Rigorous estimates for the series expansions of Hamiltonian perturbation theory

R I G O R O U S E S T I M A T E S F O R T H E S E R I E S E X P A N S I O N S O F H A M I L T O N I A N P E R T U R B A T I O N T H E O R Y

ANTONIO GIORGILLI Dipartimento di Fisica dell'Universitd, Via Celoria 16 - 20133 MILANO (Italia}

LUIGI GALGANI Dipartimento di Matematica dell'Universith, Via Saldini 50 - 20133 MILANO (Italia)

(Received 9 July; accepted 6 September, 1985)

A b s t r a c t . In the present paper we prove a theorem giving rigorous est imates in the problem of

bringing to normal form a near ly integrable Hamil tonian system, using methods of classical per tur -

bat ion theor3/, i.e. series expansions in the "small parameter" e. For any order r of normal izat ion, we

give a lower bound e r for the convergence radius of the normal ized Hamil tonian, and a greater bound

for the remainder , i.e. the non normal ized par t of the Hamil tonian. As an applicat ion, we consider

the case of weakly coupled harmonic oscillators with highly nonresonant frequencies and show how,

by optimizing, for fixed e, the order r of normalizat ion, one gets for the remainder a greater bound

- * namely an exponent ia l es t imate of of the form de (~'1/~)~, with positive constants A,a and e l ,

Nekhoroshev 's type.

0 I n t r o d u c t i o n

In per turbat ion theory for nearly integrable Hamiltonian systems, one is typically concerned with a Hamiltonian of the form H(p, q, ~) - h(p)+ef(p, q), where ~ is a small real parameter , and p, q are action-angle variables; namely p - ( p l , . . . ,pn) E ff c R '~, q - ( q l , . . . , qn) E T n, where .~ is an open subset of R n, and T n is the n-dimensional torus. For e -- 0, the unper turbed Hamiltonian being independent of the angles, the

0h system is integrable, with solutions p(t) - p(0), q(t) - q(O)+ w(p(O))t, where w 0p

for p(0) C q(0) C T ". Classical per turbat ion theory aims at determining suitable canonical changes of

variables, defined by series expansions in the 'small parameter ' ~, in such a way as to normalize the Hamiltonian, for example by eliminating the angles, to all orders in e. As is well known, such a program in general fails. Indeed, it seems tha t it is in general impossible to have informations for all times and for open sets of phase space. In Kol- mogorov's theorem [1] one looks for informations for infinite times, renouncing to have informations for open sets; and in fact one finds invariant tori defining an invariant

Celestial Mechanics 37 (1985) 9 5 - 112.

�9 1985 by D. Reidel Publishing Company.

96 A.GIORGILLI and L.GALGANI

set, whose complement is open and dense. Otherwise one looks for informations in open sets, renouncing to have informations for all times. The latter point of view is closer to tha t of classical per turbat ion theory, because, by normalizing up to a finite order r, the variation of the actions turns out to be of order e up to times of the order r From this point of view, the deepest and somehow startl ing result, obtained rather recently by Nekhoroshev [2-3] using a technique borrowed from Arnold's proof of Kolmogorov's theorem, is tha t the variation of the actions is found to be small up to times which increase exponentially as the per turbat ion tends towards zero. As Arnold says [4], in commenting on Nekhoroshev's theorem, "La vitesse moyenne ~ laquelle les variables d 'action s'~loignent de leurs valeurs initiales dans des syst~mes hamiltoniens g~n~riques quelconques est si petite qu'elle n'est pas localis~e par aucune approxima- tion de la th~orie des perturbat ions (c'est-s cet ~loignement n'est pas perceptible au bout d 'un temps de l 'ordre de 1/r r quel que soit r, r ~tant le param~tre de la per turbat ion" .

Now, in order to go to the heart of the analytical part of Nekhoroshev's theorem, it seemed to us that it would be useful to go back to the methods of classical perturbat ion theory, with their series expansions in e, even in the simple case in which the per turbat ion f contains only a finite number of harmonics in the Fourier expansion in the angles q. However, to our surprise, we found that no general theorem appears to be known which gives a lower est imate e~ for the convergence radius of

* for the Hamiltonian normalized at order r. So we give here such a lower est imate er the convergence radius of the Hamiltonian normalized at order r, and also a greater bound for the remainder. As an application, considering the part icular case of weakly coupled harmonic oscillators with highly nonresonant frequencies, we show how, by optimizing for fixed r the order r of normalization, one obtains for the remainder an exponential est imate of Nekhoroshev's type. For what concerns the normalization, we consider however the general resonant case, i.e. the case in which, being interested in a region of phase space near a resonance w-k - 0 for some integer vectors k E ~ , where

is a given subspace (module) of Z n, one keeps in the normalized Hamiltonian terms of the form e ik'q, k E J~(. In particular, for ~ -- (0} one recovers the nonresonant case, with elimination of all angles.

As one will see, the t rea tment is elementary, and only requires a good control of the terms of a recurrent nature which appear in classical per turbat ion theory. The key point possibly consists in the fact that we work with rather simple closed expressions for such terms, as the ones provided by the Lie method. The fact that , as far as we know, such a simple but far from trivial result as the one given here cannot be found in the literature, al though requiring only elementary methods, provides, in our opinion, an evidence of how cumbersome was felt the s tandard method of defining canonical transformations, which makes use of generating functions involving mixed variables.

In section 2 we recall first the formulae defining the canonical t ransformations without inversions, and add then a corresponding theorem with the necessary estimates (Theorem 1, on canonical transformations). In section 3 we show how the generating function of the canonical t ransformation normalizing the Hamiltonian up to order r is determined at a formal level. The corresponding rigorous estimates are

RIGOROUS ESTIMATES FOR THE SERIES EXPANSIONS. . . 97

given in section 4 (Theorem 2, estimates for the generating function). In section 5 the results of theorem 1 and of theorem 2 are collected in our main theorem (Theorem 3, on Hamiltonian normal forms). In section 6 we give the application to the exponential est imate for the remainder for the case of weakly coupled harmonic oscillators, while section 7 is devoted to some technical estimates required in the proof of Theorem 1.

We thank G.Benett in for very useful discussions.

0 C o n v e r g e n c e e s t i m a t e s for c a n o n i c a l t r a n s f o r m a t i o n s .

Let us recall first how canonical transformations can be defined without any in- version, at a formal level; the complete t rea tment with the necessary convergence estimates will then be given below. If g is any dynamical variable, i.e. defined on a domain P of phase space with coordinates p, q, and

X(P, q, e) = ~ e jx j (P, q) j_~l

a real function

(1)

a generating function defined for (p, q) E P and e in an interval of R around the origin, t ransformed function Txg will be defined by

Txg -- ~ eJ gi ,

then the correspondingly

(2)

where the coefficients gj

j>o

are defined by the basic recurrent formula

g o = g

g r - - - Lxj gr-j ,

r j - - 1

r = 1, 2 , . . . (3)

Lxi = (Xj, "} being the Poisson bracket with Xj. It is easy to verify directly that the t ransformation thus defined is canonical,

i.e. preserves Poisson brackets, as was done in a very similar case in ref: 5. An equivalent way is to show tha t such transformation is the one induced by the solutions at ' t ime' e of the canonical differential system with a suitable Hamiltonian )~, namely the system dv O2 d q _ 0~ de -Oq'de Or' as occurs in the s tandard Lie method [6] Indeed, it

was shown by Henrard[7]and Rapapor t Is], that the two transformations coincide if one Ox Clearly, in the equivalent s tandard Lie method the recurrent formulae takes :~ - - 0~"

analogous to (2)-(3) would be given directly in terms of ~ instead of X, and in fact the results obtained in the present paper in terms of X can be very easily t ransla ted in the more familiar language of the s tandard Lie transform in terms of the corresponding ~:. The unusual choice made here is just due to our personal affection to our formulae (2)-(3).

Let us now come to a complete t rea tment of such transformation, with the necessary convergence estimates. To this end we need some notation. As usual in perturbation theory, one works with action-angle variables, namely one has (p, q) C .q • T ",


where ff is a bounded open subset of R ~ and T n the n-dimensional torus. In order to use Cauchy's inequality for the derivatives, one needs a complexification of the domain. In fact, at variance with most treatments, we will need only a complexification of the domain of actions, and so will work with domains Pe .~e • T~, where .~e is the union of polydisks of radius 8 centered on any point of .~, namely

{ -- . . . , p';- _ p ' e ff , j - 1 , . . , n } . (4) fie p (p~, p=) e C=;Ip~ - l < ~ ,

For a given positive 6 < 8 we will denote by .~e - 6 the set of points of fie which are contained in fie together with a &neighborhood; by abuse of notation we will also indicate Pe - 6 (.~e - 6) x T ~.

On such domain P e = ~e x T ~ we will consider functions g which are real for real values of p, admit a Fourier expansion in the angles q

g(P,q)-- ~ gk(P) eik'q , k6Z'*

(5)

with coefficients gk(P) analytic in fie, and have a finite norm IlgllD~ defined by

I~klg~- ~up I~k(p)l �9 peg~

Such a norm turns out to be much more convenient than the usual one defined by sup(p,q)ep~ Ig(P,q)l, because it provides better estimates, particularly in controlling the dependence on the number n of degrees of freedom.

A special role will be played by those functions which have only a finite number of harmonics; thus, for a given nonnegative integer K we will say g E 3rK if the sum in (5) is restricted to Ikl <_ K, where

I k l - Ik, l + - . - + Ik, I (7)

(in particular :To is the space of q-independent functions). One immediately checks the following properties: let g E 3rg,g ' E 3rg , , then g + g' E 3rm~x(K,g,), gg' C 3rg+K,, Lgg' -- {g,g'} E Y'K+I,:'. From this one easily deduces in particular that , if Xj C ~ 'g and g E 3rK ,, then for the function gr defined by the recurrent relation (3) one has gr E 3r~K+g ,. We will say that g is of class 3rK on Pe if g C 3rK on Pe, has Fourier coefficients gk(P) analytic in fie and is real for real values of p.

The main result of the present section is then the following

T h e o r e m 1 (on c a n o n i c a l t r a n s f o r m a t i o n s ) . Consider a domain P e = ~e x . Tn with a positive 8 and ~e defined by (4) with ~ c R n open and bounded. Consider functions Xj of class ~'K on P e (j > 1) /or a given positive integer K , with ~'K defined as abo.ve, and assume IlxillD~ < ~ j _ l with positive constants q~,~ (so that the power ser, es X(P,q,e) : E]>_I sJXi(P,q) converges for Is[ < ~). For any g of


class irk on Pe, consider the power series Txg - ~i>_o ei gi, with the coefficients gi defined by the recurrent relation (3). Then, for any positive 5 < O, the coefficient gj

7(:+1)K o~ D~-~ . ~ .d th~ ~ , i ~ T~9 i . ~ ~ o ~ 9 ~ ab~ol~t~l~ Io, I~l < ~*, is of class w h e re

(3K~ ~_~) -1 ~* = - - 7 - + ~ ;

furthermore, for the r-th remainder Rr(g) defined by Rr(g) - Txg has the estimate

(8)

r - }--~j=o r one

IIR~(g)IID~-~ ~ IlgllD~ (1 5")--1 (--~-) rq-1 . ( )9

P r o o f . In order to prove the theorem we will find a majorant for Txg , or more precisely a sequence of suitable positive constants C i such that [[gj][p~_6 -< Cj (j > 0), with

Co - Jig[Ivy. We look now for a recursive formula for Cr, assuming that C o , . . . , C~-1 have already been found. To this end, use will be made of the (nontrivial) inequalities

[Igil[D~_6 -< B~., j - 1 , . . . , r - 1 , (10)

with positive constants Bj- defined recursively by

B o - [[g[lD~

B r K(I)

- r S ~ j= l j (2 j + 1 ) ( r - j + 1 ) ~ J - l B r _ j , (r >__ 1) .

(11)

For a proof, see section 7. We will then show that, from (11), one can estimate

B~ < ( 3 K ~ 1 0 ) _ ~ F - ~ B ~ - I , (12)

so that, to obtain the majorant, one can take the sequence Cr defined by

( 3 K r }~ ) r ( r > l ) c ~ - X + Z 1191ID~, _ , (13)

and the estimates (8) and (9) of the theorem immediately follow from s tandard results on geometric series. Let us thus come to the deduction of (12) from (11). One immediately checks directly that (12) is true for r - 1, so that we can consider the case r >_ 2. By isolating in the right hand side of (11) the term with j - 1 and replacing j by j + 1 in the remaining terms, one can write

3KO Br - Br-1 -t (j + 1)(2j + 3)(r - j ) f l i - l B r _ l _ j . (14)

5 r5 3"=1 Now one easily checks that for any positive integer j it is

10 (j + 1)(2j + 3) < - - j ( 2 j + 1) (15)

- 3

lo0 A.GIORGILLI and L.GALGANI

so that the second term in the right hand side of (14) is less than ~-q/3Br-1 from the definition (11) with r replaced by r - 1, and (12) follows.

, a s i s s e e n

0 F o r m a l r e d u c t i o n to n o r m a l f o r m .

One has now to determine the generating function X in such a way that the transformed Hamiltonian TxH be in a suitable normal form up to any fixed order r, i.e. one has

TxH - - Zo + . " + r + r f(r) , (16)

where Z 0 , . . . , Zr are independent of the angles q (nonresonant case), or contain them only through terms of the type e ik'q, with k E N, for a given module 3~ c Z n (resonant construction), as briefly recalled in the introduction; no condition is instead imposed on the remainder f(r). Now, from H - h + c f and the linearity of Tx, one has

j > l d (hi + (17)

so that, by comparison of (16) with (17), one gets

hj -3v f j--1 -- & �9 (18)

Using also

and Lx. ho -

h s - L x . h o + ~ ! L x s h ~ - i , i= l

--LhoXs, one has then Z0 - h0 : h and, for s - 1 , . . . , r,

(19)

LhX8 + Z 8 - ~g8 , (20)

where ~ is given by

f o - f i (21)

fits -- f s - I + ~ J Lxsh~-j S

i=l

for _> 2 . (22)

A more convenient expression is

gl , _ _1 f s--1 "3V ~ j- Lxj Z,_ i for s _ 2 , S S

j = l

(23)

which is of the same form as factor �89 in front expression

of f~- l .

(22) with Zs - i in place of h s - j , but with the gain of a To obtain such a formula, one inserts in (22) the recurrent

8-1 j

fs--1 -- E LXi f s - l - j , S--1 i=l

(24)


and uses equation (18) in the form h s - i - Z8- i - fs--j--1; SO one can write

8--1 J

98 = ~ s ( s - 1) Lxj [ ( s - 1)Zs-a" + f s - j -1 ] , j = l

namely (23). One also has for the remainder f(~) the expression f(~) - )-~j>_l eJ-l(h~+i +

f~+j-1); but this will not be used below, because the estimates for the remainder will be obtained through the general result of theorem 1.

One considers (20) as an equation in the unknowns X~ and Z~, while ~I,~ is known once X l , . . . , X s - 1 are given, because f j , hj and Zj (j - 1 . . . , s - 1) depend only on X l , . . . , Xs-1. As the unper turbed Hamiltonian h is q-independent, one clearly has

c3X8 LhX8 - w(p) . c3q ' (25)

with w Oh (20) reads 3~, so that eq.

. c3x8

Oq (26)

Introducing Fourier expansions for Xs, Zs, ~8, with coefficients )~8,k, Zs,k, ~8,k tively, one has thus the equations

respec-

N N

i w. k )~,k + Z~,k �9 ~ , k �9 (27)

Having assigned a module ~ c Z =, if in our domain w(p ) -k #- 0 for k r Jr[, one can there solve (27) by

.q of the space of actions one has

8,k - - 0 Z s , k - - ~ 8,k for k C ~

. (2s) ~8,k Z s , k - - 0 for k ~

Xs,k iw . k '

so tha t X8 and Z8 are defined in .~ • T =. Now, as already pointed out by Poincar6, the manifolds with w(p) �9 k - 0 are

generically dense in the space of actions if k is unlimited, so that the formal construction given above would be useless. However, the key point here consists in the remark that , if the per turbat ion f has only a finite number of harmonics, say f E 7K, then also the relevant functions appearing in (26) for any finite s have a finite number of harmonics; to be precise, one easily checks that ~I'8, Zs, X8 E Ysg (see for example ref. 9). In such a way one is provided with what could be called a "natural ultraviolet cutoff", if one restricts oneself to a given finite order r of normalization, because one has then to take into account only the finite number of all resonances w. k - 0 for k ~ M, with [k[ _< rK . In order to exclude such resonances one can for example assume that , for a positive a t , in our action space .qe one has

kl > k Ikl r K . (29)


So, at variance with the generic case considered by Poincarg, in which it was impossible to perform even a single per turbat ive step, in the present case with a finite number of harmonics in the per turba t ion f , one can perform any finite number r of per turbat ive steps in suitable open domains. One then only remains with the problem of giving rigorous estimates on Xz , . . - ,X~ in order to properly define the corresponding canonical t ransformation; indeed, in order to completely determine the

~" for example make the simplest choice, generating function X (r) - ~--~i>_1 Xi, one can namely Xj = 0 for j > r + 1.

0 E s t i m a t e fo r t h e g e n e r a t i n g f u n c t i o n

r ~3" with The generating function X (~) - ~ ' = z Xj, X~

defined by the formulae (28) for 1 < j < r in the domain = ~ l k l < j K XJ, kei k.q is thus

Pe(.M,o~r,r,K) - ~e x T n , (30)

where .~e is defined by (4) with .~ a bounded open subset of R n, where inequalities (29) are also satisfied. Due to our choice (6) of the norm, and to the bound I w. k I > ~r for k ~ ~ , one immediately sees from (28) that , if one can est imate ~s on a domain P' c Pe, then in the same domain one has also

IIxolID, _< 1 II ,IID, O~ r

IIZ liD, 1,. . . ,r) , (31)

so tha t we just have to est imate 1198 I1P'-

In est imating 98 from formula (23), we will need a repeated application of estimates for Poisson brackets, and this requires, as usual with Cauchy's formula for the derivatives, a restriction of the domain at each step. Indeed the main formula, which is proven in section 7, is

][Lgg'llp'-a K+K' 6 Ilgllp'llg'llD', (32)

for s > 1. one has

if g,g' are of class 3rK,3rK, respectively on any common domain P ' c P e. So we

introduce 6 - 6 and define domains D8 (s - 1, , r) by igz - P e and 1)8 - P s - z - 6 r - - l ' ' " " "

Then we look for two positive sequences r /1, . . . , r /~ and 0 z , . . . , 0 ~ such tha t

I[f~-x[lD~ ~ o~[Ifl[D~ (33) II~IID~ 0~llfllD~ , ( 8 - x , . . . , r ) ;

from f0 -- f and (21) one sees tha t one can take r/1 - /91 - 1. From (23), (24), (31),

RIGOROUS ESTIMATES F O R THE SERIES EXPANSIONS. . . 103

(32) and (33), using Xj, Zj, f j - 1 C ~'K, one has then, for s _> 2,

8-1

IIf -lllD. < c - - 8 1

j = l J0s~-sIIfIID~

8--1

8 j=l jOsO~-~ll f[ lD~ ,

(34)

with

C __

and this gives the recurrent relations

(r- 1)K (35)

8 - - 1

E rib - C jOjrls-j S - - 1

1"=1

8 1 S - r l : + C jOjO:__j . 8

3"=1

(36)

One is thus led to look for two sequences #8 and v8 with the properties

rib ~ N88C 8-1

08 < v s C 8-1 (37)

and one immediately sees (using 8-] < 1 for 1 < j < s) 8 - - 1 - - - - - -

defined by ~1 - ul - 1 and tha t such sequences can be

8--1

~8 -- E jv j l~8-j j=l

v8 -- #8 +

8--1

~ j v s v ~ - 1 �9

j= l

(38)

In conclusion we have the estimate

I]XsIID~-6 --< v8 C 8 - 1 1 IIJ'IID~ , (s -- 1 , . . r) ~ r

(39)

where v8 is defined recursively (together with #8), by (38), and C is given by (35). On the other hand, in order to apply theorem 1, we need IIXsI]p, <- ~ 8 - 1 , and this requires a sui table e s t imate on vs.

Let us thus come to the system (38). First of all, from the second one, one finds immediately v8 2>_ #s, and so, using it in the second one and recalling the first one,

8 - - 1 even v8 >_ 1~8 + Y'~j=I j l~ jv s - i -- 2#8, i.e. ~8 _ v8/2 for s _ 2. Again from the second


one, for v8 one then gets the recursive relation v8 < 2 ~-~8-1 _ i= 1 j v i v s _ i , or equivalently

8 - - I

v~ <_ s ~ vjv~_y j= l

(40)

for s _> 2. By induction one immediately checks that

/'Is ~ d2(a--1)

one has for example

(41) for any s _> 1; thus one has also

us < r 2(8-11 ( s - 1, r) (42) which, inserted in (39), gives the required inequality. So we have proven the following

T h e o r e m 2 (on t h e g e n e r a t i n g f u n c t i o n ) . Let the Hamiltonian H(p ,q , s ) = h(p) + ~f(p,q) be defined on the domain P e - [~(34, a r , r , g ) defined in (30) (with w - Oh) for positive ~, a~, positive integers r, K and a given module ~ c Z ~ Assume

Op ~

h is of class jro and f of class irK on Pe, e being a real (or complex} parameter. Then the generating function X(r)(p,q,~) - ~-~;=1 eJXi(P,q), which brings the Hamiltonian in the resonant normal form up to order r adapted to the module ~ , is defined in P e - 5 for any 5 < ~/2 with X1 of class ~ 'g , and one has the estimate

(j 1 ) ~ . . . ~ r (43)

with 1

IlI[ID O~r ( r - 1)r2K

ILflID (44)

0 T h e m a i n t h e o r e m .

Collecting the results of the theorem on the generating function and of the theorem on canonical transformations, we can now come to our main theorem.

T h e o r e m 3 (on H a m i l t o n i a n n o r m a l f o r m s ) . Let the Hamiltonian H(p ,q ,e ) - h(p) + e f (p ,q) be defined on the domain Pe - P e ( ~ , a r , r ,K) defined in (30) ,(with

Oh w - ov )' for positive O, ar, positive integers r, K and a given module 34 c Z ~. Assume

h is of class jro and f of class ~ai< on De, " s being a real (or complex) parameter. Then, for any 8 < ~/2, there exists canomcal transformation from [ e - 2 8 to P~ which


brings the Hamiltonian H in the the module ~4,

H (~)

resonant normal form H (~) up to order r adapted to

r

- E sJ Zj + R~(H)

j=o (45)

E kE )vt;Ikl<_jK

.

The transformed Hamiltonian H (~) w h e re

is a power series in e which converges for r < cr,

, _ _ cc~r (46) g r - - r3 ,

and for

3 6 C - -

10 K IlflID '

the remainder Rr(H) one has the estimates

(47)

1 - - ,,,,llhllD + I1 fll (4s)

P r o o f .

I]R (H)IID _2 r r (49) _ < 1 , IIHIID . ~r

Using Theorem 2, in D e - 6 there exists the generating function X (~) - r ~

~ j = l e~Xj which brings the Hamiltonian into normal form up to order r, with Xj

of class ~ g and [IXjllp~_6 < (~/~j-1, (I) and fl being given by (44). One can then

apply theorem 1, with D e - 6 in place of Pe, taking both h E :To c 3rr and f E 3rg in place of g. As a consequence, TxH - Txh + e T x f converges for lel < e*, with e~ given by (8), so that , using (44), one gets (46). For what concerns the remainder, one clearly has Rr(H) = R r ( h ) + eRr_ l ( f ) , which, using (9), gives estimate (48). The second rougher estimate, which however explicitly exhibits the dependence on []H]lp~ , is obtained by using IlRr(h)llp_26 < IIRr_l(h) llp_26, i.e. llRr(H)llp_26 <_ ]]R~-l(h) + eRr- l ( f ) ][p_26. The theorem is proven.

0 E x p o n e n t i a l e s t i m a t e s for t h e r e m a i n d e r .

* with c given by (47), one can perform at least So, for any fixed r _< e I - - C O ~ I ,

one order of normalization, and it is then quite natural to ask which is the optimal order ropt(~), that minimizes the remainder for a fixed r < ~ . In dealing with this question, one is in general forced to enter into a geometrical discussion related to the arbi t rary quantities a t , which should be chosen in such a way that the corresponding regio_~ of action space defined by (29) be nonempty. However, as was pointed out by Gallavotti [l~ the discussion is exceptionally simple for systems of harmonic oscillators, where h(p) - A . p with a constant vector A C Z n, i.e. the frequency w(p) - A is p - independen t , because in such a case there is no geometrical discussion at all. More

106 A.GIORGILLI and L .GALGANI

precisely, we can consider the case usually s tudied in connection with Kolmogorov's theorem, where A is taken to be highly nonresonant , i.e satisfying

I:~. kl >_ ~ lk l - ~ , k ~ z ~ , k r 0 , (5o)

for a given positive 7. Thus (29) is satisfied with the choice

"~ (51) (Kr)

This gives in par t icular * c ~ ,__ el , _ _

e r -- r3+n , el K n , (52)

c being given by (47). For the remainder let us take the simplified expression

er3+n ) ]lR,.(H)]lpe_2,~ _< 2 IlHllp~ ,

e I

r

(r > 1) , (53)

which is obtained from (49), (46) and (52), when e _< e r /2 . Considering the right hand side of (53) as a function of r, for fixed e, one has to

look for the minimum of ~ . For real r this gives ~(e) defined by

e r_3+ n _ e _ ( 3 + n ) ,

e 1 (54)

1

i.e. ~ e ; so we take

* 3~-n

rop t (e ) - - -I e 1_ , (55) e e

where [...] denotes the integer part . Using ropt < V and (54), we es t imate

3-}-n / eropt

e I

ropt / eY3+n / 7-- 1 ~ ~ -- e 3+he -(3+n)~ < e 3+he

e I

1

so tha t , denoting

we finally have

~ ( H ) - IIRrop~(~)(H)llp _26 ,

1

~ ( H ) < Ae -(~;/~)3+"

(56)

(57)

with - 2e 3 + ~ IIH[ID~ �9 (58)

On the other hand one h a s * ~i e" -- ee 3+• eropt ($) --- ropt ($)3+ r~ > ~($)~-{-r~ > 2e, so tha t

1 e* is satisfied. For any given e < e 1 one can the consistency condition e <_ ~ rop~(~)

also consider the m a x i m u m order rmax(e) of normalizat ion, which is clearly given by

rmax(e) = e l / e ) a--*-~ �9 So, by comparison with (55), one has ropt(e) < rmax(e), and,


in fact, for small e the optimal order turns out to be roughly one third of the maximal one. Finally, one has also to require the condition ropt(e) > 1, which, from (55), gives

1 �9 the condition (stronger than e < ~el) ~k

< . (59) - - e 3 + n ,

thus the exponential estimates (57) and (58) only hold if (59) is satisfied. In conclusion, we have shown how by classical per turbat ion methods one is quite

natural ly led for the remainder to exponential estimates of Nekhoroshev's type, at least for the case of highly nonresonant harmonic oscillators.

0 T e c h n i c a l e s t i m a t e s .

We prove here two inequalities on Poisson brackets that were used above, namely inequality (32) of section 4 and inequalities (10) and (11) of section 2. Such inequalities strongly depend on the choice of the norm of functions given in (6) and on the fact that we deal with t runcated Fourier expansions, i.e. with functions belonging to the spaces ~'g defined in section 2. We still the form (4) and positive numbers ~ < 6. P e - P and .~ - .~.

consider domains P~ - .~ • T '~ with . ~ of For notational simplicity we will write here

In estimating the norm of derivatives, Cauchy's inequality will be used only for the variables p, while the derivatives with respect to the q variables will be dominated using the fact that they are applied to functions belonging to spaces of the type t g . For a function f of the variables p only, analytic in .~, Cauchy's inequality reads

OY' + +~- f] �9

0 ~ ...Opt" 9-e ji, lil <6o)

< 6i~ +.. .+i, , 9 '

N

where [f[o = supv 9 If(p)l with the notation of (6). For a given function h will be of interest to consider a corresponding function g defined by

E t g , it

Oil +...+j,.,+sx +...+8,., - - h ,

g Op{ ~ ...Opa~'~Oq; 1 . . . O q ~ ? (61)

which evidently inequality

also belongs to ~rg. From (60) one quite easily

I[gl]D-6 c

deduces, for g, the

(62)

with j l ! . . . j . !

G - 63~ +.- .+i~ ]kl k

�9 ..Ik,,l"~ (63)

From (62) and (63) one could also deduce the g e n e r a l i z e d C a u c h y ' s i n e q u a l i t y

Oil +...+j,~+81 +...+8,,

0 ~ 1 �9 �9 . OpZn" Oq~' . . . Oq 8"n f] P-6 j l ! . . . j,,, ! K ~ + ' ' + ~ " <

6J~ +--'+J,~ [If liD , (64)


but in fact only the stronger inequality (62) will be used below.

Let us first come to formula (32), namely

I IL:gll~,_~ -< K+K'

IlfllD Ilgll~, , (65)

if f E .FK and g E FK,. In consideration of the fact that one can see here the role played by our choice of the norm, we give the detailed proof. Recalling Lfg -

01 0a let us estimate the first of such two terms. One has o~ o~ _ E;'=~ oq , ,

Of Og E Opt Oq~

l

-- E otAeiA'q A6Z ~

oi

k+k =A

on the other hand one has ~ k, Opz 9-6 < I ',1 [AI910 ,19 ~o t h a t

E, Ik',l IAI910 ,19 = K ' IlfllD IIgllD

Thus (32) is proved treating analogously the second term, which is just symmetrical with respect to the first one.

With very similar calculations one also proves the following inequality, which will be used below.

L e m m a 1. For a given function h E irK, on P, consider the function

Oil +. . .+j . +sl + . . .+s . -- h , (66)

g , . . . O q 3 o

which evidently also belongs to :TK,. Then, for any f E irK, one has

IlL:gllD_a (jl + " " + j,~ + 1)K + K '

6 a IlfllD , (67)

w h e re

JII...J,~I E G - - (~j~ +...+j,., ]kll ~1

k

�9 . . I k , , I s o l h ~ [ ~ . (68)

is the (strong) generalized Cauchy estimate for g.

P r o o f . One evidently has Ill j 0 / Opt II P--5 II Oq~

I < P


Opt Ip-5 I1 11 < - - , O q , p - - 5 h + " ' + i , ~ E k , Ik'~ I "~- . - Ik',~ I "~ Ik'~llhkl~ . So one has

IILfgllD_~ <- 1 jll...jn! ~ ~ Ifkl~/4: :~: ~ Ik ' l l "1

1 k k I

~ (J' § lf~la I Summing over l and using E l Ik'zl _< K' , E l ( J l + x)lk~l _< K ( j l + . . . + jn + 1) one has

IiLfgllD_8

then K ' (jl + . . . + j= + 1)K

[IfllD a + 5 [IfllD a

namely (67).

Our main goal is now to obtain an estimate for the repeated applications of Poisson brackets of the type L i t . . . Lfx fo for any r = 1 ,2 , . - . . The estimates that would be obtained by repeated application of relation (65) prove to be non convenient. Indeed, this would require a restriction of the domain by a quanti ty 5r at each step, with 5r ) 0 as r ~ c~, if the final domain has to be nonempty, and the convergence of such a procedure is dubious. Such a procedure works instead when a finite number of iterations is required, as occurred in section 4.

So we proceed in the following way. First we prove the

L e m m a 2. Let {Kj}j>_o be a sequence of positive integers and (fJ}j>__o a sequence of functions f j C 3rKs on a common domain P, and consider the sequence {hi}j> o of functions hj E 3rKo+...+Kj defined by

h o - f o

hj - L f j h j _ l , j - 1 , 2 . . .

Then one has

where the sequence Aj is

IlhjiID_, ~ Aj ,

defined recursively

Aj = (2 jK j + K j - 1 +

j - - 1, 2 , . . . ,

by Ao - Ilfo]l D ~ . d

�9 - - + K0)II/~IID Aj-1 , j - 1 , 2 , . . . O

(69)

P r o o f . Using the very definition of Poisson bracket, one immediately one has the decomposition

realizes that

h i -1 - E ( D ~ j _ I O~

fj-1)...(Do~ofo) (70)

where each symbol D,~, denotes a combination of derivatives with respect to p and derivatives with respect to q, in such a way that the total number of derivatives of


each type in each term is j - 1. So one has

Ilhs'-i lip_6 <_ IlD s , L-l[I ,_6,--liD ofoliD_ <-- or

E G a s _ l . . . Gao , (:It

where Ga'. is defined by I[Da'.fr[[D_6 < Go,,. according to the estimate provided by the (strong) generalized Cauchy inequality (63). Denote now L - Lfs , and make use of the properties L ( f + g) - n f + Lg, L ( f g) - (L f )g + f (ng) . So one has

hi - - L h i - 1 E( (LDO`s_I f j -1)(Do`s_~f j -2) . . . (Do`ofo)+ O`

. . . + (Do`s_, f.i-1)(Do`s_2f.i-2) . .. (LDo`ofo)) ;

on the other hand, hj has necessarily a form analogous to (70) with a suitable set of indices/3, namely

hs" -- E ( D ~ j f j - 1 ) . . . (D~ofo) , (71)

and correspondingly one can find, by lemma 1, suitable quantities G~ wich estimate the single terms appearing in the right hand side of (71). So, using lemma 1 one has

E G;~j . . . G~o <

E [(mo`s + 1)K s- + K ] - I a~j_~ . . . G~o + - - - +

o`

(m o + 1)Ks + Ko 5

G o`s-1 "'" Gao] ,

where mo`s denotes the number of p-derivatives in Do` s fj, and one concludes

E Gpj . . . G~o < (2j - 1)Kj + K j - 1 -Jr-... + 1410 E Gaj_,. Gao �9

Thus the lemma is proven with the factor 2j in (69) substituted by 2 j - 1. We prefer however to use the weaker inequality (69) because, without altering the results, it turns out to simplify somewhat the computations in section 2.

Finally we come to

L e m m a 3. For given positive integers K , K ' , consider a sequence {Xj}]> 1 of functions Xj E ~'K and a function f E irK, on a common domain D. Define the sequence {fr}r>o of functions fr E jrrK+K, by f o - f and

r Lxj f~ - j " j=l

(72)

Then one has

]IAIID_ -< B, (73)


w h ~ r ~ B o - IlfllD and

Br - r6 j {[(2j + 1)(r - j + 1) - IIK + K'} IlxjiiD 3"--1

(74)

P r o o f . One evidently has fr = ~-~a caEaf for a suitable set of indices c~, where each Ea is a composition of at most r operators Lxs and ca is a suitable coefficient. Proceeding by induction, supposing to know Br-1 we want now to find a suitable B~,

r J Lxj f r - j So, one has to estimate Lxj f r - j -- ~-~ c~Lx s E~f which can e s t i m a t e E j = I r "

for a suitable set of indices ft. On the other hand, by Lemma 2, it is I]Epfl] p-6 <- AZ, where A# could be computed explicitly by the recurrent formula (69), once the actual factorization of E~ is known. On the other hand, by the same recurrent formula one has also

m

2 ( r - j + 1)jK + ( r - j ) K + K' 5

[(2j + 1)(r - j + 1) - l l K + K '

5

Iix~]ipA~

IIXy[IDAz �9

Here, use was made of the fact that EZ contains at most r - j compositions and of the fact that in the recurrent formula (69) the only relevant informations required are how many operators Lxs enter in the factorization of E~ and the fact that f~_j E ~'(r--j)gwg" So one has

[(2j + 1 ) ( r - j + 1 ) K -

5 1] + K ' IIxjlID [ceIAe.

By the definition of fr and by ~--~Z c~Af~ given by (73), and lemma 3 is proven.

< B~_j one has thus II f ll Br, with Br

In particular, in the case where K ' -- K and IIxjIID <_ oZ j - l , which section 2, one can take the sequence Br defined by

B , - - ~ j= l

j (2j + 1 ) ( r - j + 1) /~J- lBr_j

so that the inequalities (10) and (11) of section 2 are proven.

occurs in

R e f e r e n c e s

[11 Kolmogorov A.N.: 1954, Dokl. Akad. Nauk. SSSR, 98, N.4, 527-530. Arnold V.I.: 1963 Russ. Math. Surveys 18, N.5, 9-36, and N.6, 85-191; Moser J.: 1969, in Proc. Int. Conf. on Functional Analysis and related 60-67, University of Tokyo Press;

topics,


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Rigorous estimates for the series expansions of Hamiltonian perturbation theory

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