A Nekhoroshev theorem for some infinite dimensional systems

Manus ript submitted to AIMS journals Volume X, Number X, XX 200X

Website http://aimS ien es.org pp. X{XX

A Nekhoroshev theorem for some in nite{dimensional systems

Paolo Perfetti Dipartimento di Matemati a, Universita di Tor Vergata via della ri er a s ienti a 00133 Roma , Italy

(Communi ated by Aim S ien es)

P

Abstra t. We study the persisten e for long times of the solutions of some in nite{ dimensional dis rete hamiltonian systems with formal hamiltonian 1 i=1 h(Ai ) + V ('); (A; ') 2 RN  TN: V (') is not needed small and the problem is perturbative being the kineti energy unbounded. All the initial data (Ai (0); 'i (0)); i 2 N in the phase{spa e RN  TN; give rise to solutions with jAi (t) Ai (0)j lose to zero for exponentially{long h (Ai (0)) unbounded for times provided that Ai (0) is large enough for jij large. We need A i i ! +1 making 'i a fast variable; the greater is i; the faster is the angle 'i (avoiding the resonan es). The estimates are obtained in the spirit of the averaging theory reminding the analyti part of Nekhoroshev{theorem. .

In the study of hamiltonian ordinary dierential equations, two of the main problems are: 1) to prove the existen e of the solutions for a time as long as possible 2) to understand the qualitative properties of the solutions found. As a model problem let's onsider the hamiltonian H (A; ') = h(A)+ "V ('); o N N ' 2 T ; A 2 U = U  R ; (N  2 integer), h; V analyti fun tions, " real. The

anoni al equations are of ourse A_ = "V'; '_ = hA whose solution for " = 0 is 1. Introdu tion

A(t)  Ao ;

'(t) = 'o + t!(Ao )

h o (A ) 2 RN !(Ao ) =: A

(1:1)

As well known by the theory of quasi{periodi motions, f'(t)gt2R = TN if and only N X if the omponents of !(Ao) are non{resonant over ZN i.e. !i(Ao)i 6= 0 for any  2 ZN and k=N

N X i=1

i=1

ji j 6= 0: Otherwise we have f'(t)gt2R = TN k ; 1  k  N 1 (for

1 the solution is periodi ). By the elebrated KAM theorem, under some  h

onditions whi h are essentially: i) the determinant of the matrix A dierent 1 P from zero, ii) !(Ao ) non{resonant over ZN : Ni=1 !i(Ao )i  C j jN for all 2

2

2000 Mathemati s Subje t Classi ation. Primary:70K65, 70K70, 70K43, 70K40; Se ondary: 34C29, 34C27. Key words and phrases. Nekhoroshev theorem, stability result, in nite{dimensional systems Supported by Istituto Nazionale di Alta Matemati a \Fran es o Severi" and Ministero dell'Universita e della Ri er a S ienti a e Te nologi a, resear h program \Metodi variazionali ed equazioni dierenziali nonlineari" 1

PERFETTI PAOLO

2

N X i=1

ji j 6= 0; i 2 Z; C suitable (!(Ao ) is said Diophantine),

iii) j"j  "0

small

enough1 ,(1.1) an be ontinued into A(t) = Ao + " (t!(Ao )); '(t) = 'o + t!(Ao ) + " (t!(Ao )); t 2 R; (" and " are analyti fun tions of t!(Ao) and " su h that j" j + j " j "!!0 0; hen e (A(t); '(t)) are de ned for all times in spite of " 6= 0 (see [6℄ for a proof). The set of ve tors ! satisfying ii) has full Lebesgue measure but is nowhere dense and the theorem annot avoid its presen e even weakening the non{resonan e ondition, see [14℄, [7℄, [13℄. For in luding all the ve tors ! and then all initial data by the dieomorphism h ; one is for ed to give up the solutions globally de ned in time. This is !(A) = A essentially the ontent of the Nekhoroshev Theorem (see the pioneering work [11℄, the papers (plenty) of the \Milan group" of Bambusi, Benettin, Galgani, Giorgilli et . and see also [3℄, [1℄ with the referen es therein). Let's onsider again the hamiltonian h(A) + "V (') with the onditions i) and iii). Roughly speaking, all the solutions are shown to exist for jtj  T exp( "1 ) and jA(t) A(0)j  Ao "b : No subset of the phase{spa e is ex luded; a; b are positive onstants depending on N su h that a N !+1!0: This is the nite{dimensional situation. As far as we know, there are few papers on the extensions of the stability results for in nite{dimensional dis rete systems (not originating from PDE's). The rst one is [2℄ where an array of oupled harmoni os illators over Zd is onsidered. The hamil3 X X X V(i;j)k qik qj3 k ; tonian is H1(p; q) = K + V; K = 21 !j (p2j + qj2); V = a

j 2Zd

i;j 2Zd k=0

3 X

(pj ; qj ) 2 R2 : The oeÆ ients V(i;j)k satisfy jV(i;j)k j  Ue (1+dist(i;j)) ; (U; ; k=0

onstants, Æ  1). A

ording to our de nitions V is of long{range type; see (2.2). Roughly speaking they prove that if the energy of the initial datum is of order " (small) and on entrated in one point (say the origin), the variables (pi (t); qi (t))i2Z (ln " ) (faster than any remain lose to their initial value as long as jtj  exp alnln " power of " but slower than an exponential). Ea h !j is a gaussian random variable with the same varian e  and the measure of the set of ! = f!j gj2Z ex luded hamiltonian system is O(  ): In [3℄ the authors onsider an in nite{dimensional X 1 1 2 like (1.2) with h(A) = A ; V1 (') = " (1 os('i 'j ))  > 1; Æ

d

1 2 1

d

2

i;j 2Z;i6=j ji

j j

a small parameter. When " 6= 0 and small, the exponential stability for those quasi{periodi solutions whose ve tor{frequen y ! has an arbitrary, nite number of omponents is proved. In [1℄ the author shows the exponential stability for the so alled breathers, i.e. time{periodi , spatially lo alized solutions of perturbed sysX 1 " X 1 (q q )2 ; 2 tems whose hamiltonian is H2 = ( 2 pk + V (qk )) + 4 j  i i;j 2Z;i6=j ji j j k 2Z V 0 (0) = 0; V 00 (0) > 0;  > 1; " small as usual. In all these models the kineti "

" goes to zero when C and/or N goes to in nity. Up to some te hni alities, it an be stated an  h ) is equal to zero (for example harmoni os illators analogue theorem for systems su h that det( A where h(A)=: PNi !i A or elestial me hani s systems where some of the Ai 's are not present in h(A)). 1

0

2

=1

1

2

A STABILITY RESULT FOR INFINITE SYSTEMS

3

and the potential energy are nite and the thermodynami limit does not follow. Loosely speaking, a ommon feature of the previous results is the fa t that \most of the energy is ontributed" by few variables (hen e the variables pla ed far away

arry a small amount of energy). Here we generalize the on lusions of the Nekhoroshev theorem too but in our model most of the energy is ontributed by the variables far from the origin. We onsider a lass of in nitely many ODE's i 2 N ; Ai 2 R; 'i 2 T; T = R=2Z (1:2) A_ i = fi (') '_ i = hA (Ai ); i

1 2 2x

V 'i

for a suitable fun tion V; (1.2) 1 X would be the anoni al equations of the formal hamiltonian 12 A2i + V ('); ' = i=1 ('1 ; '2; '3 ; : : : ) 2 TN2. A Nekhoroshev{like theorem is proved. More spe i ally if h (Ai (0)) suÆ iently we suppose in (1.2) that 0 < a0  jhA A j  a < +1; and A i large for i  l 1  1; we prove the existen e of an in reasing sequen e of time{s ales ftl+k gk0 (tl+k+1  tl+k ) su h that the a tion{variable Al+k (t) remains very lose to its initial value Al+k (0) as long as jtj  tl+k : The larger Al+k (0) is, the loser Al+k (t) remains to it. It follows that for jtj  tl the variables Al ; Al+1 ; Al+2 ; : : : and 'l ; 'l+1 ; 'l+2 ; : : : \do not ae t" the motion of the system whose ee tive hamiltonZ l 1 X  (l 1) V ( '1 ; : : : ; ' l 1 ) = dJ fi (') h(Ai ) + V (l 1) ('1 ; : : : ; 'l 1 ); ian is: 'i i=1 and J = Nnf1; 2; : : : ; l 1g (the possibility of doing the average respe t to the in nite set of variables 'i i 2 J; is due to the weak topology introdu ed in the on guration spa e TN; see se tion 2). If Ah =: !lo+k ; we have jAl+k (t) Al+k (0)j = l+k O(j!lo+k j 1 ) for jtj  tl+k  expfO((!lo+k )1=2 )g and 'l+k (t)  'ol+k + !lo+k t: The frequen y !lo+k+1 is bigger than !lo+k in su h a way to determine a strong non{ resonan e ondition. A tually we nd !lo+k+1  expfO((!lo+k )1=2 )g: Without doing any hypotheses on the size of (!1; : : : ; !l 1); all we an say about the variables Ai (t) i = 1; 2; : : : ; l 1; is jAi (t) Ai (0)j  C; where C is a onstant and  is the size of the analyti ity of the domain of the fun tion h(A): If Ai(0) is not great for jij ! +1 we annot say that jAi (t) Ai (0)j is small for large t: This result is in agreement with [13℄ (see also [5℄, [12℄, [4℄) where the same system of equations is onsidered and proved that if: 1) jj is dierent from 0 and small enough, 2) !ko large enough and the ve tor (!1o; !2o; : : : ; !jo); j  1 suitably non{ resonant, then
the solution of (1.2) for any t is !ot = ( ! t ; ! t ; : : : ; !  t ; !lot; ; !lo+1 t; : : : ; ) 'i (t) = 'oi + !io t + i (!o t); Ai (t) = Aoi + i (!o t); i.e. almost{periodi . Hen e our theorem an be viewed also as a result about the persisten e for long times of almost{periodi motions. For instan e if h(x) =

and fi(') =

i i

o

1

o

2

o l 1

2 We ould have onsidered a system de ned over Zd and work on N after applying a bije tion of Zd onto N:

PERFETTI PAOLO

4

(1.2) an be viewed as a model of rystal latti e although we annot perform a kineti part (needed for applying the perturbative and averaging methods). The paper is organized as follows. In se tion 2 we give the setup and some de nitions. Se tion 3 ontains the main results while the intermediate results and all the proofs, sometime sket hed, are in Se tion 4.

thermodynami limit yet due to the very high energy per degree of freedom of the

2. Setup{De nitions Metri s

([9℄, [8℄) The distan e on TN is w ('; '0 ) 

X

X

i2N

('i ; '0i ) wi

wi >

0; wi < +1; (TN; w ) =: Tw is a ompa t spa e.  is the standard ( at) metri i2N on T  Ti: ([a℄; [b℄)  inf n2Zja b + 2nj a; b 2 R and [
℄ denotes equivalen e (mod. 2) lass. X The distan e on RN is w (A; A0 )  wi ar tan jAi A0i j: (RN ; w ) =: Rw is a i 2N

omplete Bana h spa e. With the metri s given, the onvergen e is equivalent to the weak

onvergen e (\ omponent by omponent"): '(n) n!+1!' means that 8 i 'i(n) n!+1!'i (no uniformity in the omponents) and the same o

urs for the spa e Rw : For es{Perturbations We onsider two examples of maps ffig (the for e). The rst one is so alled short range; x L  1 fi 

X

kj ikL

(2:1)

'i gj

 gj ('(L) ) ; '(L)  f'k gk2Bj (L) ; Bj (L)  fk : kk j k  Lg; ki j k is the Eu lidean distan e on N: gj are real{analyti fun tions from TjBj (L)j ! R and for some positive M we have supj;' L 2TjBj L j jgj ('(L))j  M: gj

( )

( )

The system (1.2) with su h fi is alled a nite range system of in nitely many

oupled variables. A parti ular ase, often onsidered, is given in d = 1 by L = 1; gj = os('j 'j 1 ) os('j+1 'j ): Note that ea h variable is oupled only with a nite number of dierent variables The se ond example is so alled long range as ea h variable is oupled with any other variable. In d = 1 it is given by fi  os 'i

X

j 2N

aj

Y

k6=0

(1 + aj+k sin 'i+k ) ;

X

j 2N

j aj j < 1

(2:2)

We point out that we don't need the existen e of a fun tion V : TN ! R su h V = f ('): For instan e we ould take V (') = P1 mg (1 os ' ) + that i i i=1P P1 ' 1 e ji jj (1  (1

os( ' ' )) for the short range

ase and V ( ' ) = i+1 i i=1 i;j =1

os('i 'jP )) 1for the Q1long range. In this sense (2.2) would be the derivative of the \fun tion" n=1 m=1(1 + am os 'n+m ): What we need well de ned are ertain averages des ribed here i

A STABILITY RESULT FOR INFINITE SYSTEMS Averages ([Ha℄ se tion 38, [8℄) the fun tions g[I ℄ by means of g[I ℄ : TjI j ! R; g[I ℄ =:

Z

5

For a measurable fun tion g: Tw ! R; let's de ne

g(')dJ ; I  N ; J = N nI; dJ

=

O

i2J

di

In Tw there exists a unique probability measure de ned over the {algebra, R; generated by the ylinders RI =

O

i2I N

Ui

O

j 62I

o

Ui = Ui  Ti ;

Ti ;

jI j < 1; (RI ) =

Y

i 2I

i (Ui )

where i is the normalized \Lebesgue measure" on Ti. If jI j < 1, g[I ℄ is a measurable fun tion on TjI j and g[I ℄ ! g a:e: on T as jI j ! N: For the examples in (2.1) and (2.2) the onvergen e is uniform For the in nite{dimensional ve tor ffig we shall suppose that for any nite I  N there exists a C 1(TjI j; R) fun tion, V (I )('), su h that fi[I ℄(') = 'i V (I ) (');

8i2I ;

8 ' 2 TjI j:

We shall speak of g{gradients.

De nition A g-gradient f is said uniformly weakly real-analyti if there exists a real number  > 0 su h that for any nite set I  Z, V (I ) (') is real-analyti on TjI j and an be ontinued: analyti ally to the set fz 2 C j I j : Re zi 2 T; jIm zi j <  g;

ontinuously on the losure

Fun tion{spa es We shall work in the analyti lass. Let f : V  Tl ! R; V = o V  Rl be an analyti fun tion. f = f (A; ') an be extended to an holomorphi o fun tion on the omplex domain D 
  C l  C l where D = D = [x2V fz 2 C l : jz xj < g  V;
 = fz 2 C l : Re zi 2 T; jIm zi j <  g; 0 <  < 1: The extension (whi h is alled f too) is ontinuous on the losure D 
 : This lass of fun tions is denoted by C ! (D 
 ; C ) \ C (D 
 ; C ) and its elements an be

de omposed as

f (A; ') = P


' = li=1 i 'i ; j j = Pli=1 ji j

X

 2Nl

ei
' f;k (A) =

X

 2N l

ei
'

1 Z d'e (2)l T l

i
' f

(A; ')

kf k; =: P 2Nl ej j supA2D(Ao ;) jf (A)j =: P 2Nl ej j kf k ;

kf'k; Æ  eÆ1 kf k; ; kfAj k kfA;'k

r; Æ

r;

 1Æ rl kf k;

 1r kf k; ; kfAk

r;

 rl kf k;

kf gk;  kf k; kgk;

PERFETTI PAOLO

6

For a ve tor valued fun tion whose omponents are fun tions in C ! (D 
 ; C ) \ C (D 
 ; C ); the norm is the sum of the norm of their omponents. For a matrix valued fun tion fM (x; y)gki;j=1 ; : D 
 ! C 2k we set (only for the tensorial

omponents) kM k =: sup jM (x; y)vj = sup k k v2R ;jvj=1



k k X X

j

v2R ;jvj=1 i=1 j =1

Mij (x; y)vj j



1 M2 For a generi square matrix M = M M3 M4 (ea h blo k is a p  p matrix), thinking of it as a linear operator over Rp  Rp and a ting over the ve tors v= k Mv k p (v1 ; v2); v1;2 2 R with metri jvj = jv1j + jv2j; we have kM k = supv6=0 jvj = jM v +M v +M v +M v j  kM1k + 1 kM2k + kM3k + kM4 k: jv j+jv j  We shall make use also of the following notations: v = (^v; vd) where v^ 2 Rd 1 or v = (v; vd 1 ; vd ) with v 2 Rd 2 : The idea of the proof in Theorem 3.1 is of \breaking" (1.2) in a sequen e of nite{ dimensional systems and then work in a nite{dimensional setting. Nevertheless, for obtaining the solution of (1.2), we have to make ertain limits in suitable in nite{ dimensional fun tion{spa es whi h we are going to des ribe. Let be : 1) D~ N(Ao ) = i2NB(; Aoi)  Rw ; B(; Aoi)  R is an interval entered in Aoi of length 2 2) D~ (k) (Ao(k) ) = ki=1 B (; Aoi )  Rk 3)f : D~ N(Ao )  Tw ! R; f = f (A; ') a fun tion integrable respe t to measure over Tw and ontinuous on D~ N(Ao ); Let's all A the ve tor{spa e of fun tions de ned in 3). f 2 A an be given the Fourier P series f  P2ZNf (A)ei
' where the P means that f (A) are zero unless 
' = kj=1 i 'i for some k 2 N : The Fourier oeÆ ients f determine f almost{everywhere and vi eversa when f is integrable (everywhere when f is ontinuous) R : ( k ) ( k ) f (A; ' ) = f (A; ')dJ J = N nf1; 2; : : : ; kg is well de ned on D~ N(Ao )  Tk : By what said before, f (k)(A; '(k) ) an be extended to an holomorphi fun tion of the variables ('1 ; '2; : : : ; 'k ) on the domain D~ N(Ao ) 
 (
  C k ): If f (k) 2 A depends only on a nite number of A0i s; (A1; : : : ; Ak ) for instan e, f (k) an be extended to an holomorphi fun tion also respe t toPthese variables. The spa e A an be endowed with the norm kf k = 2ZNsupA2D~ N(A ) jf (A)jejj whi h makes it a Bana h spa e For a g{gradient ffig we de ne kV (I )k  V (jI j) and let's suppose that V (jI j)  V (jI j+1) (otherwise V (jI j+1) = maxfV (jI j); kV (I +1) k g). 1 1

2 2 1

3 1

4 2

2



o

Great denominators fhAi gi=1;::: ;l: =: (hA^; hA: l ): Rl 7! Rl ; h: A^ =: fhAi gi=1;::: ;l 1 For Ao 2 Rl we will write hAo = hA(Ao) = fhAi (Aoi )g = !o: The initial data of the system (1.2) are (Ao; 'o) 2 RN  TN and A: o is su h that for any k  l 1  1; the d{dimensional ve tor (hAo1 ; hAo2 ; : : : ; hAok ) = (!1o; !2o; : : : ; !ko )

veri es the relation j!^ o
^ + !ko k j 1  j! o j j j  K (!o); k

k 6= 0; > 1 K = O(j!ko j1=2 )

(2:3)

A STABILITY RESULT FOR INFINITE SYSTEMS

7

3. Results Let be l  2; lk = l + k; kV (lk ) k  V (lk ) for V (lk )('1 ; : : : ; 'lk ) Theorem 3.1 Let's onsider the system (1.2) and let Ao 2 RN be su h that the ve tor !io = hAoi i = (l; l + 1; l + 2; : : : ) satis es (k  0 integer, C and C 0 universal

onstants)

C V (lk ) (k +1)3 ln6(k +2)+ C j!ao j < 1;  j!lok j lk

C

k!lok k V (lk )
1=2 1=2 V (lk ) j!lok j   1 (3:1) 1

r

 o (V (l ))2 j!o j exp C (3:2) k  1; 2 (l ) j!l j (V (l ))2 l V k ln (k + 1) There exists a transformation R1 : DN (Ao ) 
N ! DN (Ao ) 
N su h that in the new variables (v; u) de ned by (A; ') = R1 (v; u); (1.2) be omes k

j!lok j  C

k 1

k 1

k 1

0





2

2

1 d  X G(lj ) ui = hvi + dt vi j=0 nlj

(3:3)

1 d  (l 1) X Gn(lljj) ) ( V + vi = dt ui j =0

d v dt lm

1  X G(lj ) ulm j=m+1 nlj

=

i=l

il

1 + m;

1

(3:4)

m>0

(3:5)

s

j!lo j (V (l ))2 exp C 0 j  1 kGn(l ) kDN
N  Cj 2 (ln4 j ) j!lo j j ln2 (j + 1) V (l ) (3:6) V (l ) )2 j=0 kGn(l ) kDN
N  C ( j! o j j 1

j lj





2

j 1

j 1

j 1

2

0

0

l0

 2

 2

l0

The fun tions Gn(lljj) depend on the variables (v1 ; v2 ; : : : ; vlj ; u1 ; u2 ; : : : ; ulj 1 ) and (lj ) N onverges. R1 = limN !+1 Ce(nl0 ) Æ : : : Æ Ce(nlN 1 ) Æ Ce(nlN ) : C~(lj ) j =0 kGnlj kDN  


P1

2

2

is anoni al of in nitely many anoni al variables but it is the identity when a ts on the variables (Alk ; 'lk ) k > j:

Corollary 3.2 There exists a sequen e of time{s ales ftlk g; k  0; s

j!lo j C 1 = VC exp (l ) (k + 2) ln2 (k + 3) (k + 1) ln2(k + 2) V (l ) su h that jAi (t) Ai (0)j  C; 1  i  l 1; jAl (t) Al (0)j  C (2k + 4)2(ln4(2k + 4)) jV! j k  0 for jtj  tl Remarks i) If l = 1 all the variables 'i are fast (not only those one with index i  l) ii) Theorem 3.1 needs the ondition !lo large and then Al (0) large enough with k ! +1: Otherwise it would la k the perturbative hara ter of the problem tlk



k

k

0

(lk )

o lk

k

k

k

k

k

PERFETTI PAOLO

8

To prove Theorem 3.1 we make some steps. In Theorem 4.1 we start with the P hamiltonian H0(P A; ') = li=1 h(Ai )+ V (l) ('); and end with the hamiltonian given by H1(A0 ; '0) = li=1 h(A0i ) + V (l 1)(^'0 ) + R(l) (A0; '0 ); where the important point is that the fast variable 'l has been on ned in R(l) (A0; '0 ) whi h is of order j!loj 1 With the hamiltonian Hn(l) of Theorem 4.2, the separation of 'l has been pushed p 1 to O(j!loj e Cj! j): This is a hieved with n (integer) anoni al transformations and n in reases with j!loj (see (4.4)). In Corollary 4.3 we give an estimate of the size of the anoni al transformation onstru ted between Theorems 4.1 and 4.2. We emphasize that kC~(n) Idk n!+1!0 and this fa t, ru ial in Theorem 4.8, is a hieved be ause the analyti ity loss in the rst transformation of C~(n) (Theorem 4.1) is large if ompared with the analyti ity looses in the other transformations of C~ (n) (a tri k already used by A. Neishtadt in [10℄). In the next step C~(n) is brought inside the hamiltonian with one more d.o.f. H0(l ) ((4.7)) and the separation of the fast variable 'l+1 is repeated (Theorem 4.6 and Corollary 4.7) (exa tly as for 'l ). Now we an ontinue adding more and more d.o.f. and obtain ea h time a anoni al transformation C~(n ) : Finally in Theorem 4.8 we show essentially that under some hypotheses on the frequen ies (see Corollary 3.2) the omposition of all the (C~(n ))0 s admits the limit de ning the solution of our in nite{dimensional system o l

1

lk

lk

4. Intermediate theorems, orollaries and proofs P Theorem 4.1 Let's onsider the hamiltonian H0 (A; ') = li=1 h(Ai ) + V (l) ('); (A; ') 2 C l 
 and let Ao 2 Rl be a point su h that the ve tor hAo =q!o = j!lo j 6V (l) ; (^!o; !lo) 2 Rl satis es (2.3) with k = l and K  21Æ ln Æer a 2

V (l) ; 0 <   r < 3 : If (l) j!lo j  4eÆ (ar + Vr ) (4:1) : then via a suitable, anoni al transformation (A; ') = C (0) (A0 ; '0 ); (H0 ÆC )(A0 ; '0 ) = Pl o 0 ( l 1) 0 ( l ) 0 0 0 0 0 0 (^' ) + R (A ; ' ); (A ; ' ) 2 D1 (A ) 
1 ; H1 (A ; ' ) = i=1 h(Ai ) + V 3

( l ) ( l ) 2 kR k1 ;1  eÆrj!lo j (V ) ; (1 =  3r > 0; 1 =  3Æ > 0)

i) The size of !lo makes 'l a fast variable and (^!o; !lo) non{resonant up to order K ii) we write D (Ao ) instead of D(l)(Ao(l) ) (see se tion 2 Fun tion{ spa es) be ause there is no ambiguity on the number of dimensions. iii) writing the equations of H1(A0 ; '0); one an note that dtd A0i (t) = ' V (l 1)(^'0 )+O(j!lo j 1) i = 1; : : : ; l 1 whi h means that Ai (t) Ai (0) = O(1) be ause ('1 ; : : : ; 'l 1 ) are slow variables Proof V (l) (') = (V (l) (') V (l 1) (^ ')) + V (l 1) (^ ') where P P ( l ) (V (l) (') V (l 1)(^')) = jjK ei
'V + jj>K ei
'V(l);  6=0  6=0 Let's de ne the generating fun tion P S~(A0 ; ') = A0
' + S (A0 ; '); S (A0 ; ') = ei
' S (A0 );  2N Remarks

1

1

i

l

l

l

j jK;l 6=0

(l) S (A0 ) = ihVA0
 ; that allows us to eliminate the harmoni s V(l)

of order j j  K:

A STABILITY RESULT FOR INFINITE SYSTEMS

9

!l j jA0 Ao j  2ja K implies j!(A0 )
 j  j!(Ao)
 j j(!(A0) !(Ao))
 j qj! l j o j(!(A0 ) !(Ao ))
 j  12 j! l j for all j j  K and l 6= 0: The ondition   6Va l j!lo j   implies 2j!a lojq 21Æ ln Æer 2 V l provided that r < 3 (see the end of the proof). r < 3 !lo j as well. By standard al ulations (use max e tÆ t = 1 guarantees 6Va l  2ja K t0 eÆ and the exponential de ay with j j of the oeÆ ient f of an analyti fun tion on a omplex strip; see [6℄ for instan e) we have kS'k; Æ  2eÆ jV!lolj ; kSA0 k r; Æ  2 Vol ; kSA0 ' k r; Æ  2 V lo : The ondition rj!l j eÆrj!l j o

o

( )

( )

( )

( )

( )

( )

4 V (l)  1 eÆrj!o j

(4:2)

l

is guaranteed by (4.1) and allows us to de ne the anoni al transformation (use the analyti {impli it fun tion theorem (see [6℄)) (A; ') =: C (A0; '0 ) = (A0 +(A0 ; '0); '0 +
(A0 ; '0)) and (;
): D r (Ao ) 
 2Æ 7! D (Ao ) 
 Æ : The inverse trans: formation is (A0 ; '0) = C (A; ') = (A + (A; '); ' +
(A; ')); (;
): D 2r (Ao ) 
 Æ 7! D r (Ao) 
 kk

2Æ  kS'k; Æ ;

r;

k
k

r;

2Æ  kSA0 k

r; Æ :

(4:3)

C Æ C = C Æ C = Identity on the domain D 3r (Ao ) 
 3Æ Putting (A; ') = C (A0 ; '0) into H0(A; ') we obtain H1(A0 ; '0 ) with R(l) = f1 + f2 + f3 and P ^ V (l 1)(^'0 ); f1 = ( li=1 (h(A0i + i ) h(A0i ) hA0i
i ); f2 = V (l 1) (^ '0 +
) P 0 f3 = j j>K ei
(' +
)V(l) l 6=0 kf1k 3r; 3Æ  a Pli=1 ki k2 3r; 3Æ  a Pli=1 kS'j k2 3r; Æ  a( eÆ2j! lo j V (l) )2

kf2k 3r; 3Æ  k Vl'^ k 2Æ k
^ k r; 2Æ  rj2! lo j V (l) 21Æ V (l 1) kf3k 3r; 3Æ  e 2ÆK V (l) 1 Æerj!lo j (4.3) has been used. If eÆ4a r j!lo j  1 (guaranteed by (4.1)) and K  2Æ ln 2 V l we have kR(l)k 3r; 3Æ  eÆr3j !loj (V (l))2 (

1)

( )

j! j V 1 1 e The relation 2j!a j  21Æ ln eÆr 2 V is equivalent to f (x) = x r 2a  2 ln x 2  0 where x = ÆrVj! j : The fun tion f (x) has a minimum at x = a r V and the value of q ar  6 V e f (x) is 1 ln 2 V whi h is positive for   a provided that 0 < r < 3 o l

o l

(l)

(l)

o l

(l)

(l)

(l)

(l)

In Corollary 4.3 we will need the following estimates on the quantities
'0 = (Id + SA0 ' ) 1 SA0 ' ;
A0 = (Id + SA0 ' ) 1 SA0 A0 ; A0 = SA0 ' + S''
A0 ; '0 = S''(Id +
'0 ); (4.2) implies k(Id + SA0 ' ) 1 k r; Æ  2; kSA0A0 k r; Æ  (l) 8

V 4 2 V (ol) ; kS'0 '0 k; Æ  e2 Æ2 j!lo j ; (use maxt0 t2 e tÆ = e24Æ2 and r =
3 ; Æ =
3 ). r j!l j 4 V (lo) = 36 V (l) o  36 1 2 = 2 1 2 k
'0 k r; 2Æ  eÆr j!l j e

j!l j 81 2(n 1) 9 (n 1)

PERFETTI PAOLO

10

8 V lo Æ e  4e
 1 k
A0 k r; 2Æ  8r jV!lol j = erÆ j!l j r 9
 (n 1) l l 8 V o r  8
 1 k'0 k r; 2Æ  e16Æ Vj!lo j  erÆ j!l j Æe 9e
 (n 1) l 2

V 8

V kA0 k r; 2Æ  eÆrj!lo j + e Æ j!llo j 4r jV!lol j  91 (n 11) + ( 92 )2 (n 11) = 13 (n 11) ( )

( )

2

2

( )

( )

2 2

2

( )

( )

2 2

( )

2

2

4

2





Let's de ne  C = Id
+A0 A0 Id +'
0 '0 : The metri in the phase{spa e Rl  Tl; is jA0 A0 j + j'0 '0 j: Then we have k Ck ;  kA0 k ; + 1 k'0 k ; + k
A0 k ; + k
'0 k ; k (Id C )k ;  (n 11) (1 + 3

 +

 1 ) P '0 )+ < R(l) (A0 ; '0 ) >l +(R(l) (A0 ; '0 ) H1 (A0 ; '0 ) = li=1 h(A0i ) + V (l 1) (^ ( l ) 0 0 < R (A ; ' ) >l ) 1

1

1

1

1

1

1

1

1

1

1

2

1

Starting with H1(A0 ; '0 ); we perform a nite number n of anoni al transformation and further redu e the perturbation to order O(j!loj n): As usual n depends on j!loj in su h a way that the perturbation is exponentially small respe t to some power of j!loj; ( 12 in our ase) Let be: j+1 = j 3rj ; rj = rj+1 j = 0; : : : n; 0 =: ;  1 = 1 n =:
; j+1 = j 3Æj ; Æj = Æj+1 j = 0; : : : n; 0 =: ;  1 = 1 n =:
; The anoni al transformations C and C are re alled C (0) and C~(0): Theorem 4.2 Let's onsider the hamiltonian H1 (A0 ; '0 ); (A0 ; '0 ) 2 D (Ao ) 
 : If 1  81 V (l) (n 1)2 < 1
 (4:4) a j!lo j 1  1;
 4 e

 j!lo j 2 there exist n 1 anoni al transformation (A0 ; '0 ) = C (1) Æ : : : Æ C (n 1) (A(n) ; '(n) ); (A(n) ; '(n)) 2 D (Ao) 
 su h that (H1 Æ C (1) Æ: : :ÆC (n 1))(A(n) ; '(n)) =: Hn(l)(A(n) ; '(n)) = Pli=1 h(A(in) )+V (l 1)(^'(n) ) +G(nl)(A(n) ; '^(n)) + (Rn(l)(A(n) ; '(n)) < Rn(l)(A(n); '^(n) ) >ql); 6 V ; kRn(l) k ;  3 (V ) exp ln 2 j!o j e

 kG(nl) k ;  Æer l (162) V j! j eÆr j! j 1

1

n

n

(l) 2

(l)

n n

o l

o l

n n

(l)

i) In the spirit of Nekhoroshev theorem Rn(l); whi h depends on the fast variable '(l n); is exponentially small in j!loj1=2 pii) Another feature of the Nekhoroshev theorem is the fa t that n depends on j!loj and the bigger is j!loj; the bigger is n: Proof The al ulations are analogous to those employed for H0 (A; '): We apply n times the same pro edure, ea h time redu ing the size of one order respe t to j!lo j 1 : The variables (A00 ; '00 ) play the role of the variables (A0 ; '0 ): The generating fun tion is S~(A00; '0 ) = A00
'0 + S (A00; '0 ) where Remarks

S (A00 ; '0 ) =

X

2 6=0 j jK1 Nl ;l

ei
'

0 R(l) (A00 )

i!(A00 )


(4:5)

A STABILITY RESULT FOR INFINITE SYSTEMS 11 j!lo j ; K  1 ln Æ er j!lo j ; (see at the end of j!(A00 )
 j 1  j! lo j if jAq00 Ao j  2 aK 1 2Æ 2 V l oj l j ! 6 V 00 o l the proof that 2a K  a so that jA A j  1):  ! 1 ; 1 ! 2 ;  ! 1 ; 1 ! 2 ; K ! K1 C (0) ! C (1) ; C (0) ! C (1) ;  ! 0 ;
!
0  ! 0;
!
0 k
0 k ;  kSA00 k r ;  r 2j! lo j kR(l) k ;  Æ1 k0 k ;  kS'0 k ; Æ  eÆ 2j !lo j kR(l) k ;  r1 kSA00 '0 k r ; Æ  er Æ2 j!lo j kR(l) k ;  12 3 V lo  4 3  1 k
0'00 k r ; 2Æ  eÆ r4 j!lo j kR(l) k ;  eÆ4Vr lj! lo j eÆr j!l j 19 4 6 l l 3 V o  4 eÆ 3  1 Æ k
0A00 k r ; 2Æ  r 4j! lo j kR(l) k ;  4r jV!lo j eÆr j!l j 19 r 4 2 r l 3 V lo (1 + 1 )  4r 3 7  1 r k0'00 k r ; 2Æ  e Æ4 j!lo j kR(l)k ;  e4 Æ Vj!lo j eÆr j!l j 6 19eÆ 4 6 10 Æ k0A00 k r ; 2Æ  eÆ r2 j!lo j kR(l)k ; + e Æ4 j!lo j kR(l)k ; r 4j! lo j kR(l)k ;   21 34 + 101 rÆ 21 rÆ  21 9Æ 9Æ 1 r 3 2 1 r k (Id C )k ;  17 40 +  10Æ + 2 19 r + 19  3 +  10Æ + 2 19 r 1

1

1

1

( )

( )

1

2

2

1

2

1

2

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1 1

1

( )

1

1

1

1

1

1

1

1

1

1 1

1

( )

( )

2 1

1

( )

1 1

1

1

1

1

1

1

1

1 1

2 2 1

1

1

2 2 1

1

1

1 1

1 1

( )

( )

1

1 1

2 1

1

2 2 1

1

1 1

1

1

1

2 1

1

1

1 1

1

1 1

1

1

2

1

1 1

1

2

1

The new hamiltonian is

P '00 )+ H2(l) (A00 ; '00 ) =: H1(l) Æ C (1) (A00 ; '00 ) = li=1 h(A00 ) + V (l 1) (^ + < R(l)(A00; '^00) >l + < R1(l)(A00 ; '^00) >l +(R1(l)(A00 ; '00) < R1(l)(A00; '^00 ) >l) P R1(l) (A00 ; '00 ) = 5i=1 fi (A00 ; '00 ) ^ 0) V (l 1)(^'00)); kf1k2;2  k V('^l00 1) k1 2Æ1 k
^ 0k2 ;2  f1 = (V (l 1) (^ '00 +
V (l 1) 2 o kR(l)k2 ;2 2Æ1 e r1 j!l j ^ 0 ) >l < R(l)(A00 ; '^00) >l f2 =< R(l) (A00 + 0 ; '^00 +
(l) kf2k2 ;2  k l k1 2r1 ;2 k0 k2 ;2 + k l ) and a set of

anoni al transformations C (j) su h that (A(j) ; '(j) ) = C (j) (A(j+1) ; '(j+1) ); 1  j  n; C (j 1) (A(j) ; '(j) ) = (A(j) + (j 1) ; '(j) +
(j 1) ): H1 Æ C (1) Æ : : : Æ ( l ) C (n 1) )(A(j) ; '(j) ) = Hj (A(j) ; '(j) ) G(0l)  0; G(jl) = Gj(l) 1 + < Rj(l) >l ; R0(l)  0 1  j  n If S (k 1)(A(k) ; '(k)) is the fun tion involved in the onstru tion of the anoni al transformation (see (4.5)), we have the estimates kSA(j j 1)' j kj rj ;j Æj  eÆj r2j j!lo j kRj(l) 1 k k
(j 1) kj ;j  rj 2 j!lo j kRj(l) 1 kj ;j ; k(j 1) kj ;j  eÆj 2 j!lo j kRj(l) 1 kj ( )

(

1)

1

1

1

1

1

1

1

1

1

1

Taking Æj = 3(n
 1) ; rj = 3(n
1) ; for any j > 0; it follows 81 V l (n 1)
n 1 3 (V lo ) kRn(l) kn;n  Ære j!l j e( n )( n )j!lo j o o l je

  (n l je

 we have By the se ond of (4.4) j!324 1)2 < j!162

V l

V l ( ) 2

( )

1

1

;j

1

2

1

( )

( )

(l) 2

3 (V ) exp ln 2 j!oj e

 kRn(l) kn ;n  eÆr l (162) V (l) j!lo j 



1=2

 o 1 a

 j!l j  1 implies (4.6) when the se ond of (4.4) is used to repla e the produ t Æ1 r1 in terms of j!lo j: Moreover the se ond of (4.4) implies kRn(l) kn;n  183 V (l) : 3 (V (l) )2 1o (V (l) 9 o )j 2 j 2 (Æi ri ) 1  6 (V (l)o)2 2 j+1  kRj(l) 1 kj 1 ;j 1  Æer i=1 j!l j ej!l j eÆrj!l j  32 (nV (l1)) 2 2 j 2 (l) k
'(j(j)1) kj ;j  eÆj 1 r2j 1 j!lo j kRj(l) 1 kj 1 ;j 1  4( 9(en
1)
jV!lo j ) 32 (n2 1)j 2   184 23 (n2 1)j 2 = 274 (n2 1)j 2 k
A(j(j)1) kj ;j  rj2 81 j!lo j kRj(l) 1 kj 1 ;j 1  272
81e rÆjj 11 32 (n2 1)j 2  (n2 1)j 2 rj 1 2 j k'(j(j)1) kj ;j  e2 Æj2161 j!lo j kRj(l) 1 kj 1 ;j 1  eÆrjj 11 216
81
9e 23 (n2 1)j 2 = 16 27 eÆj 1 (n 1)2 rj 1 2 j 2 j Æj 1 2 2 j kA(j(j)1) kj ;j  272 (n2 1)j 2 + 16 27 eÆj 1 (n 1)2 (n 1)2 rj 1  3 (n 1)2 k (Id C (j 1) )kj ;j  (n 1)1 2 2j (1 + 1 rÆjj 11 +  rÆjj 11 ) The presen e of (n 1)2 at denominator will be essential (see Theorem 4.8) and is due to the fa t that Æ0; r0; is mu h greater than respe tively Æj and rj for j > 0 (in fa t they are n independent) ) k ;  3 (V (l) )2 1o Pj 9 V (l) o
k kG(jl+1 j j Æer j!l j k=0 e Æ1 r1 j!l j

A STABILITY RESULT FOR INFINITE SYSTEMS 13 ) k 3 6 6 1 Pj 1 1 kG(jl+1 j ;j  Æer (V (l) )2 j!lo j k=0 ( 2 )k  Æer (V (l) )2 j!lo j  18 V (l) (using the se ond of (4.4)) ) k j 3 9 1 kRj(l+1 j ;j  Æer (V (l) )2 j!lo j (V (l) ej!lo j )j k=1 (Æk rk ) 1 : j!lo j  j!lo j : It is equivalent to K  K and then (n Now we prove that 2a K 1 2a K 1) ln( 9 Ve
l (
n 1) j!loj)  ln( e
9
Vjl!loj ): Let's all B = 9 e
V l
(nj!l1)oj ; 18 < B  36 1

( )

2

( )

( )

2

and the inequality be omesq(n 2) ln B  2 ln(n 1) whi h q is true if n  2 j ! j The last proof is 2a Kq 6Va whi h is equivalent to r1  23 Va : The following

hain r1 < r < 3 < 31 6Va Corollary 4.3 Let's onsider the anoni al transformation C~(n) : D (Ao )
 ! (Tl is a D (Ao ) 
 of the Theorem 4.2. We have k C~(n)k ;  e o l

(l)

(l)

1

(l)

2Tl

(n

n n

onstant depending only on l0 ; ;  ).

We make use of: 1) k(0)k ;  2eÆ jV! j and

n

n

0 1)2

0

(l)

Proof

1

o l

1

3 (V lo ) 9 Vol
k k (Æh rh ) 1 = 2) k=11 eÆk2j !loj kRk(l)kk ;k  Pkn=11 eÆk2j !loj Æer h=1 j!l j ej!l j l Pn 1 Pn 1 6 (V l )  9 V l 9n k+1
 ej!lo j ) V l (
 )

V = k=1 e Ærj!loj ej!lo j

 3n 9 V l  9neÆrj!loj k=1 2 k = (
9neÆrj!lo j  ) V l = V l  (
9eÆr j!lo j 3eÆj!lo j (0) k C k ;  1 + (n 11) (1 + 3 e

 + e

 1 ) (see after Theorem 4.1). k C (j) kj ;j  1 + (n 1)1 2j (1 + 1 rÆjj +  rÆjj ) = 1 + (n 1)1 2j (1 + 1

 + 

 ) Tl k C~(n)kn ;n  jn=01 k C (j) kj ;j  e n ; Tl = (1 + 3e 1

 +  e

 ): Now we must pass from the hamiltonian with l d.o.f. to the hamiltonian with l + 1 d.o.f.; then to l + 2 d.o.f. and so on. Let be lk = l + k: Change variables (A;^ '^; Al ; 'l ) ! (A(n); '(n) ; Al ; 'l ); (A;^ '^) = C (0) Æ C~(n)(A(n); '(n) ); (A(n) ; '(n)) 2 Dn (Ao ) 
n are those of the hamiltonian Hn(l) in Theorem 4.2. Al 2 R; 'l 2
n ; The hamiltonian H~ 0(l )(A; ') =: Pli=1 h(Ai ) + V (l )(^') + (V (l )(') V (l )(^')); bePn

2

( ) 2

2

( )

2

2

+1

2

2

+1

1

(

+1

i=1

0

1

1

omes

0 1)2

1

1

l0 X

( )

( )

1

+1

( )

( )

2

2

( )

1

( )

( ) 2

1

1

1

0

1

0

'(n) ) + G(nl0 ) (A(n) ; '^(n) ) + (Rn(l0 ) (A(n) ; '(n) ) h(A(in) ) + h(Al1 ) + V (l 1) (^

(4:7)

'(A(n) ; '(n) )) '(A(n) ; '(n) ); 'l1 ) V (l0 ) (^ < Rn(l0 ) (A(n) ; '^(n) ) >l0 ) + (V (l1 ) (^ Let's re all (A(n) ; Al1 ) =: (A1; : : : ; Al1 ); ('(n); 'l1 ) =: ('1 ; : : : ; 'l1 ); we rewrite (4.7)

as

H0(l1 ) (A; ') =

l1 X

^ ') + (Rn(l0 )(A;^ '^) h(Ai ) + V (l0 1) ( ') + G(nl0 ) (A;

i=1 ^ ') >l < Rn(l ) (A; 0

0

) + (V~ (l )(A;^ ') V~ (l )(A;^ '^)) 1

(4:8)

0

The following theorem is analogous to Theorem 4.1 but with one more degree of freedom.

PERFETTI PAOLO

14

Theorem 4.4 Let's onsider the hamiltonian (4.8) with (A; ') 2 Dn (Ao ) 
n . !o ; !lo1 ) satis es j!o
 j 1  j! lo j Let Ao 2 Rl1 be a point su h that the ve tor hAo = (^ 1 rÆrej!lo1 j 1 0 0 for j j  K ; l1 6= 0; K  2Æ ln 2 V (l1 ) : If (l1 ) (4:9) j!lo1 j  4 (ar + V r ) eÆ then via a suitable anoni al transformation (A; ') = C (0) (A0 ; '0 ); (A0 ; '0) 2 D1 (Ao) 
1 ;  = n;  = n ; 1 =  3r; 1 =  3Æ l1

X '0 ) + G1(l ) (A0 ; '^0 )+ H1(l ) Æ C (0) (A0 ; '0 ) = H1(l ) (A0 ; '0 ) = h(A0i ) + V (l 1) ( i=1 + R1(l )(A0 ; '0) < R1(l )(A0 ; '^0) >l (4:10) ( l ) ( l ) ( l ) ( l ) 0 0 0 0 0 0 0 0 ^ ^ ^ G1 (A ; '^ ) = Gn (A ; ' ) + Rn (A ; '^ ) < Rn (A ; ' ) >l + + < R1(l )(A0 ; '^0 ) >l P (l 1) (  V (l 1)('0 ); f2 = (G(nl )(A^0 + R1(l ) = 10 '0 +
) i=1 fi and f1 = (V  G(nl )(A^0 + ^ ; '0)) ^ ; '0 +
) ^ Rn(l )(A^0 + f3 = (Gn(l ) (A^0 + ^ ; '0 ) Gn(l ) (A^0 ; '0 )) f4 = (Rn(l ) (A^0 + ^ ; '^0 +
) ^ ; '^0 ))  >l < Rn(l )(A^0 + ^ ; '0 ) >l ) f5 = (< Rn(l ) (A^0 + ^ ; '0 +
) f6 = (Rn(l ) (A^0 + ^ ; '^0 ) Rn(l ) (A^0 ; '^0 )) f7 = (< Rn(l ) (A^0 + ^ ; '0 ) >l < Rn(l ) (A^0 ; '0 ) >l ) P i
('0 +
) V~(l ) (A^0 +  ^) f8 =  2Nl 0 e l 6=0 j j>K P i
('0 +
) (V~(l ) (A^0 +  ^ ) V~(l )(A^0 )) f9 =  2Nl 0 e l 6=0 j jK P f10 = li=1 (h(A0i + i ) h(A0i ) hA0i i ) kR1(l ) k ; = eÆr6j !o j (V (l ) )2 0

1

1

1

1

1

0

1

0

0

0

1

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

1

1

1

1

1

1 1

1

1

1

1

l1

The variable '0l is present in Rn(l )(A^0 ; '0) and < R1(l )(A0 ;1'^0 ) >l : But p the rst is exponentially small in j!loj while the se ond is O(j!lo j ): This for es us to take j!lo j exponentially small respe t to j!loj in order to get 'l (t)  'ol + !lot for a time exponentially{long. Proof The proof is similar to that of Theorem 4.1 so we omit it. We rewrite the hamiltonian (4.10) as l+1 X '0 ) + Gn(l ) (A^0 ; '0 ) + Rn(l ) (A^0 ; '^0 ) H1(l ) (A0 ; '0 ) = h(A0i ) + V (l 1) ( Remarks

0

1

1

1

1

1

0

0

i=1
( l ) 0 0 ^ < Rn (A ; ' ) >l + < R1(l ) (A0 ; '^0 ) >l + R1(l ) (A0 ; '0 ) < R1(l ) (A0 ; '^0 ) >l l+1 X '0 ) + Gn(l ) (A^0 ; '0 ) + G1(l ) (A0 ; '^0 ) + R^1(l ) (A0 ; '0 ) H1(l ) (A0 ; '0 ) = h(A0i ) + V (l 1) ( i=1 (4:11) 0

1

1

1

1

0

1

1

1

1

A STABILITY RESULT FOR INFINITE SYSTEMS

15

For a generi fun tion G(A; ') we set G^(A; ') =: G(A; ') < G(A; ') >l ' =: ('1 ; : : : ; 'm ) m integer. 6 (V (l) )2 1 6 (V )  3 V (l ) using (4.9) kGn(l ) k ;  Æer kR1(l ) k ;  eÆr j! j 2 j! j ( l) ( l ) ( l ) kG1 k ;  2kRn qk ; + kR1 k ;   : P (l ) + 6 (V ) 2 (V ) o e

 + 6 (V ) = eÆr j! j exp ln 2 j!l j (162) V eÆr j! j eÆr j! j P (l )  0: m

(l1 ) 2

1

1

1

1 (l) 2

1

1

0

0

1

o l1

n n

o l

o l

n n

1

1

1

(l1 ) 2

(l1 ) 2

1

o l1

(l)

o l1

0

9V (l1 ) 1 Theorem 4.5 Let's onsider the hamiltonian (4.11). If e6j ar !lo jÆ 1  1 and eÆ 1 r1 j!lo1 j  1; there exists a anoni al transformation (A0 ; '0) = (A00 + 0; '00 +
0) =: C (1)(A00; '00) (A00; '00 ) 2 D2 (Ao ) 
2 su h that l+1 X '00 ) + G(nl0 ) (A^00 ; '00 )+ H1(l1 ) Æ C (1) (A00 ; '00 ) =: H2(l1 ) (A00 ; '00 ) = h(A00i ) + V (l 1) ( i=1 ( l ( l ) 1) 1 00 00 00 00 + G2 (A ; '^ ) + R^2 (A ; ' ) P where G2(l1 ) = G(1l1 ) + < R^ 2(l1 ) >l1 ; R2(l1 ) (A00 ; '00 ) = 12 i=1 fi and ( l 1) 00 0 ( l 1) 00 f1 = V (' +
 ) V (' ); ( l0 ) ^00 ^ 0 00  0 f2 = Gn (A +  ; ' +
) Gn(l0 ) (A^00 + ^ 0 ; '00 ) f3 = Gn(l0 ) (A^00 + ^ 0 ; '00 ) Gn(l0 ) (A^00 ; '00 ) ^ 0) Rn(l0)(A^00 + ^ 0; '^00) f4 = Rn(l0 ) (A^00 + ^ 0 ; '^00 +
f5 = Rn(l0 ) (A^00 + ^ 0 ; '^00 ) Rn(l0 ) (A^00 ; '^00 )  0 ) Rn(l0)(A^00 + ^ 0 ; '00 ) >l f6 =< Rn(l0 ) (A^00 + ^ 0 ; '00 +
f7 =< Rn(l0 ) (A^00 + ^ 0 ; '00 ) Rn(l0 ) (A^00 ; '00 ) >l ^ 0 ) >l1 < R1(l1)(A00 + 0 ; '^00 ) >l1 f8 =< R1(l1 ) (A00 + 0 ; '^00 +
f9 =< R1(l1 ) (A00 + 0 ; '^00 ) >l1 < R1(l1 ) (A00 ; '^00 ) >l1 P i
('00 +
0 ) R(l1 ) (A00 + 0 ) f10 = 1  2Nl1 0 e l1 6=0 ;j j>K1 P i
('00 +
0 ) (R(l1 ) (A00 + 0 ) R(l1 ) (A00 )) f11 = 1; 1;  2Nl1 0 e l1 6=0 ;j jK1 P f12 = li1=1 (h(A00i + 0i ) h(A00i ) hA00i 0i ) Proof It is the same as that one of Theorem 4.2 Theorem 4.6 Let's onsider the hamiltonian H~ 0(l1 ) ; together with (4.9).If 1  81 V (l1) (nl1 1)2 < 1 a
 j!o j 1  1 4 e

 j!lo1 j 2
 l1

there exists the anoni al transformation

(A; ') = Ce(n )(A(n (H~ 0(l ) Æ Ce(n ))(A(n l1

1

l1

) ; '(nl ) ); (A(nl ) ; '(nl ) ) 2 D (Ao ) 
su h that  nl nl : ( l ) ) ( n ) ( n ) ( n ) l ;' l ) = H nl (A l ; ' l ) =

l1

1

1

1

1

1

1

1

1

1

1

1

PERFETTI PAOLO

16

(n ) (l 1) ( = Plj+1 '(n ) )+ =1 h(Aj ) + V +Gn(l )(A^(n ); '(n )) + G(nl )(A(n ); '^(n ) ) + Rn(l )(A(n kGn(l ) k ;  P (l ) + 18eÆr (Vj! j) l1

0

l1

l1

l0

1

l1

nl

kRn(ll ) knl 1 1

1

;nl1 1

kRn(ll ) knl 1 1

Proof

l1

l1

;nl1 1

1

l1

l1

l1

) ; '(nl ) ) 1

(l1 ) 2

1

nl

1

1

l1

o l1

6V l ( 9V lo )nl nl 1 (Æ h rh ) 1  eÆr h=1 j!lo j ej!l j ( 1)

( 1)

1

1

1

1

q

6 (V lo ) exp ln2 j!o j e

l  eÆr l (216) V j!l j 

( 1) 2



( 1)

1

1

Apply enough times the Theorem 4.5

Corollary 4.7 Let's onsider the transformation C~(nl1 ) : Dnl (Ao ) 
n 1

l1

2Tl

!

(Tl1 = D (Ao ) 
 of Theorem 4.6. Then we have k C~(nl1 ) kn ;n  e (nl1 l1 l1 (1 + 6e 1

 +  e

 ): Remarks We impose nl1  2 and this explains the 6 in pla e of 3 (see Corollary 1 1)2

4.3).

Same as in Corollary 4.3. It hanges slightly only Tl respe t to Tl Let's de ne some quantities we are going to use. 1) lk =: l + k; l0 = l; 2) i(l ) 0  i  nl ; 3) i(l ) 0  i  nl ; 4) H~ 0(l ) (A; ') =: Pl (l ) ('); ' 2 Tl ; 5) nl 2 N nl =: n (the n of Theorem 4.2), 6) i=1 h(Ai ) + V Ki(l ) 0  i  nl (K0(l ) =: K of Theorem 4.1, K1(l ) =: K1 of Theorem 4.2), 7) C~(n ) =: Cl(0) Æ Cl(1) Æ Cl(2) : : : Æ Cl(n 1) (Cl(0) is the transformation of Theorem 4.1, Cl(1) is one of the transformations of Theorem 4.2). Tl  Tl of Corollary 4.7. Let's de ne for k  0; P (l ) n(l ) = 1 1 1 (l ) (l ) S= 1 k=0 (k+1) ln (k+2) ; n (k+1) ln (k+2) 2 S = 2
 ;  1(l ) = 41  S (k+1) ln1 (k+2) =
 (l ) ; Æj(l ) = 3(
n 1) ; 1  j  nl 1 ; j(l )  1  (l ) Æ(l ) = 1
 (l ) ; 0 2 3 ( l ) ( l ) n n = (k+1) ln1 (k+2) 21  S = 2
(l ); (l ) (1l ) = 41  S (k+1) ln1 (k+2) =
(l ); rj(l ) = 3(
n 1) ; 1  j  nl 1; j(l )  12 (l ) ; 0(l ) = 31
(l ); Proof

k

k

k

k

lk

k

k

k

k

k

k

0

0

k

k

lk

0

k

0

(l0 )

k

0

k

k

k

k

lk

(l0 )

k

2

(lk )

k

k

lk

0

1

1

1

(lk ) (lk )

(lk )



1

j

j

j

(l0 )

k

k

k

For example  = 0 = 0(l );  = (0l ); 1(l ) ;
 = (l ) (1l ) : Kj(l ) = Æ 1 ln eÆ 2 rj! Vj k

(lk )

k

k+1 lk+1

k

k

2

k

2

1

(l0 )

k+1 lk+1

k lk

2

k

k

k

0

k

k lk

0

1

2

k

0

= 0 = 0(l ); 0

k

= 0(l );
 = (l )



1

1

(lk )

o lk

For ea h k  0 we have the relations 14  e81
 V
(n j!1) j < 12 ; a

 j!lo j 1  1; j!lo j  eÆ4 (ar0(l ) + Vr ); k C~(n ) k ;  e ; Tl = Tl for any k  1: (lk )

ll

(lk )

k

(lk ) 0

k

(lk )

(lk ) 0

lk

nl k

(lk )

2

o lk

(lk )

(lk )

2Tl

(nl

nl

k

k

k

k

1)2

k

1

A STABILITY RESULT FOR INFINITE SYSTEMS

17

It follows that there exists a universal onstant B0 su h that (being nl  2) V (l ) (k + 1)3 ln6 (k + 2) + B a 0(l ) 1 < 1 B0 0 k

0

k

0(l0 ) 0(l0 )

and

0(l0 ) j!lok j

j!lok j

"

#

(B1j!lo j0(l )0(l ))1=2  [B (k + 1)1=2 ln(k + 2)℄ nl  1 + 2 (k + 1) ln2(k + 2) (2.3) sets a strong restri tion on how small the frequen ies !lo ould be. 0

0

(4:12)

k

k

k

Hn(llkk) (A(nlk ) ; '(nlk ) ) =

+

k X j =0

l+k X j =1

h(Aj(nlk ) ) + V (l 1) ('1(nlk ) ; : : : ; 'l(nl1k ) )+

Gn(lljj) (A1(nlk ) ; : : : ; Al(jnlk ) ; '1(nlk ) ; : : : ; 'l(jnlk1) ) + Rn(llkk) (A(nlk ) ; '(nlk ) )

(4:13)

l kGn(lljj) knlj ;nlj  P (lj ) + eÆ l18j r lj (Vj!lojj j) (

(

)

lj

(

)

( 0

lj

lj P (lj ) = 2l (V l )

) ( 0

) 2

)

r

 
 j !lo j e
(162) exp ln 2

V eÆ r j! j q  ( l ) ( V ) 
 6

kRn k ;  eÆ r j! j exp ln2 j!lo j e
(216)

V In the variables (A(l ); '(l )) the system is (

+1

k lk

( 0

nl k

) 2

j) ( j) 0

(lk ) (lk ) 0 0

nl

k

(lj )

(lj )

(lj )

j

o lj

k

k

(lk ) 2

(lk )

o lk

k

(lk )



(lk )

k d (lk )  X  Gn(lljj) + (lk ) Rn(llkk) 'i = hA(lk ) + (lk ) i dt Ai j=0 Ai

(4:14)

k X  d (lk ) ( l 1) = (lk) (V + Gn(lljj) + Rn(llkk)) A dt i 'i j =0

(4:15)

k X d (lk )  Ai = ( Gn(lljj) + Rn(llkk) ) ( lk ) dt 'i j=m+1

(4:16)

If i  l 1 we have

if i > l 1; i = l 1 + m; m  k we have

Let's onsider the sequen e of transformations fCe(n )g; k = 0; 1; : : : whose domain: is DN (Ao) 
N  C N  C N : De ne RN : DN (Ao ) 
N ! C l  C l ; RN = Ce(n ) Æ : : : Æ Ce(n ) Æ Ce(n ) ; RN = (R(NA) ; R(N') ); R(NA) = fR(NA) gi ; R(N') = fR(N') gi i = 1; : : : ; lN : Note that the fun tions (R(N') )i and (R(NA) )i are analyti on the larger domain D (Ao(l )) 
 and ontinuous on D (Ao(l )) 
 ; Ao(l ) = (Ao1 ; Ao2; : : : ; Aol ): lk







2

2

2

lN 1

l0

lN

N

N



N

2

lN

N

lN

lN

N

lN

N

Theorem 4.8 For ea h i 2 N the following four limits are de ned uniformly in DN2 (Ao )
N for N ! +1: 1) (R(NA) )i 2) (R(N') )i 3) dtd (R(NA) )i 4) dtd (R(N') )i : 2

PERFETTI PAOLO

18

Remarks i) Observe that R1 =: limN !+1 f(R(NA) )i ; (R(N') )i gi2N =: (A; ') 2 (C N ; C N ) de nes the a tion{angle variables of the equations (1.2); (A; ') = R1 =: (P (v; u); Q(v; u)). There is no uniformity respe t to i ii) ui =: limN !1 '(ilN ) and vi =: limN !1 Ai(lN ) iii) In (4.17) be omes apparent that without the fa tor (nlk 1) 2 the limits do not exist and the presen e of the fa tor is due to the dierent hoi e of the rst analyti ity loss: Æi(lk ) and ri(lk ) i  1; mu h smaller respe tively than Æ0(lk ) and r0(lk ) (see after Corollary 4.7)

We show that (R(NA))i and (R(N'))i are Cau hy sequen es de ned in DN (Ao )
N and then de ne (Pi; Qi ) being in the spa e A whi h is omplete

Proof

 2



2

) )i (R(A) )i k    k(R(A) ) (R(A) ) k k(R(NA+1 N N +1 i N i lN ;lN  ;  Nk=0 k Ce(nlk ) klk ;lk j(Ce(nlN ) )(iA) A(inlN ) j  2Tlk g)(expf 2TlN g 1)  B3 (expf 2TlN g 1)  (1 k=0 expf (n (nlN 1)2 (nlN 1)2 lk 1)2 +1

2 2

+1

+1

+1

+1

(4:17) ) )i (R(') )i k : The se ond  is due to Theorem and the same o

urs for k(R(N'+1 N ; 4.2. (4.12) implies that fR(NA)gi and fR(N')gi are Cau hy sequen es in the spa e A whi h is omplete (see se tion 2) Moreover we have dtd (R(N') )i = hA ((R(NA))i ) and by Lagrange theorem, using that jhA A j  a; we an do the limit N ! +1 at left. The uniform onvergen e respe t to time allows us to inter hange the limits N ! +1 with the derivative so that limN !+1 dtd (R(N'))i = dtd limN !+1 (R(N'))i = dtd Qi = hA (Pi): (l ) (') (l ) (') (l ) (') d (A) dt (RN )i = V' (RN ) = fi (RN ) = (fi (RN ) fi (Q)) + fi (Q) = = [fi(l )(R(N')) fi(l )(Q(l ))℄ + [fi(l )(Q(l )) fi(Q)℄ + fi (Q) (Q(l ) = (Q1 ; Q2; Q3; : : : ; Ql )). The rst dieren e goes to zero be ause of the regularity properties of the fun tion fi(l ) and RN(') Q(l ) N !+1!0: The se ond dieren e goes to zero by the fa t that the fun tions fijI j: Tw ! R onverge uniformly to fi for jI j ! N (see Se tion 2, Averages). A tually fijI j would be de ned over TjI j but it does not matter be ause fijI j does not depend on the variables 'i for i 62 I: It follows that limN !+1 dtd (R(NA))i = fi(Q) and the same inter hange as before of the limit N ! +1d with the derivative an be performed here. Then we get what expe ted namely dt Pi = fi(Q) Proof of Theorem 3.1 Let's onsider the hamiltonian (4.13). kGn(l ) k ;  P (l ) + 18 (Vj! j) eÆ r +1

+1

  2 2

i

i i

i

N (lN ) i N

N

N

N

N

N

N

N

N

N

N

j lj

j

P (lj ) =

(lj ) 2

o lj

(lj ) (lj ) 0 0

2 (V llj ) (

(lj 0

eÆ

1) ( 0

r

1) 2

j 1)

j

!loj 1

r


 exp ln 2 j!lo j e
(162)

V j 

(lj

j 1

1)

(lj

(lj

1)

(lj ) nl j

(lj ) nl j



1)

Remember that P (l ) is the estimate of a term ontaining the angle{variables ( n ) ( n ) ('1 ; : : : ; 'l ) (see after Corollary 4.7) j

lk

lk

j 1

A STABILITY RESULT FOR INFINITE SYSTEMS

19

q (lk ) (lk ) ln2 ; S = P1 1 B3 = supk0 (lÆk0+1 ) r0(lk+1 ) > 1; B4 = 2e 36 k=0 (k+1)ln2 (k+2) ; S

Æ0 r0 q (lj ) 2  ( V ) (l0 ) (l0 ) j! o j 18 (V (loj ) )2 B4 o j!lj j  B3 (V (lj 1 ) )2 j!loj 1 j 2 (lj ) (lj ) j!lj j j ln (j +1) V (l0 ) lj 1 eÆ0 r0 ( l ) j P P1 P1 (lj ) (lj ) j =0 kGnlj kn(lj ) ;n(lj ) j =0 P lj lj ( lk ) (nlk ) (nlk ) Rn A ; ' k

exp then  and by virtue of (3.1) (in parti ular the power 6 of the logarithm) the series is onvergent. It follows that the series is onvergent too while ( ) goes to zero when goes to in nity. Then the equations (4.14){(4.16) admit the limit for k ! +1 and (3.3){(3.6) follow Proof of Corollary 3.2 Iterating theqpro edure of Corollary 3.1, (3.2) is implied by  (l ) (l ) j!lo j  (B3 )k ((VV )) j!loj exp lnkB(2)  V j! j ; P1 j =2 kGn k ;  kGn k ; provided that q   j! j B  ln(1 + B5 ) where B5 = supj2 (lnj ln2((jB+1)) ln (2) ) : By the same onV P ) ( l ( l ) 1 dition on j!loj we get j=j +1 kGn k ;  kGn k ; being j +1)) supjj +1 (j ln(j ln(j (+1))( B )  B5 : In parti ular P1 k j=p+2 Gn(l ) k ;  2kG(nl ) k ;  2kG(nl ) k ;  If

lk

(lk ) 2 (l0 ) 2

k

2

(l0 )

(l0 )

2

0

j lj

1

 

l1

2 2

 

2 2

2

j lj

0

0

j lj

2

2

0

o l

(l0 ) (l0 )

4

o l

(l0 ) (l0 )

4

2

3

j0 lj 0

  2 2

3

j

  2 2

j

p+1 lp+1

  2 2

r

p+1 lp+1

  2 2

(lp+1 ) (lp+1 ) nl nl p+1 p+1

 lp
 lp and then  eÆ 2l p (rV llpp j)!o j exp ln 2 j!po j e
(162)

V lp (

( 0

) ( 0



) 2

)

(

)

(

)

(

lp



)

jvlp (t) vlp (0)j  jtje(n(llpp ) 2 ) 1 kG(nllpp ) knlp ;nlp  lp lp lp ) ( lp ) ( lp ) 1 (lp ) ( V  jtje(nlp nlp ) 2P  jtj  l ( l ) j!lop j 1152S 2(p + 2)3 ln6 (p + 3)
+1

+1

+1

+1

+2

+1

+2




exp (p+1)ln (p+2) B4

2

(

+1

r

(

+1

( 0)

+1 ) +1

(

+1 ) +1

) 2

( 0) 2

j!lop j(l0 ) (l0 ) V (lp )

being vl (t) vl (0) = R0t u P1j=p+1 Gn(l )
'l ul = Ql (v; u) ul = (Ce(l ) Æ Ce(l ) Æ Ce(l ) Æ : : : )(v; u) (l ') ul
Al vl = Pl (v; u) vl = (Ce(l ) Æ Ce(l ) Æ Ce(l ) Æ : : : )(v; u) (l A) vl For what written after Corollary 4.7,
we have 2T k (Ce(l ) Æ Ce(l ) Æ Ce(l ) Æ : : : )(v; u) (l ') ul k ;  1 k=p expf (n 1) g 1 = expfP1k=p (n2T 1) g 1: By qQj! j k p V 2 j!l j  B3 V ) expf(k p)B4 (p+1)ln (p+2) gj!lo j; qQj! j Q =  V  ; Tl  T; B3 expfB4 (p+1)ln (p+2) g =: B6  2; the nfollowing holds o n +2 (2p+4) ln (2p+4) B t + exp P1k=p (n2T 1)  exp 2T 324e (Vj! )j  (4S) Ppt=0 6 V
o P1 (2 t ) ln (2 t ) t + t=p+2 V B6 The sums are bounded by V 2 (2p + 4)2 ln4(2p + 4) + VB ; P 2 4 t B7 = 1 t=1 (2t) ln (2t)(B6 ) p

p

p

p

p

p

p

p+2

lk

p+1

p

p

p

lk

p

p

p

p+1

j lj

lp

p+1

p

2

(lp )

2

lk

2 2

p

2

k

lk

 

o lp

(l0 ) (l0 )

lk

(p) 2

o lp

2

4

o lp

2

2

(l0 ) (l0 )

4

(lp )

(lp )

(lp )

p

p

2

(lp )

p

p

p+2

p

(lk )

k

p+2

7 (lp )

lk

2

PERFETTI PAOLO expfP1k=p (nl2kTlk1) g 1  expfT 648e (4j!Slop)j  lV lp l (B7 +2(2p +4)2 ln4(2p +4))g l T B8(2p + 4)2 ln4 (2p + 4)) o V l p l 20

2

2

(

)

( 0) ( 0)

(

1

)

j!lp j( 0 ) ( 0 ) Then 'lp ulp goes to zero when j!lop j goes to +1 and the behavior depends on the hoi e of fj!lok jg with k  p: The same happens for Alp vlp : As a onsequen e, being Alp (t) Alp (0) = (Alp (t) vlp (t)) + (vlp (t) vlp (0)) + (Alp (0) vlp (0)); if

(l0 )  1 B4 jtj  B9 V (l0 ) exp 2 (p + 2) ln (p + 3) (p + 1) ln2 (p + 2)

s

j!lop j(l )  (l ) V (lp) 0

0

we have jAl (t) Al (0)j  3T B8(2p +4)2 ln4(2p +4)) j!V j : For the variables Ai (t); i = 1; : : : ; l 1 it is valid what written in the remarks of Theorem 4.1 namely jAi (t) Ai (0)j  C: The onstant C is the greatest of the Bi0 s i = 0; : : : ; 8: p

(lp )

o lp

p

(l0 )

Referen es

[1℄ Bambusi D.: Exponential stability of breathers in Hamiltonian networks of weakly

oupled os illators, Nonlinearity. 9 (1996), 433-457 [2℄ Benettin G., Froli h J., Giorgilli A.: A Nekhoroshev{Type Theorem for Hamiltonian Systems with In nitely Many Degrees of Freedom, Comm. Math. Phys. 119 (1988), 95-108 [3℄ Bambusi D., Giorgilli A.: Exponential stability of states lose to resonan e in in nite dimensional hamiltonian systems, Journal of Stat. Phys. 71 (1993), 569-606 [4℄ Chier hia L., Perfetti P.: Maximal Almost-Periodi Solutions for Lagrangian Equations on In nite-Dimensional Tori in \Seminar on dynami al systems" (S.Petersburg 1991), Progr. Nonlinear Dierential Equations Appl. 12 Birkhauser, 1994 p.203{212 [5℄ Chier hia L., Perfetti P.: Se ond Order Hamiltonian Equations on T1 and Almost-Periodi Solutions, J. Dierential Equations Vol.116 No.1 1995, 172-201 [6℄ Gallavotti, G. The Elements of Me hani s Springer{Verlag [7℄ Giogilli A., Lo atelli U. On lassi al series expansions for quasi{periodi motions, Mathemati al Physi s Ele troni Journal, Vol.3 1997 pp.1{25 [8℄ Jessen B.: The Theory of Integration in a spa e of an In nite Number of Dimensions, A ta mathemati a 63 1934, 250{323 [9℄ Kelley J.: General Topology, Van Nostrand{Reinhold, Prin eton, New Jersey, 1955 [10℄ Neishtadt A.I.: The separation of motions in systems with rapidly rotating phase Jour. Appl. Math. Me h. 48 No.2 1984, 133-139 [11℄ Nekhoroshev N.N., An exponential estimate of the time of stability of nearly integrable Hamiltonian systems, Russ. Math. Surv. 32:6, 1{65, 1977 [12℄ Perfetti P.: Hamiltonian Equations on T1 and almost{periodi solutions, pro eedings of the \International Conferen e on Dynami al Systems and Dierential Equations" May 18{21 2000 Kennesaw State University (GA) USA, An Added Volume To Dis rete Contin. Dynam. Systems 2001, 303{309 [13℄ Perfetti P.: A KAM theorem for in nite{dimensional dis rete systems, Math. Phys. Ele tron J. volume 9, 2003 1{15

A STABILITY RESULT FOR INFINITE SYSTEMS

21

[14℄ Pos hel, J.: Small Divisors with Spatial Stru ture in In nite Dimensional Hamiltonian Systems, Commun. Math. Phys. 127 (1990) 351-393 E-mail address : perfettimat.uniroma2.it Re eived XX; revised XX.