ADIABATIC INVARIANT OF THE HARMONIC OSCILLATOR,COMPLEX MATCHING AND RESURGENCECARLES BONET, DAVID SAUZIN, TERE M. SEARA, AND MARTA VAL�ENCIAAbstract. The linear oscillator equation with a frequency depending slowly on timeis used to test a method to compute exponentially small quantities. This work presentthe matching method in the complex plane as a tool to obtain rigorously the asymptoticvariation of the action of the associated hamiltonian beyond all orders.The solution in the complex plane is aproximated by a series in which all terms presenta singularity at the same point. Following matching techniques near this singularity oneis led to an equation which does not depend on any parameter, the so-called innerequation, of a Riccati type. This equation is studied by resurgence methods.1. IntroductionWe consider one degree of freedom Hamiltonian system depending on a parameter thatchanges slowly with time modelled by a Hamiltonian of the form (see [1])H(I; '; "t) = H0(I; �("t)) + "�0("t)H1(I; '; "t) ;where �("t) is a function with de�nite limits at �1 and such that �k)(~t) ! 0 when~t! �1, for all k 2 N . The equations of the motion are given by� _I = �"�0 @H1@'_' = @H0@I + "�0 @H1@I :(1.1)This is a quasi integrable system in the sense that we can apply the classical averagingprocedure looking for a change of variables, close to the identity in powers of ",� I = J + "u1(J; ; t) + "2u2(J; ; t) + :::' = + "v1(J; ; t) + "2v2(J; ; t) + ::: ;(1.2)in order to eliminate the angle variables of the Hamiltonian.If we truncate the formal series (1.2) at order n the system obtained is of the form� _J = "n�0("t):::_ = @H@I (I; ") + "n::: :(1.3)Key words and phrases. Adiabatic invariants, exponentially small, matching theory, resurgence theory.This work forms part of the Projects PR9614 of the UPC, PB95-0629 and PB94-0215 of the DGICYTand the EC grant ERBCHRXCT-940460. 1

2 CARLES BONET, DAVID SAUZIN, TERE M. SEARA, AND MARTA VAL�ENCIAPoincar�e proved that even though the series (1.2) are divergent they are asymptotic. Inour case that means that the actions of systems (1.1) and (1.3) satisfyjI(t)� J(t)� "u1(J(t); (t); t)� :::� "n�1un�1(J(t); (t); t)j � K"n ;for all t 2 R. As a consequence, I(t) is an adiabatic invariant for system (1.1), in thesense that its variation is small for a long time interval. Moreover, due to the asymptoticproperties of � it happens that un(J; ; t) ! 0 and vn(J; ; t) ! 0, as t ! �1. Then,one can see that I and J have limits at �1 verifying I(�1; ") = J(�1; ") +O("n), forall n 2 N. Moreover from (1.3) and taking into account that �("t) is bounded one hasthat J(+1; ") = J(�1; ") +O("n), 8n 2 N. Hence, it follows that�I(") := I(+1; ")� I(�1; ") = O("n); 8n 2 N:(1.4)That is, I(t; ") is a perpetual adiabatic invariant at all orders.Nevertheless, this discrepancy is nonzero (otherwise system (1.1) would be integrable)but it cannot be viewed directly from the asymptotic series (1.2). The goal of this paperis to present a method to compute the asymptotic expansion of the adiabatic invariant\beyond all orders".1.1. Matching and Resurgence. In fact we have an asymptotic development of I uni-formly valid for all t 2 R and the problem is to catch the part of I invisible in theseries (1.2). Matching Theory Principle says us that in order to see the hidden propertiesof a function de�ned by an asymptotic series we must go to the regions, called boundarylayers, where these series are no longer asymptotic. Boundary layers can be found by twofundamental methods: the �rst one is an a-priori knowledge of its location provided byheuristic arguments and the second one is to look for the singularities of the series terms.If we follow the second method for (1.2), we see that the terms of these series do nothave singularities in R (due to the asymptotic properties of �) and therefore, we are ledto look for the boundary layers in C. This is the principal reason why these problems areformulated using complex numbers and why the equation requires analyticity properties.Furthermore, working with analytic functions and complex asymptotic theory gives usmore chances to obtain re�ned results.Among others, we use in this paper as a basic tool Resurgence Theory for under-standing the nature of the divergence of the series. But instead of analysing the outerexpansion (1.2), we apply Resurgence Theory to the inner expansion (the series near theboundary layer) to compute �I(") given in (1.4). These techniques have been used by V.Hakim and K. Mallick in [6] to compute formally the separatrix splitting of the standardmap.In the present paper we use their aproach to compute the behaviour of the adiabaticinvariant for a simple oscillator �x + �2("�)x = 0;(1.5)

ADIABATIC INVARIANT 3obtaining rigorously an asymptotic expression for the adiabatic invariant �I(") beyondall orders. This problem is quite well understood but we think useful and clarifying totreat it joining matching techniques and the resurgence theory. We have followed closelyWasow [17] and Meyer [11] formulation reducing (1.5) to a Riccati equation.1.2. Wasow formulation and reduction to a Riccati equation. Following [17] andtaking t = "� in (1.5), let us consider the equation"2�u+ �2(t)u = 0; t 2 R(1.6)where �(t) satis�esH1: �(t) > 0, 8t 2 R,H2: limt!�1 �(t) = �� > 0,H3: � 2 C1(R) and �k) 2 L1((�1;+1)), k 2 N, (i.e. _� is a gentle function).Now, given any solution u(t; ") of (1.6), let us denote by I(t; ") the functionI2(t; ") := �(t)u2(t; ") + "2 _u2(t; ")�(t)(when � is a constant, I(t; ") is the action variable of the integrable Hamiltonian systemassociated to equation (1.6)). Littlewood proved in [8] that for any solution u(t; ") thelimits I(�1; ") exist, and�I2(") = I2(+1; ")� I2(�1; ") = O("n); 8n 2 N:Moreover, Wasow proved that �I2 satis�es�I2(") = 2"Re 24 p�(0)u0 + i "p�(0) _u0!2 p(+1; ")35 (1 +O(")) ;(1.7)where (u0; _u0) are the initial condition of u(t; "), and p(t; ") = e�2i=" R t0 �(s)dsp(t; "), withp(t; ") being the solution of the Riccati equation( " _p = 2i�(t)p+ _�(t)2�(t)(1� "p2)p(�1; ") = 0 ;(1.8)for all " > 0. Looking for the solution as a power series in ", one can prove Littlewood'sresults, but in order to obtain more acurate estimates for �I2(") we will need to extendour problem to the complex domain for the variable t.By the change of variable x = R t0 �(s)ds equation (1.8) becomes"w0 = 2iw + f(x)(1� "2w2);(1.9)

4 CARLES BONET, DAVID SAUZIN, TERE M. SEARA, AND MARTA VAL�ENCIAwhere f(x) = _�(t)2�2(t) . Now, due to hypotheses H1, H2, H3, on �, it is clear that f(x) is areal function with gentle properties. But in order to study the problem on C let us makethe following extra hypotheses on f :H4: f is real analytic in ��� fx0g, where x0 2 C, such that Im (x0) < 0 and � = fx 2C : Im (x0) < Im (x) � 0g, and for jx� x0j � 1 one hasf(x) = 16(x� x0) h1 + ~f((x� x0)2=3)iwith ~f(u) being an holomorphic function such that ~f(0) = 0.H5: f is C-gentle in the sense that for all x 2 ��� fx0g one haslimRe x!�1ZC�(x) ��fk)(s)�� ds = 0; k 2 N ;uniformly on x, whereC+(x) = ft 2 C : Im (t) = Im (x); Re (t) � Re (x) gand C�(x) = ft 2 C : Im (t) = Im (x); Re (t) � Re (x) g :Although our hypotheses H4 and H5 of f can seem capricious, they are deduced fromthe more natural hypotheses on � made by Wasow in [18], namely, �2 has an analyticcontinuation to the complex domain and has a simple zero inC noted t0, with Im (t0) < 0,such that x0 = R t00 �(s)ds (the case Im (t0) > 0 can be studied in an analogous way).The aim of this paper is to compute w(+1; "), where w(x; ") is the solution of theRiccati equation (1.9) such that w(�1; ") = 0. The rest of this paper is structured asfollows: First of all in section 2 we seek for w(x; ") as a power series in ", for complex valuesof x. We will study its asymptotic validity until some neighbourhood of the singularityx0 which is called the inner region. As it is usual in matching methods, in the innerregion a change of variables will be needed in order to enlarge the validity of the solution.This is done in section 3, obtaining as a �rst aproximation in this region the solution ofthe so called inner equation. This inner equation is studied by the help of resurgencetheory in a self contained way in section 5. In the inner region we can catch some termsof our solution hidden in the power series, and in section 4 we prove they are going tobe exponentially small on " (but not zero!) at +1. Finally, in section 6 we make someremarks for more general non-linear inner equations. We postpone for a next paper thegeneral study of (1.1) in a hamiltonian form (see [15]). Recently, Ramis and Sch�afke [13]have obtained upper bounds for �I(") showing the Gevrey-1 character (see footnote 1 ofsection 5) of the series (1.2) in the general case.All of this is summarized in the following

ADIABATIC INVARIANT 5Theorem 1.1. (Main Theorem) Let w(x; ") be the solution of the Riccati equation (1.9)such that limw(x; ") = 0 when x! �1. Then, if hypotheses H1,...,H5 are satis�ed onehas limx!+1 w(x; ") = � i"e�2ix0" (1 +O("2 =3))where w(x; ") := e�2ix="w(x; ") and is any number verifying 0 < < 1=2 . Moreover,the variation of the action of the Hamiltonian system associated to equation (1.6) is givenby �I2(") = �2�(0)u20e 2Im (x0)" sin(2Re (x0)" )(1 +O("2 =3));therefore, it is a quantity exponentially small in ".2. The solution in the outer left domainIn this section we prove the existence of the solution w(x; ") of the Riccati equation(1.9): "w0 = 2iw + f(x)(1� "2w2) ;such that limw(x; ") = 0, for Re (x)! �1 and x 2 � (where � is de�ned in hypothe-sis H4), and we give an asymptotic expression of the solution in a suitable subdomain of�.First of all, we look for a formal solution of (1.9):Proposition 2.1. There exists a series Pn�0 "nwn(x) that satis�es formally the Riccatiequation (1.9). The functions wn(x),i. verify the recurrence8>><>>: w0(x) = �f(x)2iw1(x) = w00(x)2iwn(x) = w0n�1(x)2i + f(x)2i Pi+j=n�2wi(x)wj(x); n > 1 ;(2.10)ii. are C-gentle functions (see hypothesis H5),iii. are analytic functions in ��� fx0g with a singularity at x = x0 such thatjwk)n (x)j � Cn;k jx� x0j�(n+k+1) ; if jx� x0j � 1; k 2 N :(2.11)

6 CARLES BONET, DAVID SAUZIN, TERE M. SEARA, AND MARTA VAL�ENCIARemark { Due to the fact that wn(x) are C-gentle functions uniformly bounded forjx� x0j � 1, we can choose the constants Cn;k such thatjwk)n (x)j � Cn;k ; if jx� x0j � 1; k 2 N :(2.12)Proof : The recurrence is obtained directly by the substitution of the series into (1.9) andthe properties of wn(x) follow from hypotheses H4 and H5 on f(x).Now we will prove that if we are not close to the singularity x0, the formal series ofProposition 2.1 is asymptotic to a C-gentle function w(x; "). Unfortunately, w(x; ") willnot be a solution of (1.9) but, nevertheless, it will help us to prove the existence of thesolution of (1.9) and its asymptoticity to the formal series.Let �" be the following subdomain of �, for a suitable > 0:

-6 '6?r�" " x0Proposition 2.2. Let wn(x), n � 0, be functions de�ned in � verifying ii) and iii) ofProposition 2.1. Then, for 0 � < 1 and " > 0 su�ciently small, there exists an analyticfunction w(x; ") de�ned for x 2 �" , such thati. for any � > 0 and k 2 N,jwk)(x; ")� NXn=0 "nwk)n (x)j � CN;k"�( +�)"(N+1)(1� )� k ;for all x 2 �" ; where CN;k are constants independent of " and �,ii. w(x; ") is a C-gentle function for Im (x) � Im (x0)� " .

ADIABATIC INVARIANT 7Remark { Let us note that if = 0 (this means that we are far away the singularity)this proposition says that the series Pn�0 "nwn(x) is asymptotic to the function w(x; ")on �1. In this sense we can look at i) as a weak form of asymptotic expansion near thesingularity.Proof : First of all, let us de�neKn(") := max0�k�nfsup�" jwk)n (x)j; sup�" ZC�(x) jwk)n (s)dsjgFrom bounds (2.11) and (2.12), as x 2 �" it follows thatKn(") � Cn"� (2n+1) ;where Cn := maxfCn;k : 0 � k � ng are independent of ". Secondly, let us de�ne, forany � > 0, �n(") := 1� e� 1"�Cn ;and note that �n(") < 1"�Cn .Then, let us consider !k)(x; ") =Xn�0 �n(")wk)n (x)"n;and let us de�ne Lk := maxf1; C0;kC0 ; :::; Ck�1;kCk�1 ;C0; :::;Ck;C0;k; :::;Ck;kg :Using bounds (2.11) and (2.12) it follows thatj�n(")wk)n (x)j � Cn;kCn "�(�+ )"� (n+k):Thus, for any k � 0 and n � 0, we have thatj�n(")wk)n (x)j � Lk"�( +�)"� (n+k):(2.13)So, from (2.13) and taking " small enough we obtain thatjwk)(x; ")j � 1Xn=0 j�n(")wk)n (x)"nj � 2Lk"� ��"�k ;(2.14)and thus, that wk)(x; ") converges uniformly in �" , for 0 � < 1, k 2 N and " su�cientlysmall. Furthermore, if we de�ne w(x; ") := w0)(x; "), we have that wk)(x; ") are the k-derivatives of w(x; ").

8 CARLES BONET, DAVID SAUZIN, TERE M. SEARA, AND MARTA VAL�ENCIANow, in order to see i) let us take N > 0 and let us use again the bounds (2.11), (2.12)and (2.13). It followsjwk)(x; ")� NXn=0 wk)n (x)"nj = jwk)(x; ")� NXn=0 �n(")wk)n (x)"n + NXn=0(�n(")� 1)wk)n (x)"nj� 1Xn=N+1 j�n(")wk)n (x)"nj+ NXn=0 j(�n(")� 1)wk)n (x)"nj� Lk 1Xn=N+1 "n� (n+k+1)�� + e� 1LN"� LN NXn=0 "n� (n+k+1)� CN;k"�(�+ )"(N+1)(1� )� k :(we have used that e� 1LN"� is exponentially small in ").By an analogous argument, using the integrals of w(x; ") on C�(x), we can prove ii).Finally, the following theorem proves the existence of the solution w(x; ") and give usestimates on its domain of de�nition. In this domain we will prove also that the series ofproposition 2.1 is weakly asymptotic to w(x; ") .Theorem 2.1. Let us take 0 < � < 1 and 0 � < 1� �. Then, if " > 0 is small enough,the Riccati equation (1.9) de�ned for x 2 �" , has a unique solution w(x; ") such thatlimw(x; ") = 0, when Re (x)! �1. Furthermore, the solution w(x; ") satis�es thatjwk)(x; ")� NXn=0 "nwk)n (x)j � KN;k "�( +�)"(N+1)(1� )� k;(2.15)for all x 2 �" and k;N 2 N, where the KN;k are constants independent of " and of �.Proof : Let us take w(x; ") = w(x; ") + Q(x; "), where w(x; ") is the gentle functionobtained in proposition 2.2. Then, w(x; ") will be the solution of (1.9) if Q(x; ") is thesolution of the equation"Q0 = 2iQ� f(x)"2(wQ�Q2) + q(x; ") ;(2.16)where q(x; ") := 2iw + f(x)(1� "2w2)� "w0 is an analytic C-gentle function in �" suchthat q(x; ") = O("n), for all n 2 N and for all x 2 �" . Let us note that this implies thatq veri�es that, for any n 2 N ZC�(x) jq(s; ")" j � Kn"n(2.17)

ADIABATIC INVARIANT 9for some constant Kn.Let us now consider the operatorT (Q) = ZC�(x) e2ix�s" �q(s; ")" � f(s)" �w(s; ")Q(s; ")�Q2(s; ")�� dsde�ned in the Banach space of continuous bounded functions, with the supremum norm.Then, using bounds (2.14) and (2.17), and hypothesis H4 we have, for " small enough,that1) if jjQjj � 1, jjT (Q)jj � Kn"n + " ln " (2L0"�( +�) + 1) � 1 ;2) if jjQijj � 1, for i = 1; 2,jjT (Q1)� T (Q2)jj � jjQ1 �Q2jj"(jjwjj+ 2) ZC�(x) jf(s)jds� jjQ1 �Q2jj"(2L0"�( +�) + 2)j ln " j � 12 jjQ1 �Q2jj :So, by the �xed point theorem the integral equation T (Q) = Q, and thus the di�erentialequation (2.16) have a unique solution Q. Moreover, using again (2.17), one has, for anyn 2 N jjQjj = jjT (Q)jj � 2jjT (0)jj � 2 ZC�(x) jq(s; ")" jds � 2Kn"n :Finally, using thatQ(x; ") = w(x; ")�w(x; ") and the bound of w(x; ") given in proposition2.2 we obtain the desired result. To obtain the bounds for the derivatives wk)(x; ") weonly have to use de equation (2.16) to see that all the derivatives of Q are asymptotic tozero.Unfortunately, with Theorem 2.1 we have proved that limw(x; ") = O("n) when Re (x)! +1,for all n 2 N, but we can not obtain more re�ned description of it at in�nity. So, if wewant to obtain an asymptotic expression for this limit, we will need to study the solutionnear the singularity x = x0 of wn(x). In order to simplify the exposition, we will assumefrom now on 0 < < 1=2.

10 CARLES BONET, DAVID SAUZIN, TERE M. SEARA, AND MARTA VAL�ENCIA3. The solution in the inner domainThe goal of this section is to obtain an asymptotic representation of w(x; ") near thesingularity x = x0 of wn(x). Of course, we can not obtain it at x = x0 but as we willsee in section 4 it will be su�cient to work at a distance of order " of this singularity.So, we will extend w(x; ") of Theorem 2.1 from a point x� such that jx � x�j = " ,Im (x�) � Im (x0) + " and Re (x�) � Re (x0) (i.e. x� belongs to the boundary of the leftdomain) up to the point ~x� symmetric of x� with respect to the line fRe (x) = Re (x0)g.>From ~x� we will continue the solution in the next section.-6 '$6?rr " x0x� "Note that, taking into account the bound (2.15), for N = 0, the asymptotic expressionof f given by hypotesis H4 and that 0 < < 1=2, the initial condition of w(x; ") in theinner domain veri�esjw(x�; ")� 112i(x� � x0) j � �K"� (" 23 + "1� ��) � K�"� =3 ;where �K and K� are constants independent of ".Then, if we consider the change of variable and the change of function� = x� x0" p(�; ") = "w(x0 + "�; ")equation (1.9) is transformed intop0 = 2ip+ "f(x0 + "�)(1� p2)(3.18)and de�ning � � = x��x0" , the initial condition for p(�; ") in the inner domain must verifyjp(� �; ")� 112i� � j � K�"1� =3:(3.19)So, we have to study the solution of (3.18) verifying (3.19) for � 2 C such that Im � � 1and Re (� �) < Re (�) < Re (~� �), where ~� � = ~x��x0" (we'll establish in Theorem 3.2 theunicity of such a solution).In order to do this, we will compare p(�; ") with the solution of

ADIABATIC INVARIANT 11p00 = 2ip0 + 16� (1� p20)(3.20)such that limp0(�) = 0 when Re (�)! �1. This equation is obtained from (3.18) when" tends to 0 where the initial condition is obtained \matching" the inner solution withthe outer solution at � �.It is easy to see, as in proposition 2.1, that there exists a formal solutionPn�0 an��n�1of equation (3.20). Moreover, looking at (2.10) and the behaviour of f assumed in hy-pothesis H4 one can see that the an are the principal parts of the terms of the outer seriesnear the singularity, that iswn(x) = an(x� x0)n+1 (1 +O((x� x0)2=3)):In the next theorem existence, analytic properties, and asymptoticity of p0(�) are de-scribed. The proof, done with resurgence theory methods, is given in section 5.Theorem 3.1. i. Equation (3.20) admits a unique solution p0(�) analytic in a sectorialneighbourhood of �1 such that limRe �!�1 p0(�) = 0 :Moreover this function is analytic in C�R+, and is asymptotic to the formal solutionof (3.20) in every proper subsector of this set.ii. Equation (3.20) admits a unique solution ~p0(�) analytic in a sectorial neighbourhoodof +1 such that limRe �!+1 ~p0(�) = 0 :Moreover this function is analytic in C�R�, and is asymptotic to the formal solutionof (3.20) in every proper subsector of this set.iii. If Re � > 0 and Im � > 0,p0(�)� ~p0(�) = �ie2i� (1 +O(��1)) :(3.21)Now, if we compare p(�; ") with p0(�) we have the followingTheorem 3.2. The problem (3.18), (3.19) has a unique solution p(�; ") de�ned for D�� =f� 2 C : Re (� �) � Re (�) � Re (~� �) ; Im � � 1g. Moreover, p(�; ") satis�es thatjp(�; ")� p0(�)j � L" 23 for all � 2 D��, where L is independent of ".For the proof of this theorem we will need the following Lemma:Lemma 1. There exists a constant B, independent of ", such that for � , �1 and �2 2 D��

12 CARLES BONET, DAVID SAUZIN, TERE M. SEARA, AND MARTA VAL�ENCIAi. jp0(�)j � B,ii. j R �2�1 p0(s)s dsj � B,iii. R �2�1 j ~f(("s)2=3)s jds � B" 23 ; where ~f is de�ned in hypothesis H4.Proof: Let us take p0(�) the unique solution of Theorem 3.1, and 0 < � < �=2 some �xedangle. Then there exists some constant C� such that for � 2 C,a: if j arg(�)j > �, jp0(�) + 112i� j � C� 1� 2(3.22)(use that p0(�) is asymptotic to the series Pn�0 an��n�1, where a0 = � 112i)b: if �� + � � arg(�) � � � �,j~p0(�) + 112i� j � C� 1� 2(use the same argument as before for ~p0(�))c: if j arg(�)j < �, jp0(�) + ie2i� � ~p0(�)j � C� 1�(use (3.21)).>From these inequalities i) follows inmediately. In order to prove ii) we only need tointegrate by parts and show that R �2�1 e2isdss is bounded for any �1, �2 in D��. Finally, iii)follows from hypothesis H4 taking into account that j�ij � " �1.Proof of the Theorem 3.2: If we consider v := p� p0, the problem that we have to studyis: 8<: v0 = [2i� p0(�)3� (1 + ~f(("�)2=3))]v � 16� [1 + ~f(("�)2=3))]v2� 16� ~f(("�)2=3)(1� p20)v(� �; ") = p(� �; ")� p0(� �)(3.23)Taking into account (3.19) and (3.22) (note that j arg � �j > �) we have thatjv(� �; ")j � K�"1� =3 + C�"2(1� ) � 2K�"1� =3 :(3.24)Now, let us consider the operator

ADIABATIC INVARIANT 13T (v) = e2i(����)e� R ��� p0(r)3r (1+ ~f)drv(� �; ")� Z ��� e2i(��s)e� R �s p0(r)3r (1+ ~f)dr 16sv2(s; ")ds+ Z ��� e2i(��s)e� R �s p0(r)3r (1+ ~f)dr 16s ~f(1� p20(s))ds;de�ned in the Banach space of continuous functions on D�� with the supremum norm,such that kvk � L" 23 with L = 12e2B=3B(1+B2). Taking into account the bounds (3.24),(i), (ii) and (iii) of Lemma 1, for " small enough, we have that1) if kvk � L" 23 ,kT (v)k � 16e2B=3 �12K�"1� + 2L2" 23 ln " �1 +B(1 +B2)� " 23 � L" 23 ;2) if kvik � L" 23 , for i = 1; 2 ,kT (v1)� T (v2)k � 2e2B=3L" 13 ln " �1kv1 � v2k � 12kv1 � v2k:So, by the �xed point theorem the integral equation T (v) = v and thus the di�erentialequation (3.23) has a unique solution. Moreover this solution can be bounded byjv(�; ")j � L" 23 ;for � 2 D��.Finally, using that v = p� p0 we �nish the proof of the theorem.As we have seen, Theorem 3.2 gives us a bound of the function w(x; ") on the right sideof the inner domain. In fact, at the point ~x� symmetric of x�, we havejw(~x�; ")� 1"p0( ~x� � x0" )j � L" 23 �1;(3.25)which will be used in the next section.4. The solution in the outer right domainIn this section we will extend the solution w(x; ") from the end point ~x� of the innerdomain up to +1. We will do this comparing w(x; ") with the solution ~w(x; ") of theequation (1.9) such that lim ~w(x; ") = 0, for Re (x) ! +1. The existence and theproperties of ~w(x; ") are analogous to w(x; ") considering now x belonging to the outerright domain ~�" :

14 CARLES BONET, DAVID SAUZIN, TERE M. SEARA, AND MARTA VAL�ENCIA-6 $6?r" ~�" x0All of this is summarised in the followingTheorem 4.1. Let us take � > 0. The Riccati equation (1.9) de�ned for x 2 ~�" ,0 < < 1 � �, and " > 0 su�ciently small has a unique solution ~w(x; ") such thatlim ~w(x; ") = 0 when Re (x)! +1. Furthermore, the solution ~w(x; ") satis�es thatj ~wk)(x; ")� NXn=0 "nwk)n (x)j � KN;k "�( +�)"(N+1)(1� )� k ; k 2 N ;for all x 2 ~�" , where KN;k are constants independent of " and of �, and wn(x) are thefunctions given in Proposition 2.1.Proof: It is analogous to the proof of Theorem 2.1.Remark { As in the previous section, in order to simplify the exposition, we will take fromnow on 0 < < 1=2.Now, let us take again ~x� = x0 + "~� �. We want to have an estimate of w(~x�; ") in orderto consider it as an initial condition to extend w(x; "). As we have seen in (3.25),jw(~x�; ")� 1"p0( ~x� � x0" )j � L" 23 �1 ;and from (3.21) with � = ~� � it follows thatjp0( ~x� � x0" ) + ie2i ~x��x0" � ~p0( ~x� � x0" )j � C�"1� :On the other hand, using theorem 3.1, theorem 4.1, the asymptotic expression of f andthat 0 < < 1=2 we obtain an analogous formula to (3.24), i.e.j ~w(~x�; ")� 1" ~p0( ~x� � x0" )j � 2 ~K�"� =3 :

ADIABATIC INVARIANT 15Considering this all together, one can obtain thatjw(~x�; ")� ~w(~x�; ") + i"e2i ~x��x0" j � 3L" 23 �1 :(4.26)Now we are willing to prove the followingTheorem 4.2. The solution w(x; ") of (1.9) exists for x 2 C such that Re (~x�) � Re (x)and Im (x0) + " � Im (x) � 0 and veri�esjw(x; ")� ~w(x; ") + i"e 2i" (x�x0)j � Ce� 2" Im (x�x0)" 23 �1(4.27)In order to prove this theorem we need the followingLemma 2. For x 2 C such that Re (x) � Re (~x�) and Im (x) � Im (x0+") the followingbounds holdi. j R x~x� "f(t) ~w(t; ")dtj � C"1� ,ii. ���R x~x� e� 2i" (~x��s)+R ~x�s 2"f(t) ~w(t;")dt"f(s)ds��� � "2� .where C is a constant independent of ".Proof: The �rst bound follows inmediately from hypothesis H4 and the asymptoticity of~w given in theorem 4.1. For the second one it is su�cient to integrate by parts.Proof of theorem: Let us de�ne z(x; ") := w(x; ") � ~w(x; "). From (1.9) and (4.26)z(x; ") veri�es "z0(x; ") = (2i� 2"2f(x) ~w(x; "))z(x; ")� "2f(x)z2and jz(~x�; ") + i"e2i ~x��x0" j � 3L" 23 �1 :Thus, noting that z is the solution of a Bernoulli equation one can obtain the followingintegral expression for z:z(x; ") = e 2i" (x�~x�)�R x~x� 2"f(t) ~w(t;")dtz(~x�; ")1 + z(~x�) R x~x� e� 2i" (~x��s)+R ~x�s 2"f(t) ~w(t;")dt"f(s)ds :(4.28)Now, using the bounds given by Lemma 2 and (4.28), we have that there exist someconstants ~Ci, i = 1; 2; 3, independent of " such thatjz(x; ")� e 2i" (x�~x�)z(~x�; ")j � ~C1"� je 2i" (x�~x�)j

16 CARLES BONET, DAVID SAUZIN, TERE M. SEARA, AND MARTA VAL�ENCIAand then jz(x; ") + i"e 2i" (x�x0)j � ~C2" 23 �1je 2i" (x�~x�)j:Now, using that Im (~x�) = Im (x0) + ", we obtainjz(x; ") + i"e 2i" (x�x0)j � ~C3" 23 �1je 2i" (x�x0)j = ~C3" 23 �1e� 2" Im (x�x0) :Finally, taking into account that z(x; ") = w(x; ")� ~w(x; ") we obtain the theorem.Now we are in a position to prove the Main Theorem 1.1 by taking the limit as Re (x)!+1 in inequality (4.27):j limRe (x)!+1 e�2i" xw(x; ") + i"e�2i" x0j � e2Im (x0)" " 23 �1:5. Resurgence of the solutions of the inner equation-Proof ofTheorem3.15.1. Introduction. In this part of the article we provide a self-contained introductionto �Ecalle's theory of resurgent functions, and we show how our inner problem (3.20) �tswithin this framework.We already know a formal solution of it : Xn�0 an��n�1, and it is easy to see that thereis no other formal solution. Our goal is to prove Theorem 3.1.After the change of variable z = 2i� ;equation (3.20) may be viewed as a particular case of singular Riccati equation of thetype dYdz = Y +H�(z) +H+(z)Y 2(5.29)where H� 2 z�1Cfz�1g (analytic germs at in�nity, vanishing at in�nity) : in our case,H�(z) = �H+(z) = 16z .Now, resurgence is a good tool for analyzing all the equations of this kind ; in fact�Ecalle's theory allows the analytic classi�cation of local equations in far more generalcontexts ([3, 4, 2] | of course resurgence is not the only possible approach, see [9, 10] foranother method of classifying singular local equations), but the study of equations (5.29)provides a nice elementary introduction to some aspects of �Ecalle's work, even if manysimpli�cations arise in the case of Riccati equations.5.2. Singular Riccati equations and resurgence.

ADIABATIC INVARIANT 175.2.1. Resurgence of the formal solution. In the sequel, H+ and H� are two �xed analyticgerms, vanishing at in�nity. As equation (3.20), equation (5.29) admits a unique solutionamong formal expansions in negative powers of the variable ; let's denote Y� 2 z�1C[[z�1]]this unique formal solution. We shall show that it is generically divergent using formalBorel transform B.The linear mapping B is de�ned by� z�1C[[z�1]] ! C[[�]]z�n�1 7! �n=n!and it induces an isomorphism between the multiplicative algebra of Gevrey-11 formalseries (without constant term) and the convolutive algebra of analytic germs at the ori-gin Cf�g, that is '1(z) 7! '1(�)'2(z) 7! '2(�)'1(z)'2(z) 7! '1 � '2(�) = Z �0 '1(�1)'2(� � �1)d�1 :Moreover, a formal series '(z) converges for jzj > � if and only if its Borel transform is anentire function of exponential type at most � : j'(�)j � const e�j�j. Hence, �nite radiusof convergence for ' (i.e. existence of singularities in �-plane) means divergence for '.We shall call resurgent function a Gevrey-1 formal series ' whose Borel transform hasthe following property : on any broken line issuing from the origin, there is a �nite setof points such that ' may be continued analytically along any path that closely followsthe broken line in the forward direction, while circumventing (to the right or to the left)those singular points. A non-trivial fact is the stability under convolution of this property.Indeed resurgent functions form an algebra which can be considered either as a subalgebraof C[[z�1]] (formal model) or, via B, as a subalgebra of Cf�g (convolutive model). TheBorel transform of a given resurgent function is often called its minor.Proposition 5.1. The formal solution of (5.29) is a resurgent function, with singularitiesin the convolutive model over the negative integers only.Proof : We start by performing Borel transform on equation (5.29) itself ; di�erentiationwith respect to z yields multiplication by �� and we obtain an equation for Y� :�(� + 1)Y (�) = H� + H+ � Y �2 ;(5.30)where H+ and H� are some entire series with in�nite radius of convergence.Let's de�ne inductively a sequence of C[[�]] by� Y0(�) = �H�(�)=(� + 1)1Let us recall that a formal power series Pn�0 an��n�1 is said to be Gevrey-1 if there exist twopositive constants M , K such that janj �Mn!Kn.

18 CARLES BONET, DAVID SAUZIN, TERE M. SEARA, AND MARTA VAL�ENCIA� 8n � 1; Yn(�) = �1� + 1(H+ � Xn1+n2=n�1 Yn1 � Yn2) :The valuation of Yn being at least 2n, the series Pn�0 Yn converges formally in C[[�]]towards the unique solution Y� of (5.30). Now we observe that H+ and H� de�ne entirefunctions of at most exponential growth in any direction ; Y0 de�nes thus a meromor-phic function with a simple pole at �1, and, by successive convolutions, we only get forthe Yn's other simple poles at the negative integers together with rami�cation (logarithmicsingularities).In particular, for each integer n, Yn is analytic in the universal covering of C n (�N�) ;with some technical but easy work, one can prove that the series of holomorphic func-tions P Yn is uniformly convergent in every compact subset of this universal covering.Therefore, Y� is convergent at the origin and satis�es the required property of Boreltransforms of resurgent functions.Remark 1 { The de�nition of a general resurgent function doesn't impose anything onthe nature of singularities one may encounter in following the analytic continuation ofits minor and visiting the various leaves of its Riemann surface. We shall call simpleresurgent function a resurgent function '(z) whose minor '(�) has only singularities ofthe form : '(! + �) = c2�i� + (�) log �2�i + R(�) ;with c 2 C and ; R 2 Cf�g. Simple resurgent functions form a subalgebra, whichcontains Y� and all the other resurgent functions that appear in the sequel.Remark 2 {When writing in details the proof of Prop. 5.1, one obtains in fact exponentialbounds in any sector S+� = f� 2 C�=� � + � � arg � � � � �g (� being a small positiveangle) : 8� 2 S+� ; jY�(�)j � const e�j�j ;where the positive number � depends on the radii of convergence of H+ and H� and on �.This allows us to apply Laplace transform in any direction di�erent from the directionof R�.Laplace transform in a direction � is de�ned byL� : '(�) 7! '�(z) = Z ei�10 '(�)e�z�d� :When applied to an analytic function of exponential type at most � in direction �, it yiedsa function '� analytic in a half-plane bisected by the conjugate direction : Re (zei�) > �.If ' has at most exponential growth and no singularity in a sector of aperture � (in�-plane), by moving the direction of integration and using Cauchy theorem, we get a

ADIABATIC INVARIANT 19function analytic in a sectorial neighbourhood of in�nity of aperture �+� (in z-plane)2 ;moreover, in this neighbourhood, '�(z) is asymptotic in Gevrey-1 sense3 to the formalseries ' = B�1' (a series which is the result of termwise application of Laplace transformto the Taylor series of ' : B�1 is in fact the formal Laplace transform).So, by chosing di�erent values for �, it is possible to associate to the formal series '(z)a family of sectorial germs f'�(z)g. When the series ' is convergent, the di�erent '�'syield the same analytic germ at in�nity : the sum of '. In general, the passage from 'to '� through L� � B may be considered as a resummation process, since multiplicationof formal series is taken to multiplication of sectorial germs, and di�erentiation w.r.t. z isrespected too. We sum up this Borel-Laplace process in a diagram :Formal series ' ' (convolutive model)Sectorial germ '�

��������1PPPPPPPPi BL�6

Applying Laplace transform L� to Y� with � 2] � �; �[, we get an analytic functionde�ned in a sectorial neighbourhood of in�nity of aperture 3� in z-plane, which is asolution of equation (5.29). In particular, we have two possible summations of the formalsolution Y� in the half-plane fRe z < 0g near in�nity : Y �� with � close to �, andY �0� with �0 close to ��. These functions correspond respectively to the solutions p0(�)and ~p0(�) Theorem 3.1 refers to.2This is a subset of C which contains, for all � 2]0; �+�[, a sector fz 2 C=j arg(zei�)j < �=2 ; jzj > �gfor some positive �.3 If '(z) =P'nz�n�1, this means that in every closed subsector �S of the domain, there exist C;K > 0such that : 8N � 1;8z 2 �S; jzjN+1j'�(z)� N�1Xn=0 'nz�n�1j � CKNN ! :

20 CARLES BONET, DAVID SAUZIN, TERE M. SEARA, AND MARTA VAL�ENCIAhhhhhh ((((((r r��0 Im � Re �'&-'&- $%Im z Re zY �0� Y �� ' $?'$?& %Im � Re �~p0 p0

The question now is to compute the di�erence Y �0� � Y �� ; we shall do it by analyzing thesingularities of the minor Y�.5.2.2. Formal integral. But before that, we study a formal object, more general than theformal solution Y�, which solves equation (5.29) too : the formal integral. We shall seethat Y� is the �rst term of a sequence (�n(z)) of simple resurgent functions such thatY (z; u) =Xn�0 unenz�n(z) 2 C[[z�1; uez]]formally satis�es the equation. This means that equation (5.29) is formally conjugated todXdz = X(5.31)through the formal di�eomorphismY = �(z;X) =Xn�0Xn�n(z) 2 C[[z�1; X]] :Due to the fact that we deal with a Riccati equation, the formal integral admits a simpleexpression :Proposition 5.2. There are formal series Y+ 2 z�1C[[z�1]] and Y0(z) 2 1 + z�1C[[z�1]]such that Y (z; u) = uezY0(z) + Y�(z)uezY0(z)Y+(z) + 1formally solves equation (5.29). Like Y�, these formal series are simple resurgent func-tions; 4 their Borel transforms have singularities over Z only, and at most exponentialgrowth at in�nity.4The de�nition of resurgent functions can be extended to allow them to have a constant term. Beingthe unity of the convolution, the Borel transform of 1 may be considered as the Dirac distribution �at � = 0. If ' = c+ is a resurgent function of constant term c, its Borel transform is B' = c� + B ,but we still call minor the germ ' = B . See section 5.3 for one further generalization.

ADIABATIC INVARIANT 21Proof : First we observe that our equation is equivalent to� ddz (1=Y ) = 1=Y +H+(z) +H�(z)(1=Y )2 :(5.32)Thus, using the same arguments that we used for proving Proposition 5.1, we see thatthere is a unique formal series Y+ 2 z�1C[[z�1]] whose inverse solves equation (5.29), andthat it is a simple resurgent function whose Borel transform has singularities over thepositive integers only and at most exponential growth.Expecting a linear fractional dependence on the free parameter, we perform the changeof unknown function Y = a+ Y�(z)aY+(z) + 1 :It yields the equation da=dz = a(1+H+Y�+H�Y+) : the general solution is a = uez+�(z),where � is the unique formal series without constant term of derivative H+Y� +H�Y+.The series � is a simple resurgent function ; its Borel transform�(�) = �1� (H+ � Y� + H� � Y+)has singularities over Z� only and at most exponential growth. Its exponential Y0 =e� inherits this property, by general properties of exponentiation of resurgent functions([3, 2] : Y0 has constant term 1 and its minor Y0(�) = Pn�1 ��n=n! is analytic in theuniversal covering of C n Z, with no singularity at the origin on the main sheet).So, we have Y (z; u) =Pn�0 unenz�n(z) with �0 = Y�, and for positive n,�n = (�1)n�1Y n0 Y n�1+ (1� Y�Y+) :If we apply Laplace transform in a non-singular direction �, we obtain a one-parameterfamily of analytic solutions of (5.29) :Y �(z; u) =Xn�0 unenzL��n = uezY �0 (z) + Y ��(z)uezY �0 (z)Y �+(z) + 1 ;de�ned for Re (zei�) � � > const:juezj (a condition meant to ensure that the Laplacetransforms of Y0, Y�, Y+ are de�ned and that the denominator in Y �(z; u) does notvanish).In the convolutive model, we can apply Cauchy theorem and move the direction ofintegration in the upper or in the lower half-plane (depending on the value of �). Thisprovides an analytic continuation of Y �(z; u) allowing z to vary in a sectorial neighbour-hood of in�nity of aperture 2�, that we call Y +(z; u) or Y �(z; u) as illustrated on thediagram :

22 CARLES BONET, DAVID SAUZIN, TERE M. SEARA, AND MARTA VAL�ENCIAhhhhhh ((((((r r r r��0 Im � Re � '& $%-�'&$%-�Im z Re zY �0=Y � Y +=Y �Thus, we essentially have two one-parameter families of analytic solutions of equation (5.29),characterized by their asymptotic behaviour in the above-mentioned domains in z-plane.There must be some connection between them : a member Y +(: ; u) of the �rst family hasto coincide with some member Y �(: ; u0) of the other family for values of z with negativereal part, and with some solution Y �(: ; u00) for values of z with positive real part. Theseconnection formulae will be computed in next sections.We are especially concerned with the functions Y +(z; 0) and Y �(z; 0) which correspondrespectively to p0(�) and ~p0(�). At this stage, the �rst two statements of Theorem 3.1are proved ; the unicity assertion for the second one, for instance, is a consequence of thefollowing easy lemma :Lemma 3. If u 2 C�, Y �(z; u) is de�ned for Re z � 0, Im z � 0, and jzj big enough,and Y �(z; u)� Y �(z; 0) = uez(1 +O(z�1)) :Indeed any solution of equation (5.29) analytic in a neighbourhood of i:1 on theimaginary axis has to coincide with some function Y �(z; u) ; only one tends to 0 as Im ztends to in�nity, and it corresponds to u = 0.And now we see that in order to prove the last statement of the theorem, we simply needto compute the value u0 of the parameter such that Y +(z; 0) = Y �(z; u0) for Re z < 0and to apply the lemma.5.2.3. Alien calculus. It is essential to be able to analyze the singularities that appear inthe convolutive model, for they are responsible for the divergence in the formal model.This is done by means of alien calculus, one of the main features of �Ecalle's theory, whichrelies on a new family of derivations : the alien derivations. Let's introduce them in thecase of simple resurgent functions.Let ! be in C�. We de�ne an operator �! in the following way : given a simpleresurgent function '(z), let's try to follow the analytic continuation of its minor '(�)along the half-line issuing from the origin and passing by ! (the minor is de�ned by theBorel transform of ' without taking into account the constant term if there is any) ; onthis line, there is an ordered sequence (!1; !2; : : : ) of singular points to be circumvented.If r � 1, we obtain in this way 2r�1 determinations of the minor in the segment ]!r�1; !r[(with the convention !0 = 0 if r = 1 | in this case, there is only one determination), andwe denote them by '"1;::: ;"r�1!1;::: ;!r�1

ADIABATIC INVARIANT 23each "i being a plus sign or a minus sign indicating whether !i is circumvented to theright or to the left :r r r r r��W � � ��Wxxxxxxxxxx xxxxxxxxxxxx direction of !' '�;+;�!1;!2;!3O !1 !3 !4!2� If ! 62 f!1; !2; : : : g, we set �!' = 0:� If ! = !r for r � 1, each of the above-mentioned determinations may have a singu-larity at ! :'"1;::: ;"r�1!1;::: ;!r�1(! + �) = c"1;::: ;"r�1!1;::: ;!r�12�i� + "1;::: ;"r�1!1;::: ;!r�1(�) log �2�i + regular function,and we set �!r' = X"1;::: ;"r�1 p(")!q(")!r! (c"1;::: ;"r�1!1;::: ;!r�1 + B�1 "1;::: ;"r�1!1;::: ;!r�1) ;(5.33)where the integers p and q = r � 1 � p are the numbers of plus signs and of minussigns in the sequence ("1; : : : ; "r�1).It is easy to check the consistency of this de�nition. In some sense, �!' is a well-balanced average of the singularities of the determinations of the minor over ! ; addingor removing false singularities in the list (!1; !2; : : : ) would not a�ect the result, whichis a simple resurgent function (the de�nitions of section 5.2.1 were formulated exactly forthis purpose).The de�nition of operators �! for more general algebras of resurgence is given in [3,4, 5]. These operators encode in fact the whole singular behaviour of the minor ; givena sequence of points (!1; : : : ; !n) in C�, not necessarily on the same line, the composedoperator �!n � � ��!1 measures singularities over the point !1 + � � �+ !n.The main property that makes these operators very useful in practice is the following :the �! are derivations of the algebra of resurgent functions, i.e.8! 2 C�; 8'1; '2 resurgent functions, �!('1'2) = (�!'1)'2 + '1(�!'2) :By contrast with the natural derivation ddz , they are called alien derivations.Alien derivations interact with natural derivation according to the ruleddz�!' = �! d'dz + !�!' ;which reads ddz ��!' = ��!d'dz(5.34)when one introduces the symbol ��! = e�!z�! (pointed alien derivation). But the(�!)!2C� generate a free Lie algebra.

24 CARLES BONET, DAVID SAUZIN, TERE M. SEARA, AND MARTA VAL�ENCIAFinally, let's state the other property that we shall use : suppose that all the singularitiesof the minor of ' in a sector f� < arg � < �0g form an ordered sequence (!1; !2; : : : ) on ahalf-line inside the sector ((((((((((((((((((hhhhhhhhhhhhhhhhhhr r r ��0O !1 !2 !3and that we can apply Borel-Laplace summation process, then'� = '�0 + Xr�1; i1;::: ;ir�1 1r!e�(!i1+:::+!ir )z(�!i1 � � ��!ir')�0 = [(expXi�1 ��!i):']�0(5.35)if we systematically use the notation � for L�B , and (e�!z )� for e�!z �.5.2.4. Bridge equation. Coming back to our Riccati equation (5.29), let's try to computethe alien derivatives of the various simple resurgent functions that appear in the formalintegral of Proposition 5.2. We shall use the generating seriesY (z; u) =Xn�0 unenz�n(z) ; �!Y =Xn�0 unenz�!�n ;and we shall assume ! 2 Z� since we already know that �!Y vanishes if it is not thecase.Of course, it is equivalent to look for ��!Y , and it turns out that one can easily derivefrom equation (5.29) a deep relation, simply by applying ��! to the equation itself : oneobtains a linear equation for ��!Yddz ( ��!Y ) = (1 + 2H+Y ) ��!Y(because pointed alien derivations commute with natural derivation, vanish on convergentseries like H� and satisfy Leibniz rule), which admits @Y=@u as non trivial solution, sothat there must be some proportionality relationship��!Y = A!(u)@Y@u ;simple arguments show that the coe�cient A!(u) must be zero if ! � �2 (because �1 6= 0,so the valuation of @Y=@u w.r.t. ez is exactly 1), that it is of the form A!(u) = A!u!+1(for homogeneity reasons), and �nally that it is zero if ! � 2 (because one can repeateverything with equation (5.32) ; it's only here that we use the fact that (5.29) is a Riccatiequation and not a more general nonlinear equation). We end up with

ADIABATIC INVARIANT 25Proposition 5.3. There exist A�; A+ 2 C such that���1Y = A�@ Y@u��+1Y = �A+u2@ Y@u��!Y = 0 if ! =2 f�1;+1g.So, the action of alien derivations on the formal integral is equivalent to the actionof some di�erential operator : this important and very general result was called bridgeequation by �Ecalle, since it throws a bridge between alien and ordinary calculus. Wheninterpreted in the convolutive model, it expresses a strong link between an analytic germat the origin and its singularities : in some ways, the germ reproduces itself at singularpoints, and this was the reason for naming \resurgent" such an object. Of course, withour de�nitions, not all resurgent functions have this property, but �Ecalle observed thatit holds for all resurgent functions that arise \naturally" (as solutions of some analyticproblem).For instance, bridge equation holds for more general nonlinear equations, but in contrastwith Riccati case, there can be then an in�nity of numbers A!; ! 2 f�1; 1; 2; : : :g ; theyare called analytic invariants of the equation, because it can be proven that two suchequations are analytically (not only formally) conjugated if and only if they have thesame set of A!'s.Thus, in our problem, inside the class of formally conjugated equations (5.29) (theyare all conjugated to equation (5.31)), analytic classes are parametrized by a pair of twonumbers.Alien derivatives can be computed explicitly in terms of the two analytic invariants A�and A+ :��1Y� = A�Y0(1� Y�Y+) �+1Y� = 0��1Y+ = 0 �+1Y+ = A+Y �10 (1� Y�Y+)��1Y0 = �A�Y 20 Y+ �+1Y0 = A+Y� :In particular, ��1Y� = A� + O(z�1), which means that A� is the residuum of Y�(�) atthe point �1. These two numbers are transcendent functions of the convergent germs H+and H� ; we shall see later how to compute them in special cases. The vanishing of alienderivatives at integer points other than �1 does not mean that there is no singularity athose points : these other singularities can be detected by iterating bridge equation.Bridge equation may be used for other purposes than analytic classi�cation : for-mula (5.35) can be justi�ed with ' = Y (: ; u), and we are now in a position to compareY �(z; u) and Y +(z; u), the two families of solutions of (5.29) we obtained by resummationat the end of section 5.2.2. In the sequel, we shall take various values of z with big enoughmodulus, and appropriated values of u in order to have Y �(z; u) de�ned.

26 CARLES BONET, DAVID SAUZIN, TERE M. SEARA, AND MARTA VAL�ENCIAIf Re z < 0, applying formula (5.35) with � < � < �0 (and these angles both close to �)yields Y �(z; u) = [exp ���1]�0(z; u) = [exp(A� @@u)]�0(z; u) = Y �0(z; u+ A�) ;that is Y +(z; u) = Y �(z; u+ A�) ;(5.36)and similarly, if Re z > 0, chosing � < 0 < �0,Y �(z; u) = Y +(z; u1 + A+u) :(5.37)Let's take u = 0 : we already knew that Y +(z; 0) and Y �(z; 0) coincide for Re z > 0(as was noticed at the end of section 5.2.1), but now, formula (5.36) and Lemma 3 showthat, for Re z < 0 and Im z > 0,Y +(z; 0)� Y �(z; 0) = A�ez(1 +O(z�1)) :Finally, when the original variable � = �iz=2 has positive real and imaginary parts,p0(�)� ~p0(�) = A�e2i� (1 +O(��1)) :(5.38)5.3. Computation of analytic invariants. To complete the proof of Theorem 3.1, weonly need to compute the coe�cient A� associated with an equationdYdz = Y + 16z (1� Y 2)(5.39)deduced from equation (3.20) by the change of variable z = 2i� , and to put it insideformula (5.38).We shall in fact compute the pair of analytic invariants for all equationsdYdz = Y � 12�iz (B� +B+Y 2) ;(5.40)where B� 2 C. The result is proved in the second volume of [3], but we present here analternative method.Proposition 5.4. The analytic invariants of equation (5.40) are given byA� = B��(B�B+) ; A+ = �B+�(B�B+) ;where �(b) = 2b1=2 sin b1=22 .Note that this implies A� = �i in the case of equation (5.39), as required for endingthe proof of the theorem.Proof : Let's begin with a few simple remarks. The coe�cient A� is the residuum at �1of the Borel transform of Y�, where Y�(z) is the unique formal solution of (5.40) andY+(z) the unique formal solution of the equation corresponding to (5.32). It is easy tosee that Y�(z) = B�y(z) ; Y+(z) = �B+y(�z) ;

ADIABATIC INVARIANT 27where y is the unique formal solution of an equation depending only on b = B�B+ :dydz = y � 12�iz (1 + by2) :If we solve the Borel transform of this equation like we did in the proof of Proposition 5.1(by expanding everything in powers of b), we �nd for the residuum of y(�) at�1 an analyticfunction �(b) such that �(0) = 1, and we obtain A� = B��(b) and A+ = �B+�(b).Rather than studying the power expansion of �(b) (this is more or less what is donein [3, vol. 2,p. 476{480], but in a very e�cient manner through the theory of moulds), weprefer to perform the change of unknown functiony(z) = 2�ib � zq0(z)q(z) ;which leads us to a second-order linear equationz2q00 + (z � z2)q0 � �2q = 0where �2 = b4�2 . We assume in the the sequel that Re � > 0 (excluding real non-positivevalues of b is innocuous since the function � is analytic).We exploit the peculiar form of this new equation, and write its unique formal solutionwith constant term 1 as the product of a monomial and of an expansion in fractionalpowers of z : q(z) = z�r(z) = 1 +O(z�1) ; r(z) 2 z��C[[z�1]] :(5.41)The series r(z) may be called resurgent if we extend the de�nition of Borel transform byz�� 7! ���1=�(�) ; if � 2 C and Re � > 0 ;and admit among resurgent functions all formal series (with possibly fractional powers)whose Borel transform, which may be now rami�ed at the origin, has endless analyticcontinuation. The convolution of minors is de�ned as before and we are still dealing withan algebra.The point is that alien derivatives of r are easy to compute, for the equation it satis�esr00 + (�1 + (2� + 1)z�1)r0 � �z�1r = 0(5.42)can be solved explicitly in the convolutive model :Lemma 4. The Borel transform r = Br is given byr(�) = ���1�(�)(1 + �)� :Proof : The Borel transform of equation (5.42)(�2 + �)r � (2� + 1)1 � (�r)� �(1 � r) = 0

28 CARLES BONET, DAVID SAUZIN, TERE M. SEARA, AND MARTA VAL�ENCIAis equivalent to a �rst-order linear equation obtained by di�erentiation with respect to �(this was the only purpose of the change of unknown function (5.41)) :�(� + 1)drd� = [(2� � 1)� + � � 1]r :Alien calculus applies in this slightly generalized context, we only need to be carefulabout the determination of �� we use (let's say it is the principal one) and about thesheet of the Riemann surface of the logarithm we look at. In particular alien derivationsare now indexed by points in this Riemann surface rather than by points in C�, and inorder to compute �ei� r we perform a translation :r(ei� + �) = ei�(��1) ���(�)(1� �)��1 ;and take the variation of the resulting singular germ (just as we were asked to retainthe coe�cient of log �=2�i in the case of pure logarithmic singularities, according toformula (5.33)) : B�ei� r = �ei��(1� e2i��) ���(�)(1� �)��1 :We deduce that �ei� r = �ei��(1� e2i��)�z���1(1 +O(z�1))and ��1 q = z��ei� r = �2i� sin(��)z�1(1 +O(z�1)):Finally we use Leibniz rule and formula (5.34) :��1 y = 2�ib z��1(q0=q) = 2b1=2 sin b1=22 +O(z�1) :The constant term of the alien derivative is the residuum �(b).6. Remarks on General non-linear inner equations: the Kruskal-SegurstrategyFormula p0(�) � ~p0(�) = �ie2i� (1 + O(��1)) obtained in Theorem 3.1 for the innerequation (3.20) has been crucial for determining w(+1; ") and thus �I2("), and thiswas the aim of previous section. There the equation (3.20) is studied by the resurgencetheory obtaining the formula (5.38): p0(�) � ~p0(�) = A�e2i� (1 + O(��1)). In order tocompute exactly the coe�cient A�(= �i) in the subsection 5.3 is essential to use thatequation (3.20) can be transformed into a second order linear equation. However inmost applications (for exemple for standard maps, see [6]) the inner equation is not aRiccati equation and the method outlined in section 5 does not give quantitative results.

ADIABATIC INVARIANT 29Nevertheless in the general case we can join the resurgence method with the Kruskal andSegur strategy, [7]. This strategy adapted to our case would have that following form:The main idea is that A� can be computed looking at the coe�cients of the formalsolution of (3.20). Following the Kruskal-Segur strategy one can see that the growth ofan is controlled comparing them with the coe�cients bn of the associated linear problem.In this sense let Pn�0 an��n�1 be the formal solution of (3.20) which vanish at �1(and in fact at +1 ), and letPn�0 bn��n�1 be the associated formal solution of the linearpart of equation (3.20) q00 = 2iq0 + 16� . We obtain that bn = (�1)n+1 112i n!(2i)n and as a �rststep in this method we would have to prove:Proposition 6.1. an = knbn, where kn = 3� +O( 1n), as n!1 .In our case we have numerical evidences of this result, and probably using the specialform of our equation it could be proved analytically (see [16]). In this paper we have notadopted this strategy because of the special form of our equation where this result is aconsequence of the computation of ��1Y� in section 5.Proposition 6.1 would be essential to control the behaviour of the Borel transform ofour solution. Let '(�) := Pn�0 an�nn! be the Borel transformation of Pn�0 an��n�1 and'0(�) := 3�Pn�0 bn�nn! = � 12� 12i+� the Borel transformation of 3�Pn�0 bn��n�1, and let uscall '1 := '� '0. Finally let be f(�) the Borel resummation of '(�), that is, its Laplacetransform in some direction of the upper plane Im� > 0. As we know exactly '0(�) in allthe complex plane this method allows us to compute its contribution to the resummationf(�). Nevertheless it would remain to prove that '1(�) contributes to f(�) only withhigher order terms. In order to prove this, it would be necessary to know the behaviourof '1(�), and, for our case, this is done in proposition 6.2.Proposition 6.2. i. '0 has a unique singularity at �2i, which is a pole with residue� 12� ,ii. '1 has logarithmic singularities at �2i;�4i;�6i; � � � ,iii. Moreover f(�) is Gevrey-1 asymptotic to the formal solution Pn�0 an��n�1 in thesector �3�=2 + � � arg � � �=2� �, when j� j ! 1 .An analogous proposition is studied in section 5 with the help of resurgent theoryfor our equation, and it seems this can be generalized to other equations, like in [14].Resurgent theory gives us the location of the singularities of '1 as well as their type andconsequently their contribution to the \resummation" f(�).Finally, as a last step of this method, putting together proposition 6.1 and 6.2 we wouldhaveProposition 6.3. i. f(�) = p0(�) if �=2 < arg � < 2� ,ii. f(�) = ~p0(�) if � � < arg � < �=2 .

30 CARLES BONET, DAVID SAUZIN, TERE M. SEARA, AND MARTA VAL�ENCIAThen, for � such that 0 � arg � < �=2, one can use the analytic continuation of f(�),and using these propositions, and the Cauchy theorem one can see that, for our equationp0(�)� ~p0(�) = Z +1�1 e��s'(s)ds= Z +1�1 e��s'0(s)ds+ Z +1�1 e��s'1(s)ds= e2i� (�i +O(��1)):Observe that following the Kruskal-Segur strategy we can compute the coe�cient A�2�ias the residue of the Borel transform '0 at its pole �2i. We are convinced that the linkbetween the resurgence approach and Kruskal-Segur strategy rests on the fact that, ingeneral, all the successive approximations of the corresponding equation (5.30) have a poleat �2i. Then '0 would be the summation of all the polar parts at �2i of those approx-imations, and its residue the sum of their residues (which corresponds to the coe�cientA� computed in proposition 5.4). References[1] V.I. Arnold, V.V. Kozlov, A.I. Neishtadt. Mathematical aspects of Classical and Celestial Mechanics.Dinamical Systems III. Encyclopaedia of Mathematical Sciences. vol. 3. V.I. Arnold Ed. SpringerVerlag, 1988.[2] B. Candelpergher, J.-C. Nosmas, F. Pham: Approche de la r�esurgence. Actualit�es Math. Hermann,Paris (1993).[3] J. �Ecalle: Les fonctions r�esurgentes. Pr�epub. Math. Universit�e Paris-Sud, Orsay (3 vol. : 1981, 1981,1985).[4] J. �Ecalle: Cinq applications des fonctions r�esurgentes. Pr�epub. Math. Universit�e Paris-Sud, Orsay(1984).[5] J. �Ecalle: Six Lectures on Transseries, Analysable Functions and the Constructive Proof of Dulac'sConjecture. D. Schlomiuk (ed.), Bifurcations and Periodic Orbits of Vector Fields, pp. 75{184, KluwerAcademic Publishers, (1993).[6] V. Hakim, K. Mallick: Exponentially small splitting of separatrices, matching in the complex planeand Borel summation. Nonlinearity 6, pp. 57{70, (1992).[7] M. Kruskal, H. Segur: Asymptotics beyond all orders in a model of crystal growth. Stud. appl. Math.85, pp. 129{18, (1991).[8] J.E. Littlewood: Lorentz's pendulum problem. Ann. Physics 21, pp. 233{242, (1963).[9] J. Martinet, J.-P. Ramis: Probl�emes de modules pour des �equations diff�erentielles non lin�eaires dupremier ordre. IHES Publ. Math. 55, pp. 63{164, (1982).[10] J. Martinet, J.-P. Ramis: Classi�cation analytique des �equations di��erentielles non lin�eairesr�esonantes du premier ordre. Ann. Sc. ENS 16 no.4, pp. 571{621, (1983).[11] R.E. Meyer: Exponential Asymptotics. SIAM Review, 22, pp. 213{224, (1980).[12] R.E. Meyer: Gradual re ection of short waves. SIAM J. Appl. Math., 29, pp. 481{492, (1975).

ADIABATIC INVARIANT 31[13] J.P. Ramis; R. Sch�afke: Gevrey separation of fast and slow variables. Nonlinearity, 9, pp. 353{384,(1996).[14] D. Sauzin: R�esurgence parametrique et exponentielle petitesse de l'�ecart des separatrices du pendulerapidement forc. Ann. Inst. Fourier, Grenoble. 45, 2. pp. 453-511, (1995).[15] A.A. Slutskin: Motion of a one-dimensional nonlinear oscillator under adiabatic conditions. SovietPhysics Jetp, 18, pp. 676{682, (1964).[16] Y.B. Suris: On the complex separatrices of some standard-like maps. Nonlinearity, 7, pp. 11225{1236,(1994).[17] W. Wasow: Adiabatic invariance of a simple oscillator. SIAM, J. Math. Anal., 4, pp. 78{88, (1973)[18] W. Wasow: Calculation of an adiabatic invariant by turning point theory. SIAM, J. Math. Anal., 5,pp. 673{700, (1974).Carles Bonet: Dpt. Matem�atica Aplicada I, Universitat Polit�ecnica de Catalunya,Diagonal 647, 08028 Barcelona, SpainE-mail address : bonet@ma1.upc.esDavid Sauzin: Bureau des Longitudes, CNRS, 3, rue Mazarine, 75006 Paris, FranceE-mail address : sauzin@bdl.frTere M. Seara: Dpt. Matem�atica Aplicada I, Universitat Polit�ecnica de Catalunya,Diagonal 647, 08028 Barcelona, SpainE-mail address : tere@ma1.upc.esMarta Val�encia: Dpt. Matem�atica Aplicada I, Universitat Polit�ecnica de Catalunya,Diagonal 647, 08028 Barcelona, SpainE-mail address : valencia@ma1.upc.es