Taking into account that the contour hi (s,z) of integration in the inner integral on the right in the latter formula vanishes at the origin point of each ray 7;, one can rewrite the latter integral in the form

f (2)

1 "2 (=)

/ eg Fy (st+2TPy, I ,Pr) ds A dp; H(z)

1 \lI/ A (=) / ere err (s,2’,p7) ds Adp;

(2.47)

H(z)

over an absolute (though noncompact) cycle H (x) in space with the coor-

dinates (s,p7). Clearly, the real part Re s increases along the (multidimensional) contour H (z), so that the latter integral converges. The right-hand part of formula (2.47) gives us the required representation.

In conclusion, we shall derive the formula for the asymptotic expansion for the function f(z) near its focal points. To do this, we deform the contour H (z) along the positive part of the real axis s for each fixed py until

Section 3. Legendre Uniformizati on os nseaieioren Otorleett eae seateloeroth

155 ed

rr

h,(s,x)

Singularities of F,

Figure 2.4. The homology class hi(s,z). it takes the form shown on Figure 2.5. We denote by A (z) the projection

of the deformed contour on the plane p; and by

(s/) = ih (s} (z',p7))

the contours in the plane C, shown on Figure 2.5. Then we obtain

n=

1

(=)

\!7|/2

= clery

H(z)

i e °F (s, x’, pz) ds / dpr.

; nse)

Applying to the inner integrals on the right in the latter formula the usual asymptotic expansion for a resurgent function with simple singularities, we obtain the required asymptotic expansion for the resurgent function f (zx) near its focal points: co

|7|/2

Tr

ary

fe=DY 271(sa) f eer si'wal, (e!,m=)ha

(248)

A(z)

It is easy to see that the functions ajI (x? , Pz) are homogeneous functions with respect to the variables x! of order |7| /2-—k-1. Clearly, in the general situation, one cannot write down a more explicit asymptotic expansion for the function f(z). However, for concrete Lagrangian manifolds, one can obtain more explicit asymptotics with the help of different special functions. This can be done by reducing the integrals involved into the right-hand side of (2.48) to the normal form. This

156

CHAPTER II. Resurgent Analysis

Figure 2.5. Projection of H(z) to the plane C,.

procedure is out of the framework of the book and we do not consider it here.

2.4

2.4.1

Investigation Near Focal Points. Connection Homomorphism

The monodromy

properties of a resurgent

function The aim of this subsection? is to investigate the relation between formal and real monodromies of the resurgent function. The statement of this problem is as follows: Consider a resurgent function f (x) with simple singularities (see Definition 2.10 above). Then, due to Theorem 2.6, this function has an asymp2This subsection A. Borisovich.

as well as the Section

2.5 was

worked

out

in collaboration

with

Section 4. Connection po reso ek See thfowlsHomomorphis bin aches sh moer Or

4 157

totic expansion of the form

Jia) = oe e7 Sk(=) yy. al*) (x) k

(2.49)

n=0

outside the set F of its focal points. Here S; (x) for different k are the values of (ramifying in general) function S (x) (an action), and the functions a‘? (x) are homogeneous functions with respect to the variable z (an amplitude functions) with decreasing in n orders of homogeneity. functions for different k are values of ramifying functions ap (z).

These

The terms

e7 Sk(z) oe al*) (x) n=0

included into the right-hand side of (2.49) correspond to the microfunctions

1 [ af (2) Qni 3 2s, — S; (x)

+ In(s — S; (z)) Ssa) (a peeereoue

(2.50)

n=0

included in the decomposition of the function F'(s,z) = B[f (z)] at the given point zx.

If we perform the analytic continuation of the function f (x) along some loop ! surrounding the set F of its focal points we shall come, in general, to another sheet of the Riemannian surface of the function f(z). Thus, the fundamental group of the complement C?\F acts as a group of transforms of the Riemannian surface of f (x); this action is called a real monodromy

of the resurgent function f (zx). From the other hand, the fundamental group of the complement C7 \ F evidently acts on the set of microfunctions determined by the hyperfunction F(s,x). Due to the relation between these microfunctions and terms of asymptotic expansion (2.49), this action can be considered as an action on the space of asymptotic series which can be included into the asymptotic expansion of the function f(z). Therefore, this action we call a formal monodromy corresponding to the given resurgent function. Due to the Stokes phenomenon, the relation between real and formal monodromies of one and the same resurgent function f (x) is not a direct one. In other words, the asymptotic expansion of the analytic continuation of the function in question along the loop / does not coincide with the analytic continuation of its asymptotic expansion. In particular, the functions

S(z) and a, (x) can have ramification along the loop / even in the case when the function f (z) itself is univalued along this loop. This situation is, in particular, typical for the applications of the resurgent analysis to the asymptotic investigation of solutions to differential equations. Thus, there arises a problem of writing down the conditions on the ramification of asymptotic expansions of resurgent functions necessary and

158

CHAPTER II. Resurgent Analysis

sufficient for this function to be a univalued one. Such conditions, usually called resurgent equations in the resurgent functions theory, can be written down it terms of the connection homomorphism defined in Subsection 2.2.2 above, and we present in this subsection a general method of their construction.

We remark here that the local structure of the Riemannian surface of the hyperfunction F'(s,x) corresponding to the given resurgent function f (x) is fixed by the requirement that this resurgent function has simple singularities. In this case, the ramification of the function F'(s,z) near each of its singular points S; (x) is given by formula (2.50). However, this requirement does not provide us with any information about the global structure of this Riemannian surface. As we shall see below, the above mentioned resurgent equations supply one with such a global information and allows us to investigate the Stokes phenomenon.

Let us proceed with the constructing of resurgent equations. For simplicity we suppose that the considered resurgent function has only the finite number of singular points DP,

{O1 (Lyre.

ON ZY

(2.51)

(that is, the function S (z) describing the set of singularities of the function F (s,z) has the ramification of finite multiplicity). Let, as above, | be a loop surrounding the set F of focal points with the origin (and the endpoint) at a certain point zo. Suppose that, for any given value of xz € I, each singular point S; (xz) can illuminate at most one other

singular point S, (2). We remark that the singular points (2.51) cannot be enumerated in one and the same way at all points x. However, since these singular points change their values in a regular way along the loop !, we can fix some numbering along this loop having in mind that numbers of one and the same singular points at the endpoints of this loop can be different. The relation between the numbers of singular points at x9 corresponding to the origin and to the endpoint of the loop | is given by a permutation a of the set {1,...N} so that the point S; (x) is taken to the point S,,;) (x) during the analytic continuation along the loop I. Clearly, the permutation a describes the formal monodromy of the considered resurgent function along l. Further, without loss of generality, one can assume that the loop l transversally intersects the Stokes surface of f (xz) at points of regularity of this surface. We denote by zo,..., 2 the points of intersection of the loop | with the Stokes surface in the order they appear when moving along |. (We suppose that the origin zo of the loop I is one of the points of intersection of this loop with the Stokes surface.) Due to the above assumptions, the decomposition of the function f(z) into the sum of Laplace transforms of

Section 4. Connection Homomorphism

159

microfunctions has the form

N

TG) = oe ely,

(2.52)

k=1 where Fy = Fi, (s,z),

k = 1,...,N is a microfunction determined at point

s = S,(xz) by Borel transform B[f] = F(s,zx) of the function f(z). Clearly, the decomposition of the form (2.52) takes place at all points z € I except for the points zo,...,x2y of intersection of the loop | with the Stokes surface. To write down the form of the decomposition at points of the Stokes surface, we remind some constructions introduced above in Subsection 2.2.2 when investigating a connection homomorphism. We shall fix the orientation of the Stokes surface such that the loop | comes from the negative to the positive side of this surface when one moves along ! in the positive di-

rection. Denote by J, = 1,(x) the ray emanated from the point S;(z) along the direction of the positive real axis. For any point z;, 7 = 0,1,...,.N we

denote, as above, by ue (xj) (correspondingly, Us, (x;)) the set of singular points of the function F'(s,r) coming from above to the ray |, when the point z approaches to x; from the positive (correspondingly, negative)

side of the Stokes surface. Let wi (x;) be the index wh (2) = (+,+,-,4+,.--),

where the — sign stands at all places corresponding to the points from

ug, and by w* (z;) the index of the same form with — signs at all places corresponding to the points from U5 . Clearly, the decomposition using the Laplace transforms of the type Luk (z,) Changes analytically when the point z moves from the point x; inside a Stokes domain to the positive side of the Stokes surface (that is, along the path | in the positive direction), and the decomposition using the Laplace transforms of the type L,«(2,) changes analytically when the point z moves of the point x; inside a Stokes domain to the negative side of this surface (that is, along the path / in the negative direction). We denote by Pe = F¥ (3,2), ki= 15250, N_ the set.of microfunctions at points s = 5S; (xz) included into the decomposition

N

A @aledep aodGaray [FF]

(2.53)

k=1

for each intersection point z;, j = 0,...,M. Thus, F;¥ is a component of the decomposition of the function f (z) at the point 7; with support at the

singular point S;, (xz) if we use the decomposition procedure corresponding to the positive side of the corresponding Stokes surface. If we denote by A; the operator of the analytic continuation along the path / from the point z;+0 to the point x;41—0 for j = 0,..., M (we denote

160

CHAPTER II. Resurgent Analysis

ZM+1 = Zo), then the microfunctions A;F¥, k = 1,...,N will clearly be components of the decomposition of the function f (x) at the point 2j41 if we use the decomposition procedure corresponding to the negative side of the Stokes surface, since we used the analytic continuation along the part of the path / coming to the point z;41 from the negative side of this surface:

N

f (2) = D7 Lak (eser) [APG]

(2.54)

k=h

Thus, comparing the formulas (2.53) for 7 replaced by 7 + 1 and (2.54), one

can see that the tuples (F},,,...,F4,) and (A;F},...,Aj;F)¥) for j = 0,...,M —1 as well as Ce ye te) and (A;Fi,,...,A;F{y) must be linked with one another with the help of the connection homomorphism das

(F}paiscesadiges Fa)

(ug

it (Ag Bp oneng Ach pated = Ort aia

AcowAte) AED Vien (Amt

eee)

(see Definition 2.5 of the connection homomorphism 7). The latter relations are obviously necessary and sufficient for the function f(z) to be a univalued one along the path / (at least up to rapidly decreasing terms). To write down system of equations (2.55) in the more explicit way, we recall that we consider the case when, for any given value of x, each singular point S; (x) can illuminate at most one other singular point S; (x). To make the further computations more clear, we shall visualize the relations between different singular points S; (rz) ,..., 5 (x) by a diagram shown on Figure 2.6 which we shall call the illumination diagram. Let us describe this diagram in more detail. The left vertical line of this diagram represents the loop /. The rest of the vertical lines represent the

analytic continuation along / of the points S; (z) ,..., S~ (x) of singularities of the function F'(s,z) = B[f(x)]. We recall that tracing along the path 1 transposes the singularity points with the help of the permutation oa. Therefore, the lower and the upper endpoints of the vertical lines (except for the first one) must be identified in accordance to this permutation. Further, if z € | coincides with any of the points zo,...,24, then there exists one or more pair of singular points S;(z) such that one of them illuminates the other. We mark each illuminating point by a small circle; the horizontal arrows connect the illuminating and the illuminated points. The singular points which are not included in such pairs are called neutral; they are marked on the diagram by small crosses. For example, the diagram shown on Figure 2.6 indicates that the point S3 (zo) illuminates the point

Si (xo) and the point S2(zo) is a neutral one. Thus, the illumination diagram describes the evolution of the points of singularity of the function F'(s,x) (the resurgent structure of the function f (z)) when tracing the loop l.

Section 4. Connection Homomorphism S e a

161

Si(%) S2(Xo) S3(X%)... Si (Xp) 02.9; (Xq) eee Mabe,

Soay(Xo)

Soa (Xo)

Soca)(Xo)

eee

Sov) (%o)

Figure 2.6. Illumination diagram.

Now we are able to write down the system (2.55) in the form of the system of the so-called resurgent equations for which the microfunctions FF are unknowns. Namely, each point S; (x;) of the diagram gives rise to exactly one resurgent equation. Due to the definition of the connection homomorphism 1, it is clear that if the point S, (z;) is not an endpoint of some horizontal arrow, then

FF = AjiFpia

(2.56)

(one must not forget about the above made identifications). On the opposite, if the point S;,(z;) is an endpoint of some horizontal arrow (for the example presented on Figure 2.6 this situation take place, for example, at point S,(z;)), then the relation between the microfunction FF and the analytic continuation of the microfunction Fey along the path I is as follows:

FF = A;_1F}_, — 6F;,

(2.57)

where F; is the microfunction corresponding to the point S;(z;) which illuminates the considered point 5; (x;). From our consideration, it is clear

that the system of equations of the type (2.56) or (2.57) written down for each point 5S; (z;) is equivalent to system (2.55) and, hence, this system gives us the necessary and sufficient conditions for the function f (x) to be univalued along the path /. Thus, we have obtained the following result:

162

CHAPTER

II. Resurgent Analysis

Figure 2.7. Excluding non illuminated points.

Theorem 2.7 The validity of the system of MN equations of the type (2.56), (2.57) over all points S,(x;) of the illumination diagram is necessary and sufficient for the corresponding function f(x) to be univalued along the given path l.

The system of equations mentioned in the statement of Theorem 2.7 is called a system of resurgent equations generated by the given illumination diagram. As we shall see below, the validity of this system imposes certain restrictions on the structure of the Riemannian surface of the function F'(s,x) provided that this system is valid for any resurgent function having

the form (2.53) at some point of the path | (say, at the point zo). The derived system of equations can be reduced to another system of less size written down for those microfunctions F¥ which correspond to the illuminated points of this diagram. To perform this reduction, we consider

the part of the diagram including the illuminated point S; (x;) (see Figure 2.7). On this figure by |Se we denoted the microfunction corresponding to the first illuminated point lying over the considered point S; (z;) (this means that 5S; (x;~+) is the last illuminated point met on the path / before

the point S,(z;)). Similar, Fj_, is a microfunction corresponding to the point S;(z;-,) which is the first illuminated point lying over the point Si (z;) which illuminates 5S; (z;). To simplify our notation, we shall write A instead of A; so that, for example, the expression A;_1 + Aj FEY

Section 4. Connection Homomorphism will be denoted simply by AFR.

k

By

Ai

163

Thus, due to equation (2.56), we have

iy =

266

=

A’

+

(Ers

and, similar, iio

F; =

l

AF;_;

=

4

=

rol |

Substituting the obtained relations into equation (2.57), we reduce it to the form

FP

AF

OA

Fe_,)

(2.58)

which contains only the microfunctions corresponding to the illuminated points. The set of equations (2.58) over all illuminated points is clearly equivalent to the system of resurgent equations obtained above.

Now there arises a question how one can solve the systems of resurgent equations of the type constructed in this subsection. We shall show that in some cases it can be done with the help of the notion of the alient derivative (see the next subsection). In short words, it occurs that the obtained system of resurgent equations can be rewritten in the form of the system of “differential” equations with respect to the new — resurgent — notion of the derivative, so that one can apply the methods worked out in the theory of ordinary differential equations to systems of resurgent equations of the above type. Such derivatives were introduced by J. Ecalle

who called them alient derivatives. This notion will be considered in detail in the next subsection. The examples of solutions to systems of resurgent equations the reader can find in the final section of the present chapter.

2.4.2

Alient

derivatives

1. In the previous subsection, we have derived the conditions of univaluedness of a resurgent function in terms of sets of microfunctions included into the decomposition of this resurgent function in different Stokes regions (see formula (2.58) of the previous subsection). These conditions (or, in the other words, resurgent equations) include two kinds of operators: the operators of analytic continuation and the “difference” operator

6=7 —id,

(2.59)

where T is a connection homomorphism (see Subsection 2.2.2 of this chapter). The two mentioned kinds of operators describe the two types of behavior of the asymptotic expansions of the given resurgent function. Namely, the operators of the analytic continuation describe the analytic dependence of the decomposition on the variables z, and the operators 6 describe jumps which the decomposition can have at points of the Stokes surfaces corresponding to the function in question. In this section, we investigate this second type of the operators involved into resurgent equations.

164

CHAPTER

II. Resurgent Analysis

We remark that operator (2.59) is quite similar to the usual difference operator (actually, the application of this operator to a direct sum of microfunctions equals to the difference of the two decompositions of the corresponding resurgent function from the two different sides of the Stokes surface); the connection homomorphism 7 plays the role of the corresponding shift operator. However, as everybody knows from the usual analysis, the difference equations are much more complicated than the differential ones. Thus, there arises the idea of rewriting resurgent equations in the

form of (in some sense) differential equations. This idea is based on the fact that the differentiation operator is strongly connected with the corresponding shift operator (or, what is the same, with the corresponding difference operator). Let us illustrate this connection on the example of the usual derivative. The Taylor formula co

f(z+h)= se, (1=) f(x) n=0

—

(valid, for example, for real-analytic functions) can be rewritten in the form

Taf (z) = f(x +h) = e*4/* f(z), so that the shift operator JT, can be represented as an exponential of the

derivative h4:

Ty = er 147

(2.60)

The latter formula shows the above mentioned connection between derivative and the shift operator. We remark that the traditional definition of the derivative is essentially based on the formula

ae= him,5(eh ~) = fim5-0) However, usage of the latter definition of the derivative is possible only in the case when we can calculate the shift operator T, for arbitrarily small values of the step h. Clearly, this is not the case for the “shift” operator

T (this operator does not contain a small parameter of the type h at all). So, one must think whether it is possible to use formula (2.60) with fized h to define the derivative d/dz. (In the usual analysis, it can be of use, for example, in the case when the function f (xz) under differentiation can be computed only at points of some lattice with the fixed step h. To be definite, we use the value h = 1 for the step; the corresponding shift operator will be denoted by T = 7;.) To construct this new definition of the derivative, we shall use the formula

d z= mT

(2.61)

Section 4. Connection Homomorphism

165

which is (at least formally) the direct consequence of the formula (2.60). The logarithm on the right in (2.61) is understood in the following sense:

nT

y Cn iy py

|

(ealye

Ms

‘

Let us check that the application of formula (2.61) gives the correct result on the example of the exponential function e**. In accordance to this formula, one has d 2p

azn

dz

=fa

= (alee n Se (T-1)"e

az

a

co

Ss (-1) n=)

n—-1

Ue

nr

(s>-0" cons ; k=0

The latter relation can be rewritten in the form nr

d

ae

(ot

|oe

=

e€

ax

—

(-1)**

Serial

k

Ak ak

paren Cie k=0

co

er

5 a

n—-1

|-1 )

(e*

—1)”

=

e*7 In e*

=

ae",

n

as required. We remark that the convergence of the logarithmic series in the latter formula takes place only for sufficiently small a, so that in application of formula (2.61) for computation of the derivative, one has to take care of the convergence. The formula (2.61) can be easily verified as well for real-analytic functions different from the exponential; for example, this formula can be verified for polynomials. We leave this verification to the reader as an exercise. 2. Let us describe the general algebraic scheme used below for the definition of the notion of the alient derivative (that is, the derivative corresponding to the “shift” operator 7). Consider a commutative and associative algebra A with the unit element? and suppose that this algebra is filtered with the set of ideals Ay, a € R,so that Ag: C Ag for a’ > a’’.We denote by A_. the intersection of the ideals A, over all real a. We define the order orda of an element a € A as the exact lower bound of numbers a such that a € Ag. Suppose that the following condition is valid: 3Below A will be the algebra of resurgent functions and Ag will be the algebra of resurgent functions with exponential type not more than —a. Alternatively, A can be identified with the algebra ®! eOMs,cont and Ag with the ideal in this algebra consisting of elements with supports in the half-plane Res > a. The isomorphism between these two algebras is given (up to rapidly decreasing functions) by the Laplace transform of microfunctions.

166

CHAPTER II. Resurgent Analysis

Condition 2.1 For any sequence a, € A such that

ord (an — Gm) — +00 as n,m — oo, there exists an element a € A such that

ord (a — an) > +00 as n — oo. The element a is defined up to the ideal A_.. This condition will be used below for verifying the convergence of the corresponding series. We need also the following definition: Definition

2.12 The linear mapping

p:AnvaA is called a mapping of strictly negative order if for any a € A the order ord (y”a) of the element y”"a tends to +oo as n > 00. Remark 2.5 The introduced notions can be reformulated in terms of usual topology in A/A_. connected with the introduced filtration. The fundamental system of neighborhoods of zero in this last topology is simply the set

{A,/A-..} of ideals included into the filtration considered. If we use

this topology, then Condition 2.1 requires simply that the algebra A/A_. is complete. Definition 2.12 then can be reformulated as follows: the linear mapping y : A — A is called a mapping of strictly negative order if the

sequence {y”"a} tends to zero as n — oo for any given element a € A. Now we can formulate the statement on connection between algebra homomorphisms and differentiations. To be short, we shall assume here that A_.. = 0; this means simply that we consider the quotient A/A_. with the filtration {A,/A_.o} instead of the algebra A with the filtration {Aa}.

Theorem 2.8 Let A be a complete algebra with filtration {A,}. Then the set of algebra homomorphisms tT : A — A such that 6 = 7-1 is an operator of strictly negative order is in one-to-one correspondence with the set of differentiations A : A — A of strictly negative order. This correspondence is given by the following relations: 7=e>, A=Inr=In(l +6). Here the functions e* and In(1+ 6) are understood as power series in A and 6, respectively. These series converge due to the completeness of the algebra A.

Section 4. Connection Homomorphism

167

Proof. If A is a differentiation of the algebra A, then we clearly have

D(a) =

CoN

ako

k=0

for any natural n and any elements a,b € A.

Hence, for the operator

T = e® we obtain

T(ab)

=

e*(ab)= X =A" (ab) = >

aceon"

=" (>:zoe) (s-=a"6) = (e“a) (e*b) = (ra) (rb), k=0

~

m=0

’

as required (the series converge since A is a differentiation of strictly negative order). Inversely, suppose that 7 is an algebra homomorphism such that 7 = 1+ 6 with some operator 6 of strictly negative order. We define the operator A by the relation

A =In(1+6) = Ss oo

(2.62)

Then usual computation with power series show that tT = e® and we must

only show that the operator A defined by (2.62) is a differentiation of the algebra A. Since 7 is an algebra homomorphism, we obtain that

ble GY SVS Crest) i(ieee for any m = 1,2,... and for any a,b € A. This means that come ye =A" (ab) n=0

oo oo | (>: Tote] (3: Tate)

=

= Sa ak SeGaN n=0

~

taheert

(2.63)

k=0

for any positive integer m. Since the operator A has strictly order, only finite number of terms of the latter relation do not the quotient algebra A/A, for any given number a. We denote the maximal value of the summation index for which terms on

negative vanish in by N (a) the right

and on the left in (2.63) do not vanish in A/A,. Then we have N(a)

a

n

d> & ar (ab) — 57 CE Ata A™*b] =0€ A/AQ.

n=0

TN:

k=0

(2.64)

168

CHAPTER II. Resurgent Analysis

Consider the system of equations (2.64) for m =

1,2,...N(a).

The ob-

tained system of linear equations have non vanishing determinant and,

hence, we obtain the relations

A" (ab) - ) Ch A*aA* *b=0€ A/A, k=0

for any a € R. Since NA, = 0, we arrive to the relations

" (ab) = 3c; AWG Aveb k=0

valid for n = 1,2,... and, in particular,

A (ab) = (Aa)b+a(Abd). This completes the proof of the theorem.

oO

3. Let us apply the above theorem to the subalgebra A of the algebra ®

sEC

Ms, cont

consisting of elements with supports in the right half-plane Re s > 0 with the filtration A,,a € R4, where the ideals A, consist of elements with their supports in Res > a. The restriction on supports of the considered elements is not essential since, as we shall see below, the corresponding derivative is invariant with respect to the shift operators in the complex plane C,. We remark that, due to Lemma 1.2 of Chapter I, this algebra is a complete one. Let zo be a point in the space of the resurgent variables C, lying on a Stokes surface (see Subsection 2.2.2 above). Consider the corresponding connection homomorphism Tare

®

sEC

MAT cont

ae

@ We

conte

We claim that the corresponding difference mapping 6 = T—1 is an operator of the strictly negative order with respect to the above introduced filtration. To show this, let us consider a microfunction F €

M, cont for some s such

that Res > 0. As above, we denote by X¥ the set of singular points of the function F' visible from the point s along the ray emanated from this

point in the direction of the positive real axis and encircling these singular points in accordance with the index w = (+,+,-,.-..). Here the + sign denotes that the corresponding singular point is encircled from the above and the — sign stands for the points of singularity which are encircled

Section 4. Connection Homomorphism

169

from below. Denote by &, the union of all ©” over all possible indices w. We recall that by =* we denote the set of singular points which approach the integration contour used in the Laplace transform of the considered hyperfunction from the above when the point z approaches zo from the positive side of the Stokes surface and by Ly the similar set defined for the case when z tends to zo from the negative side of this surface. The corresponding indices w; and w_ are defined in such a way that the — sign stands on all places corresponding to the singular points from ©t and uu, ,respectively. (We suppose that the singular points on the integration contour are enumerated in the order of their appearance on the ray parallel

to the positive real axis emanated from the point s.) Now it is clear that the support of the element 6F can contain all points from U, except for the point ¢ itself. Similar, the support of the element 6”F can contain all points from ©, except for the first two ofits points, etc. Since the set L, is a discrete set in the ray of integration, one can easily see that the minimal real parts of the points s included into the support of the elements 6"F tend to infinity as n — oo. This proves the required statement.

Definition 2.13 The derivative of the algebra A corresponding to the connection homomorphism due to the result of Theorem 2.8 is called an alient derivative. We denote by AF the alient derivative of the element F

E @ec/Ms.cont-

The following affirmation describes the main properties of the introduced notion. Theorem

2.9

The following commutation formulas are valid:

T..A = AT so, fe)

0

Os fe)

Os’ One.

ae

=

Oe?

jJ=

Laps abel
Mo,cont

>

Mo,cont

given by (2.66) determines the differentiation of the algebra

Mo ,cont-

Proof. Since A is a differentiation of the algebra ByecMs,cont for any two elements F’, G from Mo,cont;

we have,

A (FG) = GAF + FAG.

(2.67)

Taking into account that the functions F' and G considered as elements of the algebra @{¢¢Ms,cont have their supports in a single point s = 0, and taking the components of both sides of (2.67) at the point s;, we arrive at the relation

A(FG),. = GiAT)

+ E(AG)y :

Applying the shift operator T,, to both sides of the latter relation, we obtain

A,, (FG) =T;, (A(FG),,) =Ts (G(AF),,) + T., (F(AG),,) Using the relations

T,, (FG) = (Ts; F) G = F (T.;G) valid for any elements F,G € ®f¢coMs,cont (we recall that the shift operator T;, corresponds to the multiplication by the exponent e*? under the action of the Borel-Laplace transform), we obtain

A, (FG) =GA,,F + FA,,G,

172

CHAPTER II. Resurgent Analysis

as required. The proof is complete.

a)

We remark that the operator A,, can be defined even in the case when

the ray real the

point s; has nonvanishing imaginary part and, hence, does not lie on the r emanated from the original point along the direction of the positive axis. (In our case the original point is s = 0 and, hence, the ray r is positive real axis itself.) This can be done with the help of the formula AE

a

p—0X_-i6 5, (poF) 5

(2.68)

where (p9F’) (s,r) = F (e*’s,xz) and 6 = arg s; so that the right-hand side of the latter formula is well-defined (since e~*’s; is a positive real number). Clearly, the operator A,, thus defined is, as above, a differentiation of the algebra

Mo cont-

Definition 2.14 The above defined operator A,; on the space of microfunctions Mo cont Supported at the origin, is called the alient derivative at S;-

Formally, the alient derivative A,,F' is defined only in the case when the point s; is a singular point of the function F’. However, if we define A,,F = 0 if the point s; is a point of regularity of the function F’, then the obtained operator will be a differentiation of the algebra Mo cont as well. Thus, the connection between operators A and A, can be written down in

the form‘

eyo

Bf 2 sEC,

for any microfunction F € Mg cont-

Remark 2.7 It is easy to see that the introduced alient derivatives commute with the usual derivatives both with respect to resurgent variables and with respect to parameters (if the latter are present in the considered resurgent functions). 6. To conclude this subsection, we present the simple example of computation of the alient derivative. Consider the microfunction

Roe

Wl ns/x

= oe ae

E Morcent

(2.69)

(this microfunction has appeared in the consideration of the Euler example in Section 0.1 in the introduction). The only focal point for this microfunction is x = O and the Stokes line is the positive real axis. We orient the

Stokes line as it is shown on Figure 2.8a) and consider the branch of the (ramifying) function (2.69) choosing the argument arg s of the variable s 4We remark that the alient derivatives A, F do not vanish only at singular points of the microfunction F visible from the origin along rays emanated from there.

Section 4. Connection Homomorphism

173

args =27

Figure 2.8. Intersecting the integration contour.

as it is shown on Figure 2.8b). Now, due to formula (2.65), we have for real positive value of z:

eM TS ae

Te

Gee

11 (724) 271 s—2

] - (44) s=ze'0

—27i6, (s)

271

s—2

serena

(2.70)

since the singularity of function (2.69) equals zero at s = xe’? and equals

2716, (s) at s = ze?™*. Using formula (2.68), one can check that relation (2.70) remains also valid for any values of the variable z (clearly, except for the focal point z = 0).

2.4.3

Alient differential equations

Similar to usual differential equations, alient differential equations play an important role in the resurgent analysis. In particular, as it will be shown below, resurgent equations introduced in Subsection 2.4.1 can be written down in the form of alient differential equations. In this subsection®, we shall show that the theory of alient differential equations is, in great extent, similar to that of ordinary differential equa5 Here we use the results of paper [35].

174

CHAPTER

II. Resurgent Analysis

tions. To do this, we shall consider two examples of linear alient differential equations with resurgent coefficients. 1. As the first example, we consider a linear alient differential equation of the form

Sa, HS0).=vAl S$,0)eee)

(2.71)

where A(s,z) € Mo,cont is Some given microfunction and the product AF is the product in the convolutive algebra Mo,cont- First of all, we shall

prove the following statment:

Theorem 2.11 Let Fo(s,x) be a solution to equation (2.71) which is an invertible element in Mo,cont- Then the general solution to (2.71) is given

by F(s,xz)

= C(s, 2) Fo(s, 2),

where C(s,x) ws an arbitrary constant of resurgence, from Mo,cont such that A,C =0

(2.72)

that is, an element

for any s € Cg.

Proof. It is evident that (2.72) gives a solution to (2.71) for any choice of C.

Inversely, if F(s,z) is an arbitrary solution to (2.71), then it can be

represented in the form (2.72) with C(s,z) = Fy*(s,2) F(s,2)To prove that C(s, x) given by the latter formula is a constant of resurgence, we shall calculate its alient derivative at so:

A,

,ChHa(A,, F, )F + Fe Aeehi= —F5 (A, Fo) =" SAF, Fi4+AR OF =0:

FE ALF

Clearly, A,C = 0 for all s # so. This proves the theorem.

Oo

We remark that, if F(s,z) is a nontrivial solution to (2.71), then this function must have a singularity at point so (in the opposite case A,, F = 0). However, the point so is not the only point of singularity of a solution to (2.71). Actually, due to this equation, the variation of a solution F

has singularity at point so of the form A(s — so,xz)F(s — so,z). But then this solution has also a singularity at point 2s9. Recursively, F(s,2z) will have singularities at points of the lattice s = kso, k = 1,2,3,.... Thus, the singularity at so originates singularities at all points of the mentioned lattice — this situation is typical for resurgent analysis. In particular,

alient derivatives of a solution to equation (2.71) at points of this lattice will not be equal to zero.

Let us try to construct a solution Fo(s,z) to equation (2.71). For simplicity we shall assume that A(s,z) is a constant of resurgence. function ]

Boca) ay As — 89,2) Ins € Mo cont

Then the

Section 4. Connection Homomorphism

175

satisfies the relation

NG (6,0 =e Al ser). Evidently, B(s,z) is a regular microfunction at s = 0. Now we define the

solution Fo(s,xz) by the formula® co

Fo(s,x) = eB (7) = yas (Ble, 2))*.

(2.73)

k=0

(All products here are products in the convolutive algebra Mo,cont-)

The function B(s,z) has the only singularity (except for the origin) at the point s = so. Then the reader can verify that the convolution powers (B(s,zx))* will have singularities at points s = soj, j = 1,2,...,k and that

the structure of the Riemannian surface of (B(s,z))* is stable at each point $ = Soj for sufficiently large k. Thus, all the convolution powers of B(s,xz) can be defined on one and the same Riemannian surface with the following structure: This surface has ramification of logarithmic type at s = 0. Then, at each sheet of this surface over s = so, it has, in turn, the ramification of logarithmic type

and so on. Further, the usual estimates for convolutions allow one to obtain that

((Bls,2))| < Es,2)}*nl(s, 2), where [(s, x) is the distance from the origin to point (s,xz) along the above dercribed Riemannian surface. The latter estimate shows that series (2.73) converge and, therefore, the function Fo(s,z) is well-defined. The above considerations are the particular case of the following general result ((47], {29]).

Theorem 2.12 Jf B(s,z) is an endlessly continuable regular microfunction at s = 0, and if the series

e(e) =) Pace k=0

6 This formula is quite similar to the formula

y(x) = exp

fo dz 0

which gives a particular solution to the ordinary differential equation

y' =a(z)y.

176

CHAPTER II. Resurgent Analysis

converges in a neighborhood of the origin, then the function

y(B(s,z)) =~ ax[B(s,2)]* k=0

is a well-defined endlessly continuable microfunction from Mo,cont In particular, from the latter theorem, it follows that any series of the type (2.35) with nonvanishing ao determine an invertible element of

Mo,cont, and, hence, the function Fo(s,xz) given by (2.73) is invertible. Hence, formula (2.72) with such an Fo gives a general solution to equation (271) 2. Our second example is the example of the second-order system of linear alient differential equations’ Awe. =

Aj (x) F+

B, (x) G,

A,,G = A2 (rz) F + Bo(z)G, AF = O0tors 7.381, and. A,G —0 tor

(2.74) $7 1s2-

The following statement is valid:

Theorem 2.13 Let (Fi,Gi) and (F2,G2) be two solutions to (2.74) such tha def D

=>

fh G;

Ge

=

F, G2

=

FoG,

(2.75)

is an invertible element in Mo,cont. Then the general solution to (2.74) is given by eo Fy Fo where Cy and C2 are constants of resurgence, Mo,cont Such that A,C; =0 for any s € C.

that is, elements C,,C2 €

Proof. Clearly, any pair (F,G) determined by relation (2.76) is a solution

to (2.74). Let us prove the inverse statement. Since determinant (2.75) is invertible in the algebra Mo,cont, equations (2.76) are solvable with respect to the unknowns C, C2 for any pair (F,G) of elements from Mo cont- Let

us show that, if (F,G) is a solution to (2.74), then the microfunctions C and C2 determined in such a way are, in fact, constants of resurgence. To do this, we remark that the direct computation shows that the alient

derivatives of the microfunction D given by (2.75) are jay NegB)=

Aj (x) Ly.

Aged. = Bo (x) Ds:

7In what follows, we shall omit for brevity equations in resurgent systems similar to the last equation of the considered one. We suppose that all alient derivatives, except for those included into the resurgent system in the explicit way, vanish identically. 8We recall that all products below are products in the convolutive algebra Mo,cont-

Section 4. Connection Homomorphism

177

Further, we have

C, = D“'!(FG2 -GF), CiaD

(GF, — FG;),

and, hence,

A,,C, = —D-? (FG2 — GFy) A,, D+ D-* (G2A,, F— GA,, Fy) =

ie rage (x) (F'G2 - GF») +p!

-G(Ai

as required.

(x) Fo+

B, (x) G2)] =-A;

[Go (At (x) F+

B, (x) G)

(x) Ci + Ay (x) Cr

0,

The relations A,,C; = 0, A,,C2 = 0 and A,,C2 = 0 as well

as A,C; = 0 for s # s1, 82 can be proved in a similar way. This completes the proof of the theorem. Oo There is a detail in the above computations which may seem (at least from the first glance) to be a contradiction. The matter is that the products F,Gz and FG, included into the definition of the singularities at points s,, sz, and s; + sg whereas A,D does not vanish only for s = s; and s = sg. a phenomenon is that, in computation of the alient

determinant D have the alient derivative The reason for such

derivative, only those points of singularity are taken into account which are visible from the origin along straight lines. At the same time, the singularity of the function D at point s; + sg is located on the another sheet of the Riemannian surface of

this function and, hence, is not visible from the origin.

In conclusion, we remark that any solution (FG) of system (2.74) has singularities at points ks;+msz for any positive integers k and m. However, since only the points s = s; and s = Ss are visible from the origin along

straight lines, the alient derivatives of functions F' and G do not vanish only at s = s; and s = Sg, correspondingly, except for the case when s, and s2 lie on one and the same ray emanated from the origin. In this latter case, alient derivatives of F and G will not vanish at all points of the double lattice {ks; + ms2,k,m € N} and, hence, system (2.74) is not solvable (at least if we require that A,F' = 0 for s # s; and A,G = 0 for s = s2). The similar situation took place in the first example of this subsection.

2.4.4

Stokes phenomenon

and univaluedness

Consider a resurgent function f(z,q) which is a priori univalued in a neighborhood of some its focal point go. We suppose that f(z,q) is a classical resurgent function, that is that its Borel transform F'(s,z,q) is an endlessly-continuable hyperfunction homogeneous in (s, x) of order —1. Then the singularity set of the function F' is described by the equation

$=

(e.g)

(2.77)

with the (ramifying) analytic function S (z,q) homogeneous of order 1 in z, and, hence, the set of the focal points of the function f (x, q) can be considered as an analytic set in the complex plane GP For simplicity, we suppose

178

CHAPTER II. Resurgent Analysis

that the function S (xz, q) has a finite number of values S; (z,q) ,--- Sn (z,q) though the numbering of these values cannot be chosen in a natural way

for all nonfocal points q. As it was shown in Subsection 2.4.1 above, for the function f (x) to be a univalued function in a neighborhood of the focal point go, the function F'(s,z,q) must require some additional conditions which can be written down in the form of the corresponding resurgent equations. Thus, to obtain the information about the structure of the function F'(s,,q), it would be useful to find a general solution of the constructed resurgent equations. Clearly, the resurgent structure of the considered resurgent function (that is, the set of singularities (2.77) of its Borel transform) must be given beforehand, since this information is needed for writing down the corresponding system of resurgent equations. Similar to Subsection 2.4.1 above, we suppose that, for any point g lying on the Stokes surface, each singular

point S;(x,q) can illuminate at most one another singular point 5, (zx, q); this assumption is introduced for simplicity. Using the notion of the alient derivative introduced in the previous subsection, we can rewrite system (2.58) of resurgent equations for microfunctions corresponding to the illuminated points of the illumination diagram of the considered resurgent

function f (x) in the following form:

Bie A Be

Ace

AEe.)

(2.78)

(for notation, see Figures 2.6 and 2.7). Below we present several examples of solving the system of equations (2.78) over all illuminated points of the diagram of the function f (z).

2.5

Examples

In this section, we present examples showing what information can be obtained from resurgent equations derived in the previous section for univalued resurgent functions. We have seen that rewriting these equations with the help of the above introduced alient derivatives allows us to derive the general form of the corresponding resurgent function using methods similar to the methods of solving ordinary differential equations, and that is why the notion of the alient derivative occures to be useful for investigating of

resurgent functions?. °We use here strongly the results of the PhD E. Delabaere and H. Dillinger [36].

thesis by A. O. Jidomou

[68] and

Section 5. Examples

179

S(%D)

Figure 2.9. Illumination diagram for a resurgent function of the Airy type.

2.5.1

Resurgent functions of the Airy type

The first example of this section will be the example of resurgent functions of the Airy type. Let x € C, be a resurgent variable and g € C, be a parameter. We say that a resurgent function f (z,q) is a resurgent function of the Airy type if the set of singularities of its Borel transform is given by

Sa

2

SMtag) Ae

(2.79)

Clearly, the only focal point for such a function in the g-plane, is the origin

q = 0. Hence, we can use the unit circle g = e*”, » € [0,27] as the loop ! used for the construction of the system of resurgent equations in Subsection 2.4.1. Let us describe the illumination diagram for a function of the considered type. Evidently, one of the two points (2.79) illuminates the other iff y = 0, y = 27/3, or y = 4n/3 (we shall carry out our considerations for real positive values of x). The corresponding Stokes lines are drawn on Figure 2.9 a). One can easily check that the corresponding illumination diagram has the form shown on Figure 2.9 b). The points S; (z,q) and S2(z,q) change their places three times when tracing along the loop l. It is not hard to see that the corresponding system of resurgent equa-

tions for the illumination diagram of this form (written for the micro-

functions Fd, F?, and F} corresponding to the illuminated points of this

180

CHAPTER II. Resurgent Analysis

Figure 2.10. Integration contour for a univalued function.

diagram) is

Pig A EA AL.) Fe ALE (AFA) PAZ NCAR) e

(2.80)

We emphasize that, for any given hyperfunction F'(s,xz,q) with (2.79)

as ramification points, one can choose a set of microfunctions Fj, F?, and F} determined by singular points of the function F'(s,z,q) in question in such a way that the corresponding resurgent function f (z,q) is univalued in a neighborhood of the origin and, hence, the resurgent equations (2.80) are valid. To do this, it suffices to define the function f (z,q) as the integral of the form (2.38) with the integration contour I encircling both ramification points as it is shown on Figure 2.10. Then the decomposition of the obtained function will give us the required microfunctions which satisfy system (2.80). However, if we require that the analytic continuation of the Laplace transform of any microfunction determined by the function F'(s,z,q) at some point g (say, g = go) is a univalued function, then resurgent system (2.80) imposes some restrictions on the function F'(s,z,q)in question.

To be definite, let us consider the system of resurgent equations for the

microfunction corresponding to the recessive component of f (x,q) at the

point go. To do this, we set F? = 0 in (2.80). Then we have F? = AF? = 0 and, due to the first equation in (2.80), we have Fi = A?F?. Now,

Section 5. Examples

181

excluding the microfunction F? from the considered system, we arrive to

the following system of resurgent equations:

A (AF}) = —A-?F, A(A Fl) = AFI, Denoting by Fa and F’, the dominant and the recessive components at the point g = qi, correspondingly:

Pp AU, ay i = Aefy a we rewrite the latter system in the form AFy = -F,, (A“*AA) Te, 3 i

Now we notice that the operator A~!AA in the second equation is none more than the alient derivative of the microfunction F, at the point s; = —42q°/ 2 (as well as the operator A in the first equation can be treated as

the alient derivative of the microfunction Fy at s2 = $2q°/. Thus, the latter system can be considered as the system of resurgent equations

Ageha=iFy

{ANgsko=

(2.81)

—F,,

at one and the same value q of the parameter. The next step in the investigation of the obtained system of resurgent equations is to show that the classical Airy function is a resurgent function satisfying resurgent system (2.81). As it is well-known, the Airy function,

that is, the solution u(z,q) to the equation

£

—)

du

45

taal qu=0

(2.82)

can be written down in the form of the integral

u(x,q)= fee

ee) d€

(2.83)

ay

(cf. the introduction). Here y can be chosen similar to the contours shown on Figure 0.21 of the mentioned subsection. Performing the variable change e3

£ (< - 5) S99; 3

(2.84)

we reduce the latter relation to the form

d ; a(x;q)= fos 7!

ds,

182

CHAPTER II. Resurgent Analysis

where € = €(s,z,q) is an (in general, ramifying) solution to equation (2.84) with respect to €. Let us investigate the singularities of the function €(s,z,q). This function can be written down in the form €(s,2,q) = =(s/z,q), where the function = (0, q) is determined from the equation =3

=q-—=—=-9. 3 The solution to this equation has singularities at points, where

MEE

EN Ce

that is, for

ee _

,=

Aad ag

23

DlrSe wiewes eeu!

with certain choice of the sign of the function g*/?. Hence, the singular points of the function €(s,z,q) are located at -=

2 2 Baca, that is, $= zor

which coincide with formula (2.79). Further, one can easily check that the

function 0€ (s,xz,q) /Os has singularities of the type with

O09.

(3)

ar

5,219)

(x, q)(s es S;(s,q))

k-1/2

ar (k + 1/2)

at each point s = S;(z,q). Besides, directly from the definition of the function u (z,q) it follows that this function is univalued in a neighborhood of the focal point g = 0 (in fact, the function defined by formula (2.83) is an entire function of the variable g). Therefore, the two microfunctions F oo and F'“") determined by the function 0€ (s, x, q) /Os form a solution of resurgent system (2.81). Using the result of Remark 2.7, one can easily construct another solu-

tion (ese, Fu) to (2.81), where Fe) and F 76; (S,q)

(3.9)

j=0

and the function F's (s,i) is given by

Fs (s,i) =~ WH! (5).

(3.10)

j=l The following statement summarizes the obtained results:

Theorem mula

3.2 Jf the function S(xz) satisfies equation (3.8) then the for1

1

Ba) |1s ee =Hs |S,h5a,h ce r3'H |2,h=— =)

2+~+0

3

ES (0) &

~O RSA. (Sit Ge +h~ 3 (5,~(Say) (3.11)

takes place, where the symbol Hs (s,qh) is given by (3.9) and the function

Fs (s,ry)by (3.10). The functions H', (S,q) involved in the right-hand part of (3.9) are polynomials in q of order

m—j—-1.

From now on we shall omit the Feynman indices over the operators.

3.1.2

Reduction to the Volterra equation

In this subsection, we shall reduce equation (3.3) to an integral equation of the Volterra type. To do this, we consider the (ramifying) function p(z) which is defined as a solution to the equation

Ef (6) = 0: Let zo be a point of regularity of the function p(x). (We shall also include in the set of ramification points of the function p(x) such values of x for which two or more branches of the function p(x) coincide with one another.) Then there exists a neighborhood of zo such that the function p(x) decomposes

to m univalued branches p;(z),...,m(x). We shall use the notation p;(z) also for analytic continuation of the analytic element p;(x) given near the point zo.

Remark 3.3 One can on pointed Riemannian are determined by one p(x) by different choice

consider the functions p;(x) as the functions given surfaces. Then it is evident that all these surfaces and the same Riemannian surface of the function of base points.

Section 1. Ordinary Differential Equations

199

Now the formulas =z

5;(2) = |p, (2) de

(3.12)

ZO

determine exactly m different solutions to Hamilton-Jacobi equation (3.8). Clearly, there exists a Riemannian surface on which all functions (3.12) are defined as single-valued functions. We shall carry out all our considerations on this Riemannian surface. The reduction of equation (3.3) to an equation of Volterra type will be carried out with the help of successive application of transforms (3.4) corresponding functions (3.12). Each such transform will lower the order of the considered equation to unity. As a result, we shall obtain the Volterra equation (we remark that already after the first transformation we obtain the equation on the above described Riemannian surface). The subsequent investigation of the equation in question is performed in the following steps. 1. Construction of a Riemannian surface on which all terms of the Neumann series corresponding to the obtained Volterra equation are defined. 2. Proof of the convergence of the Neumann series on the constructed Riemannian surface. Let us proceed with the described program. To simplify the notation we denote def

Aah

TGs

ad = Basis 11

We introduce the operators Tij =T;

Tj, 2,7 =1,...,m.

Clearly, the following relations are valid TiTij Le Tj

Ti

=

id, T ji =

Ta

TijTjk

Later on, the right inverse R for the operator (2) a

—

=

Tik-

(3.13)

& — 1 is given by *

.

S

RG(s,5) = jigs apes (s,5) dS, 0 ~

where aA (s-5) is a shift to the value

S — S$ along the s axis:

e (S-S)@ (3,5) =G(s+5-5,5) so that the solution of the equation

(e) ae

SMa

(3.14)

200

CHAPTER III. Applications

can be represented in the form

u=RF+C (fh)eh S 6 (5)

(3.15)

with an arbitrary analytic function f(s). Here C (x) are arbitrary constants depending on the “parameter” h. The reader can consider them as polynomials in h. The solution to equation (3.3) can be represented in the form

Visa) Y ( can be rewritten as

S50.

WEN

Ge

ee

Aa oal 0 0, poe

Peep

«Nes

{nth (s.Az3- A)Ty + hF (S,h) Re + iF, (51,2) fie} Y>

=i{r, (1,%) Cy (Je AS 4 By (S2(x) , i) Ce (i) "Sa ie, (s1,%) RC R (h) e*ai} ; It is easy to see also that the functions F} are polynomials in h of degree less than or equal to m — 7. The final result of the above recurrent procedure is described in the following statement:

Theorem form

3.3 Any solution to equation (3.16) can be represented in the

with the function Yn satisfying the equation

F (2) Vn + BR Bonin = BY RCc(i) FS,

(8:19)

where the operators R,; are given by eedef F; (5;(x) ,h) 5 say (5;-1 (x) ,h) Rj-1

aeeey gs (S1(2), h) Rusk R Here F (x) is a function which does not vanish on the domain of regularity of transforms (3.4). We remark that the choice of the “constants” C; (A determines different solutions to equation (3.16). To construct m linearly-independent solutions, one must put subsequently all C; (i) except for one equal to zero.

Below we shall consider the solution corresponding to C) = 1, C2 = 0,...,

C,, = 0. Dividing equation (3.19) by F(z) we obtain for this solution

YR

eR vee),

V, tAr (2) Rony = hee ia) (1 (x) , hyeA*Si(@)_

(3.20) (3.91)

Section 1. Ordinary Differential Equations

3.1.3.

203

Analytic continuation of terms of the Neumann series

First of all, we remark that one can assume that the function F (z) equals to 1 identically (in the opposite case, we shall include the function F—! (x) into the coefficients of the operator Rin): Under such suggestion, the equation for the function Ya can be rewritten in the form

eh *Si(z) (2+ 7Rm Rr) Yn = AF (si(x) ,h) and one can construct the formal solution to this equation in the form of the Neumann series:

Yn = (14 %RnRm) BE ($1(2),8) &SE) (RpRm)’ =J>(-1) H+! (Rmkm) 2 1°F#

h) eb*S1(2) _ (3.22) Fi ($1(2) ,h)

Thus, to prove the existence of a resurgent solution to equation (3.3), it is sufficient to prove the convergence of the series on the right in (3.22) in the considered class of functions. However, the considered series is an operator and, before investigating its convergence, one must apply it to some function f(s). To construct a resurgent solution with simple singularities, we can consider the application of the operator Y,, to one of the functions from the following Ludwig

sequence: ? soln

ssn. oes.

(3.23)

For technical reason, we choose f (s) = s In s. This choice is quite unessential since all terms of the sequence (3.23) can be obtained from one another with the help of the operators h and h-!. We remark also that all the considerations below are valid for any endlessly continuable function f (s) with isolated integrable singularity at point s = 0. We shall not consider the general case and restrict ourselves with consideration of the function

f (s) = sln s only. Thus, we must prove the convergence of the series

YA (S32) ef Ve (2,h) f(s)

= 3 (-1)) 4} (RmnFn)’ i (Si(x) ,h)eb*Si(2)¢(5) (3.24) j=0 in the class of endlessly continuable functions. To do this, one must verify the validityof the following three assertions: 2We recall that f(s) is considered as a hyperfunction, so that the sequence below is, in fact, a Ludwig sequence.

204

CHAPTER III. Applications

1. All the terms of the series on the right in (3.24) are endlessly continuable functions. 2. There exists a Riemannian surface FR such that all these terms can be considered as functions on this Riemannian surface.

3. Series (3.24) converges at any point of R. To verify these three assertions, it is convenient to extend the notion of the endless continuability to analytic functions of several variables.

Definition 3.1 The function F (ahh a) given in a neighborhood of a point ro € C% is called to be endlessly continuable if, for any positive number L, there exists an analytic set © C C” in the ball Kz, (zo) such that, for any path | C K,(zo)\= with length less than L with origin at Zo, there exists an analytic continuation of the function F' along the path

i We shall prove the following statement: Proposition 3.1 The terms of the Neumann series are endlessly continuable functions with singularities at ramification points of transforms (3.4) and at those points (s,x) for which

s+S;(z)+C, =0

(3.25)

for some j =1,2,...,m. The constants C; are determined by the functions S; (x) and points S; of ramification of transforms (3.4). Proof. First of all, we note that, since the operator Rim iS a composition of the operators R;, 7 = 1,...,m, we need to investigate the action of

the operators R; to ramifying analytic functions with singularities of the described type. Thus, let F'(s,xz) be such a function. Let us describe the Riemannian surface of R,F (s,xz). This function is given by the integral

Rr

ieb (Si) S19) 5% (y) F(s,y) dy To

= fs (y) F(s + S; (x) — S;(y),y) dy.

(3.26)

zo

The Riemannian surface of the latter integral can be described in the following way: Clearly, each point of the Riemannian surface of integral (3.26) is determined by a homotopy class of a contour with the origin at zo and the endpoint at x. Since we consider the functions bounded (and, hence, integrable) at its ramification points, we can choose a representative of the 3The exact description of these constants will be given below.

Section 1. Ordinary Differential Equations

205

minimal length in each homotopy class of the above mentioned type. This representative will be an open polygon with the origin at zo and the endpoint at z with all vertexes except for zo and zx at ramification points of

the function F'(s + S; (x) — S;(y),y). We remark that the length of this polygon is exactly the distance from the point zo to the point x along the Riemannian surface of the function R;F (sha):

It is clear that the singularities of the function R;F (s, 2) can a priori be originated by the following three reasons: a) coincidence between one of the singularities of the function

F(s +5; (x) -— S;(y),y) determined by the relation

s+S;(z) -—S;(y)+

Si (y)+C. =0

(3.27)

with one of the endpoints (zo, xz) of the path of integration; b) coincidence between the two singularities of the form (3.27) if the path of integration is “pinched” between these two points; c) coincidence between one of singularities (3.27) and one of the points S, of ramification of transforms (3.4) again in the case if the path of integration is “pinched” between these points. Let us consider all these possibilities. In the case a), if y = z is a singular point of the integrand in the

right-point part of (3.26), then one has

s+S,(rz)+C, =0. Similar, if y = xo is a singular point of the integrand, then s+

S;(z)+C.

=" ()

due to the fact that S; (zo) = 0 for all values of 7. Thus, the first possibility does not enlarge the set of singular points of the function F'(s, z). Let us now consider the case b). It occurs that if two singularities of type (3.27) meet each other, then at the same time they meet a ramifi-

cation point of (3.4). Actually, if the two singularities of the type (3.27) coincide with each other, then we have y = y(z), where the function y(z) is determined by the following system of equations: s+S; (x) — S;(y) + Su, (y) + Ce, =9, s+58; (x) Say (y) + Siz (y) + Cr, = 0,

so that

Si, (y(z)) — Siz (y(x)) = const.

206

CHAPTER III. Applications

Hence, the point y coincides with one of points of ramification of transforms (3.4). Since, as it can be seen from the latter relation, the point y does not depend on zx. Therefore, for the corresponding singular point we obtain

s+ S; (2) +C' =0,

(3.28)

where C’ = Si, (y)—S; (y)+Cx,. We remark also that the considered point of singularity can appear only on those points of the Riemannian surface

of the function R;F (s,x) for which the corresponding ramification point y is one of the vertexes of the open polygon determining this point of the Riemannian surface and this vertex coincides with one of the ramification points of (3.4). Finally, if the singular point y coincides with one of the points S; (more precisely, with its image in the z-plane), then formula (3.28) takes place

once more with C’ = S; (y) — S; (y) + Ck. The singularities of the integrals along the s-axis which occurs due to the operator h- included in the terms of Neumann series (3.24) can be investigated in a similar way. We remark only that such integrals do not enlarge the singularity set of the considered functions. From these considerations, it follows that singular points of functions

23 (8,2) = (-1) B+! (Rahm) Fi (Si (a) 8) BS)

(5)

can appear at points (s,z) of its Riemannian surface for which the corresponding open polygon contains at least one point of ramification of transforms (3.4), and, hence, the number of the singular points which can be

reached by the paths of the bounded length is finite.

(Here we use the

evident fact that the length of the polygon is bounded in any finite part of the Riemannian surface of the function in question considered as a function of two complex variables.) This proves the proposition. Oo

Now let us investigate in more detail the values of constants C; involved

in expression (3.25) for singularities of terms of the Neumann series. From the proof of the above proposition, it follows that, on each step of computation of the integrals over zx involved in the expression of the corresponding term, the value of these constants can be changed by an additive term of

the form S; (y) —S; (y) for some y being one of ramification points of transforms (3.4). Thus, all the constants C; are equal to linear combinations

Ci = Dmx [S; (y) - Se (Y)] with integer coefficients m;,. Here the values of the functions S; (y) and Si (y) involved in the latter formula can be reached along the paths of the length not more than the distance from the fixed point zo of the Rieman-

nian surface of the considered term to its point x and the sum )>|m,,|

Section 1. Ordinary Differential Equations

207

does not exceed the number of points of ramification of (3.4) lying on the corresponding integration contour. The proved proposition answers the first question from those listed in the beginning of this subsection. However, the more detailed analysis of this proof can also give the answer to the second of these questions. Namely,

since, as it was mentioned above, the sum }>|m;,| does not exceed the number of ramification points lying on the integration contour corresponding to the point in question. Actually, in the process of continuation of integrals (3.26) along the corresponding open polygon, we arrive at the integrals of the same type over the contour which is the deformation of this polygon by moving points of singularity in the integrand of (3.26). This deformation is performed over paths determined by equations (3.27) for different values of k, 7, and | when the point z moves along the open polygon in question. Since we consider the compact part of the Riemannian surface of terms of the Neumann series, the length of these paths is uniformly bounded for all (z,s) lying in this compact part. This means that the obtained integration contour can contain only the finite number of singular points of the Riemannian surface of the substitutions 7; (more exactly, of the universal covering over this Riemannian surface). Since, as it is shown above, the singular points of terms of the Neumann series can arise only due to ramification points of the substitutions 7;, we see that only a finite number of singular points can arise on the part of the Riemannian surfaces of terms of the Neumann series corresponding to polygons of length less that L for every fixed value of L. These singularities will arise during a finite number of steps of the successive approximation method and, hence,

the Riemannian surfaces of terms of the Neumann series will be stable in any finite part of this Riemannian surface beginning from some number of iteration. Thus, there exists a Riemannian surface on which all terms of the considered series are defined at one and the same time.

3.1.4

Convergence of the Neumann

series

In this subsection, we shall prove the convergence of the Neumann

series

(3.24) on the above constructed Riemannian surface. To do this, we rewrite the Volterra equation (3.21) applying both parts of this equation to the function f (s): m

Yn

=

j=1 m—1 +

m—-j

m

Ds 6 fi ot") [| %

\i=1

ye

k=j

fa ym f(s+51(2)),

(3.29)

pil |

where Y,, = Yn f (s) and f;; (x) are functions regular in the domain of regularity of transforms (3.4). (We have taken into account that the functions

208

CHAPTER III. Applications

F; (s;,h) are polynomials in h of order not more than m — j.) Let us cut off all the points S; of singularity of transforms (3.4) by disks of arbitrarily small radius e possibly depending on the point of singularity. (This cut-off operation is supposed to be performed on the above constructed Riemannian surface of the solution.) Denote the Riemannian surface obtained with the help of the described cut-off operation by 2°. Clearly, one can suppose that the cut-off disks do not intersect one another. Denote by 24, the domain on the obtained part of the Riemannian surface accessible along paths in 2* of length not more than L with origin at the point zo. Due to the regularity properties of the functions f;; (x) included into equation (3.29), these functions are bounded on 25. The proof of the convergence of the Neumann series will be performed

with the help of the method close to the well-known Cauchy majorant method in the theory of ordinary differential equations. To construct a majorant equation for (3.29), we replace all functions in the latter equations with their upper bounds, the operator keby a real number k, and the integration over contours in z-plane by an integration operator J in the space R¢, where € is a real variable corresponding to the variable z. In doing so, we obtain the equation ™m

n=

m—Jj

m—-1

>i (= cok) BM-itlpm—-j+l

j=l

n+

bscut] Ax(k)

1

i=1

with respect to the unknown function 7 = n(€,k), € € [0,L], k — Here c;; are upper bounds for the functions f;; (x) on N{:

(3.30) +00.

[fig (2)| < cay on M4, B is the mutual upper bound for all the function S‘ (x):

ey G)

Bon

forg

1)

me

the constant A is supposed to be chosen in such a way that

If (s + Si (z))| < Ax(|s|) for z € NF, the operator J is given by E

(In) (€,k)

in(é’,k) dé, 0

and x(k) is a positive real-valued function equal to k In k for large values of k. (This function is a majorant for the right-hand part of (3.29); see the definition of the function f(s) above.) Equation (3.30) is in some sense

Section 1. Ordinary Differential Equations

209

majorant for equation (3.29). To formulate the exact sense of this assertion we need some notation. Let us solve equation (3.30) with the help of the successive approximation method. The obtained solution will have the form

nike) =>

sk (Ek),

(3.31)

l=0

where

Be

tio (€, k) - |=can] Ax(k)

(3.32)

zi

and the functions m (€,k) are constructed with the help of the following recurrent relations: m

m+i (€,k) =

m—-j

Dia (= cit) Bs 7=

TIL

tek)

(3.33)

lee)

As it is well-known from the Volterra equations theory, series (3.33) con-

verge for any values of (€,k). Later on, relations (3.32) and (3.33) show that the terms m (€,k) of series (3.31) are positive functions in (€,k) for all values of l. The following statement shows that series (3.32) is a majorant series

for the Neumann series (3.24). Proposition 3.2 For any j > 0 the estimate

(Re Rm)Fi ($1 (2) sh)eS) f(s) < nj (UC) (8) ts valid. Here I(x) is the length of the shortest path joining the point xo with the point x in N§, and |; (s) is the length of the shortest path joining some fized point so with the point s (it is supposed that so is not a point of singularity of the function defi

U; (s, x) =

-=

(Rm)

j

~

p-1

F, (1 (x) ,h) e”

Si(

oe

) f(s)

for zx € N{). Proof. If a function V (s,z) satisfies the estimate

IV (sa) < 5 UC) [hs (6)] C

T

t

(3.34)

for some C > 0 and some integers 2 and r then

RV (s,2) Ss < (+1)! GB [!(zr (hi (s)]’ )

(3.35)

CHAPTER III. Applications

210

lav(s2)] < Seo

oy.

(3.36)

The latter estimate is quite clear, and the former one can be obtained as follows:

JV (s,2)] < [1s" WIV (s+ 55(2) - $5). lau

g BRON Lug aw) = So te ho as required. Suppose now that the j-th term U; (s,z) can be estimated via the sum U; (€, k) of expressions standing on the right in (3.35). Then, due to the evident recurrent formulas

U541 (8,2) = Crit; (Sse) we arrive at the estimate

[i*1U,41 (s,2)| < bs(=cat) Bm l=1

a=)

(3.37) ym

U; (§,k) €=I(x), k=11(s)

This is true since the right-hand part of (3.35) can be obtained from the right-hand part of (3.34) by application of the operator BI (in the variables (€,k)) as well as the right-hand part of (3.36) can be obtained from the right-hand part of (3.34) by multiplication by k. From the other hand, the terms of the series (3.31) are also determined by the recurrent relations (3.33) having on the right the same operator as in (3.37). Now the required affirmation follows by induction. This proves the proposition.

Oo

The following affirmation is a direct consequence of the latter statement. Theorem

3.4

There exists m endlessly continuable solutions to equation

(3.3) corresponding to m functions S; (rz), 7 =1,...m, defined by (3.12). The resurgent structure of such solutions (for 7 = 1) is described in Proposition 3.1.

This theorem allows us to construct m linearly independent formal so-

lutions to equation (3.1) (that is, solutions up to rapidly decreasing at infinity terms). However, it is possible to prove that there exist m linearly independent real solutions to this equation for which the above mentioned formal solutions will be the asymptotic expansions. This will be done in the following subsection; in conclusion to this subsection, we prove a statement which will be of use for us when constructing real solutions.

Section 1. Ordinary Differential Equations Proposition 3.3

yan

The following estimate

lY (s,x)| < Cetls! as valid for a solution Y (s,x) to equation (3.3) with some constants a and Ge Proof. As a result of the previous considerations, one has

¥ (3,2) +e") (sin s)]< (BIA (U(z),4(s))

(3.38)

(see formulas (3.20), (3.21), and (3.24)). Due to estimate (3.38) to prove the proposition, it is sufficient to show that

In (€,k)| < Ce™*

(3.39)

since along any ray which comes to infinity in s-plane for fixed value of x one clearly has li (s) < Cy |s|. Inequality (3.39) follows immediately from the fact that equation (3.30) for the function 7(€,k) is equivalent to some k-differential equation with constant coefficients in the variable €. Oo

3.1.5

Resurgent solutions to ordinary differential equations

Now we can prove the main statement of this section.

Theorem tions.

3.5 Equation (3.1) has the complete system of resurgent solu-

Before proving the stated theorem, we remark that, up to the moment, we have not used the polynomial dependence of the coefficients of equation (3.1) in the variabie x. To prove the above theorem, we must take this fact into account.

Proof of Theorem 3.5. Let K be the smallest number such that the orders a; of the polynomials P; (x) satisfy the inequality

o; < K(m-j) for all 7=0,1,...,m (we put a = 0 since the coefficient P,, (x) of equation (3.1) equals 1 identically). Clearly, one can treat the polynomials P; (x) as polynomials of order ¢; = K (m —j) in which principal coefficients can vanish. Consider the equation

y™ + Pr—a (x, 2°) yO") + 20.4 P, (a, 2°) y’ + Po (x, 2°) y = 0, (3.40)

212

CHAPTER III. Applications

where x° € C is an additional variable and the coefficients P,, he 0,...,m—1

are given by

Ps (2°) = (2°)! Bs (=) oa

0\

def

0\7%

zr

Then the coefficients P; (x,x°) of equation (3.40) are homogeneous polynomials in (z,z°). Besides, it is clear that solutions to equation (3.1) can be obtained from solutions to (3.40) by putting 2° = 1. Later on, it can be checked (we leave the corresponding considerations to the reader) that all the above constructed solutions to the corresponding “dual” equation

a\"™anmy' Wee Cae

(5)

~ Senegal

rr

a\—~™t1 gm-ly! Asa Soe

ae (=)

ES EO ep yh

pros

:

are homogeneous functions in the variables (s,z, 2°) of some order M with respect to the following action of the group C,: a (aaa) oe: (NETS

Ag ADS) ,

Hence, we have the estimate

Laat

ts ale

‘:@ /

s

=(@). mu(sm Bear

oo. In each Stokes region, one has

y(z)~ Ds eSil@)+Cng., (x), jk

(3.41)

where a;, (x) are expansions of the form

ajz(z) = >) a}, (2), 1=0

the functions a!, (x) are almost homogeneous functions with orders decreasing in /. (As above, an almost homogeneous function f(z) in z is

simply a restriction to z° = 1 of some homogeneous function f (z,z°).) The set of points

s = S;(x) + C, included in the summation in (3.41)

is contained in some sector of opening less than z bisected by the direction of the positive real axis. Clearly, the technique worked out in the previous chapter allows one to investigate the Stokes phenomenon for the constructed solutions. However, to do this, one must obtain more exact information on the resurgent structure of the considered solution (that is, on the structure of the Riemannian surface of the corresponding function

Y (s,z)) than the information which is contained in Proposition 3.1. Such information can be obtained only for concrete equations. Remark 3.5 As it was mentioned above, the considerations of this section allow one to construct asymptotic solutions to 1/h-differential equations. We shall not consider such situation here; the reader can do it by himself or herself.

3.2

Partial Differential Equations

The aim of this section is to construct (formal) asymptotic solutions at infinity for partial differential equations in the complex space C” with

coordinates (z},...,2”). Similar to the previous section, we shall consider here solutions to differential equations

fu

>

P»(@)

a Ox

Pheer

(3.42)

214

CHAPTER III. Applications

with polynomial coefficients Py (x). To construct such solutions, we use again the resurgent representation of the form

u(z)=

|

| e °U(s,x) ds

(3.43)

with endlessly continuable analytic function U (s,z). Similar to the above considerations, this function must satisfy the equation

pal Pa) (Zhe (x) ve 0:

(3.44)

la

3.2.1

Asymptotic solutions to the Schrodinger equation

To simplify our presentation, we shall carry out our considerations on the most simple but rather representative example of the Schrédinger equation‘

Au(z)+ V (xz) u(x) =0

(3.45)

with a polynomial potential V (x). The corresponding equation for U (s, 2) 1S

(=) AU (3,2) + V (2) U(s,2) =0. Os

(3.46)

Due to results obtained in the previous chapter, formula (3.43) determines an asymptotic solution to equation (3.45) provided that the resurgent structure of the resurgent function u(x) determined by (3.43) is well enough. More exactly, the following affirmation takes place:

Theorem 3.6 Let U(s,x) be an asymptotically homogeneous solution to (3.46) with respect to the following action on the group C,:

N (hz) = OF 342)

(3.47)

with some natural k. Then the corresponding resurgent function u(x), being an exponential function of order k, is an asymptotic solution to equation (3.45) up to functions of the arbitrary negative type. Remark

3.6 As we shall see below, the number k will be equal to

m+ 1

if the polynomial V(z) is a polynomial of degree 2m. 4This equation is essentially the Schrédinger equation h2

—-—Au+W(z)u 2m

= Eu

with polynomial potential W(z) with V(z) = E — W(z) and special choice of units of measure (which allows us to eliminate the coefficient h?/2m).

Section 2. ee Partial ee Differential ce as ee Equations

215

Thus, our main goal in this subsection will be the investigation of the resurgent structure of the constructed solution. It is well-known that if the

equation s = S(zx) describes the singularity set of a solution U (s,x) to (3.46) then the function S (zx) must satisfy the Hamilton-Jacobi equation®

@) +V (x) =0,

(3.48)

where we used the notation (0S/0z)” = Doja1 (0S/da! 2 In what follows, we suppose in addition that the function U(s,z) has simple singularities (such solutions are of the most interest in the physical applications), that is, this function has the form

x) ~ ) a; (a) f; (s — 5(z))

(3.49)

j=0

near each point s = S(z) of its singularity except for the focal points, where

{fj(z), 7 € Z} is some Ludwig sequence: f(x) = f;-1(z). Remark 3.7 As it was mentioned above, the function U(s, x) is a function with simple singularities in the classical sense if we consider the Ludwig sequence

1 = In s, s(In s—1),

However, sometimes it is useful to consider functions with simple singularities with respect for different Ludwig sequences. For example, one of the often used Ludwig sequences is

s°td I(o+39 +1)

iON e.

for some noninteger number a. Below we shall use one of the two mentioned Ludwig sequences. Under the above assumption, one can write down the system of trans-

port equations for amplitude functions a; (x) involved into (3.49). These equations are

OH (_ as

pees

Bahan

ee

x) p; («, assOpiOOp; BerxI , * 5 DLOx*O aG esOni bes

a; (x) = F;,

(3.50)

6This fact can be easily verified if one uses the so-called 0/0s-formalism (see [150]). Actually, we remark that equation (3.46) is a 0/0s-differential one. The WKBasymptotics (by smoothness) of solution to such an equation has the form U(s, x) = e~ 5(#)9/98 A(s, x) for some asymptotically homogeneous function A(s,x). Substituting the function U(s, x) in this form into equation (3.46), we obtain Hamilton-Jacobi equation (3.48) as well as the corresponding transport equations.

216

CHAPTER III. Applications

where F; = Fj [ao,...,@;-1] are expressions containing the functions aj, 1 n+k-—1. Suppose that -1 < q¢ U(p®)\L(X), where L = {(z,p)|px = 0} C CP x CP,5, is a locally trivial bundle with the fiber (L, NU(zo), X). It is easy to see that in a neighborhood of L(X)NU(p°), h(p) is a lifting of some vanishing cycle of the corresponding quadrics.

Now we shall define hi(p) for p € U(p®)\L(X).

Note that the pair

(U(x), X) is contractible, because X is a manifold in U(z9). Consider the exact sequence corresponding to this pair:

Section 1. Transform of Homogeneous Functions

241

H,(U(x0),X)

H,(U(29), X UL»)

H,(LpNU(zx0),X)

Since H,(U(zo),X) = 0, we see that the mapping

0: H,(U(r0)X UL,) > H.(Lp NU (a0), X) is an isomorphism.

The condition

Ohi (p) = A(p)

(A.6)

now defines the class hi(p) uniquely. Let Y be an analytic set such that f(x) is regular outside X UY and X is not a component of Y. Denote by X UY the dual set for X UY. By using Thom’s isotopy theorem, one can easily prove that the projection

(cP: x CPap\ty(X UY), Lua (Xu Y)) + CP,,p\(X UY) defines a locally trivial stratified bundle. Denote by Y the union of those components of X UY which do not contain £(X). Due to Thom’s theorem we see now that the classes h(p) and hi(p) may be extended to ramifying

classes in the spaces (A.4) for all p € £(X)UY. It can be proved that both

L£(X) and Y are analytic sets (see (151], [157]). Note also that by the construction of the classes h(p) and h(p), the Riemannian surfaces of the integrals (A.2), (A.3) are pointed coverings,

and h(p) is related to hi(p) by equality (A.6) for all p € L(X)UY. Definition A.2 Let h(p) and h;(p) be the ramifying classes defined above. The transform given by formulas (A.2) and (A.3) is called the F-transform.

Remark A.1 Since the classes h(p) and h,(p) depend only on the choice of the point p € CP,,,p, formulas (A.2) and (A.3) imply directly that the F-transform of a function belonging to Ai (X*) is a homogeneous function

of degree —(n +k +1).

242

APPENDIX.

Integral Transforms

We introduce also a transform F' on the space A,(L£X) of the variables p in a similar way. Namely, we set

of functions f(p)

Riser= teense) f Regret A(z)

k >—n,

BF pe)=OP (2) f Four eee, se

n+k+1

z

hi(z)

k —1, the inclusion

Fef € AGP

(C(X))

holds. Remark A.2 A similar result is valid for the F-transform.

Theorem A.2 Suppose q > max (—1,k + ##* —1); then the transforms

Fy: ARK) Ae aX), Frith CAREX) are mutually inverse.

AFD ou)

Section 2. Associated Transforms

Remark

243

A.3 It follows from the Picard-Lefschetz formulas (see, e.g., the

book [119]) that if f(z) is a holomorphic function in a neighborhood of X, then Ff is either holomorphic on L(X*) (if n is odd) or has there the square root ramification (if n is even). If f(x) ramifies like the square root, even and odd values of n interchange their roles in the above assertion. Similar affirmations are valid for F.

Now we formulate the commutation formulas for F- and F-transforms with differentiations and multiplications by independent variables.

Theorem A.3 Suppose that gq > 0. following equalities hold:

For any function f € Ai(X), the

(Ff) = Fis (2*f), PiPy(f) = Fe-1 (-34) The corresponding result for F-transform is as follows: Theorem

A.4 Oo

If q> 0, then the relations Coed

Ded

Cd

ioe

Bz Fe S) =— Frui (pi f),

a

CN

ee

2° Fe f=Fr-1

hold for any fe Ae

A.2

Transforms, Associated the F-transform

In this section,

we present

with

the two specializations of the above intro-

duced F-transform: the R- and the 0/0s-transforms of analytic (nonhomogeneous) functions. Both of these transforms are specializations of the F-transform in the affine charts of the corresponding projective spaces.

A.2.1

The

R-transform

In this subsection, we introduce the R-transform of analytic functions which is in some sense analogous to the Radon transform in the analysis of C® functions. To do this, we perform the “affinization” of the above introduced F-transform in the projective chart x° # 0 of the projective space CP7.

244

APPENDIX.

Integral Transforms

Let us consider the open set x° # 0 in the space CPZ. We identify this set with the space C”, so that the point 2’ = (z',...,x2”) of this space corresponds to the point of CP” with the homogeneous coordinates

(1:2! :...: 2%) (under this identification, the space C” is called the affine chart in CP" determined by the relation z° = 1). Let X C CZ, be an algebraic set. It is evident that X is an intersection of an analytic set in

CP” with the chart z° = 1. As above, we shall denote this analytic set by the same letter X. The set X is assumed to satisfy the nondegeneracy condition (see Subsection A.1.1). The restriction operator

it DARK)

AX)

HL, . 2") = 2’)

(A.7)

is an isomorphism for any k. Indeed, the operator

Gatley) = (V7 (S)

(A.8)

takes any function f(z’) € A,(X) to (i,)~"[f] € Af(X) (we recall that each element of A’ (X ) may have additional singularities, so the singularities at z° = 0 which do exist for the function (A.8) when k < 0 are simply included into the corresponding set Y;; see the definition of the space At in Subsection A.1.1). The operator given by (A.8) is evidently the inverse

operator to (A.7). We put k = —n in formula (A.7) and define the R-transform as the composition

Re Foo G2)a ee

Avy n-1(LX).

(A.9)

Due to formula (A.2) of Section A.1, the operator R is given by /

rg=

| ResoA

1

= eA

n

Vv

=f (Doss Pa),

(A.10)

h(p)

where pz’ = po +piz't+...+pnz”. It is evident that the integral does not require any regularization for such choice of k. Remark

A.4

We point out that, due to the definition of isomorphism

(A.7), for each function f the set Yy must contain the projective plane

z° = 0 as one of its components (the set Ys is used in the construction of h(p), see Subsection A.1.1). Hence, h(p) does not intersect the plane

z° = 0, and we can use the chart r° = 1 for writing down the integral F_,. This leads us to formula (A.10). As follows from Theorem A.1, Section A.1, the operator Ro7= a

O1Patt Avy noi(LX) zs A,(X),

Section 2. Associated Transforms

245

is the inverse for R. This operator is given by the formula

Ro f= mai (se -\ | f pa fe h(z)

(one must use the regularization described in A.1.1 for small values of q).

The formulas describing commutation of the operator 0/8z' with the R-transform follow from the results of Subsection A.1.2 (see Theorem A.3). Namely, it is clear that the diagram —0/dz'

A,(X)

wr iL a =
n+ k+ 1.) Computing the residue under the integral sign, we rewrite the expression for the 0/0stransform in the form

Dna

f(stp’2"

(=)

i

Foyaslf] a

(A.19)

"dx",

h(s,p’) on we do not distinguish where h(s,p") stands for h(s,1,p’) (from now

(A.19) is ate oy variables s and 3). The inverse for the operator

where

;

Fool=(x)

n—-l1

h(s,z"’)

a

n+k

ff (SZ)

e- ete" ede”.

(4.20)

The commutation formulas for the transforms (A.19) and (A.20) can be easily derived from those for the transforms Fy. Namely, the formulas

of

F)

F5yas (st) Oxi = piss Fojas(f),

#

5

F5/a ( —1, we need a regularization of integral (A.20). To obtain such a regularization, we note that the operator re)

5p AalX) > Apa(X),

determined in the scale A,(X)

for g > 0, is invertible. The inverse operator

(/As)"" : Aq-1(X) > A,(X) is given by the formula

(Bio eee afVia, aide.

(A.23)

s(x)

Now we are able to define the regularization of the integrals (A.19) and (A.20) for functions of A,(X) for g > —1. Namely, we put ,

a

n+k

ih f(s + pix", x")dx";

Fojaslf] = (=)

(A.24)

h(s,p'') 4

n—1

ian [f] = (=)

r,)

n+k

x

(=)

/ f(s—p'z",p")dp".

(A.25)

n(s,2")

. Using relations (A.22) the operators (A.24) and n-+k-—1. Moreover, the instead of k are inverse to

and the operator (A.23), one can verify that (A.25) coincide with (A.19) and (A.20) for g > operators (A.24) and (A.25) with -—(n+ k + 1) each other.

The formulas for the 0/0s-transform take the most symmetric form for k = —(n+1)/2 (at least if n is odd). This is due to the fact that, for such value of k, one has —(n + k +1) = —(n + 1)/2 and the indices k in the inverse and direct transforms coincide.

In this case, we omit the index k

and denote a

(n—1)/2

Foyaslf] = (=)

f(st+p"2r",2")dz",

(A.26)

h(s,p’’)

Slatin ht (h=1)/ Fojaalf] = (=) (+ -) aif f (s— px" ,p")dp".

(A.27)

h(s,2"’)

If n is even, then the meaning of the right-hand sides of the two latter formulas is not clear. It turns out, however, that one can define these

250

APPENDIX.

Integral Transforms

expressions for even values of n as well, since the operator 0/08 possesses

a square root in the scale A,(X). ideas let us vonnides the operator Q, determined by the following formula’

Oise A

= [ser F243

7]

Gey

TN

7(s,z'’)

for f € A,(X), where 7(s,x”) is a one-dimensional contour joining the

points +,/s(z’) — AN

One can show that the operator Q acts in the spaces

OA

A eC),

and that the following relation

A

takes place (here 1 is the identity operator). operator, we have

; Cnt

2

OQ

Since 0/0s is an invertible

=I

(78) = (&) A

It is clear that Q commutes with the operator 0/0s and with all operators | pee We define

BN

Oe

O Nand deta

Calera Ge) Baler Sn soerere Thus, the first operator in (A.28) is a square root of the operator 0/0s. This fact allows us to define the half-integer powers of 0/0s and the righthand sides of (A.26) and (A.27) for even values of n. Namely, in this case we have —1/2 Fosaslf(s, os ‘)) ==

Beat

(3)

7 » 9 \1/2 coyoe ss Pe)| = (=) Fone

i)

’

[f (s,p")]-

These formulas allow us to prove that transforms (A.26) and (A.27) are mutually inverse. Thus, the following statement is valid:

Section 2. Associated Transforms

Theorem

A.6

AS

The transforms

Fojas : Aq(X) > A,(L£X),

Fajas: Ay(LX) > A,(X), determined by formulas (A.26) and (A.27), are inverse to each other. The commutation

formulas for the introduced symmetric form of the

0/0s-transform are still given by (A.22). This completes our short presentation of the theory of integral transforms in spaces of ramifying analytic functions.

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