Response solutions for arbitrary quasi-periodic perturbations with Bryuno frequency vector
aa r X i v : . [ m a t h . D S ] N ov Response solutions for arbitrary quasi-periodicperturbations with Bryuno frequency vector
Livia Corsi and Guido Gentile
Dipartimento di Matematica, Universit`a di Roma Tre, Roma, I-00146, ItalyE-mail: [email protected], [email protected]
Abstract
We study the problem of existence of response solutions for a real-analytic one-dimensional system, consisting of a rotator subjected to a small quasi-periodic forcing. Weprove that at least one response solution always exists, without any assumption on the forc-ing besides smallness and analyticity. This strengthens the results available in the literature,where generic non-degeneracy conditions are assumed. The proof is based on a diagrammaticformalism and relies on renormalisation group techniques, which exploit the formal analogywith problems of quantum field theory; a crucial role is played by remarkable identitiesbetween classes of diagrams.
Consider the one-dimensional system¨ β = − εF ( ω t, β ) , F ( ω t, β ) := ∂ β f ( ω t, β ) , (1.1)where β ∈ T = R / π Z , f : T d +1 → R is a real-analytic function, ω ∈ R d and ε is a realnumber, called the perturbation parameter ; hence the forcing function (or perturbation) F isquasi-periodic in t , with frequency vector ω .It is well known that, for d = 1 (periodic forcing) and ε small enough, there exist periodicsolutions to (1.1) with the same period as the forcing. In fact the existence of periodic solutionsto (1.1), or to the more general equation¨ β = − ∂ β V ( β ) − εF ( ω t, β ) , (1.2)with V : R → R real-analytic, can be discussed by relying on Melnikov method [5, 18]. A possibleapproach consists in splitting the equations of motion into two separate equations, the so-called range equation and bifurcation equation . Then, one can solve the first equation in terms ofa free parameter, and then fix the latter by solving the second equation (which represents animplicit function problem). This is usually done by assuming some non-degeneracy condition involving the perturbation, and this entails the analyticity of the solution. If no such conditionis assumed, a result of the same kind still holds [20, 1, 7], but the scenario appears slightlymore complicated: for instance the persisting periodic solutions are no longer analytic in theperturbation parameter.If the forcing is quasi-periodic, one can still study the problem of existence of quasi-periodicsolutions with the same frequency vector ω as the forcing, for ε small enough. The analysis1ecomes much more involved, because of the small divisor problem. However, under somegeneric non-degeneracy condition, the analysis can be carried out in a similar way and thebifurcation scenario can be described in a rather detailed way; see for instance [3]. On thecontrary, if no assumption at all is made on the perturbation, the small divisor problem andthe implicit function problem become inevitably tangled together and new difficulties arise. Inthis paper we focus on this situation, so we study (1.1) without making any assumption onthe forcing function besides analyticity. Of course, we shall make some assumption of strongirrationality on the frequency vector ω , say we shall assume some mild Diophantine condition,such as the Bryuno condition (see below).Note that (1.1) can be seen as the Hamilton equations for the system described by theHamiltonian function H ( α , β, A , B ) = ω · A + 12 B + εf ( α , β ) , (1.3)where ω ∈ R d is fixed, ( α , β ) ∈ T d × T and ( A , B ) ∈ R d × R are conjugate variables and f isan analytic periodic function of ( α , β ). Indeed, the corresponding Hamilton equations for theangle variables are closed, and are given by˙ α = ω , ¨ β = − ε∂ β f ( α , β ) , (1.4)that we can rewrite as (1.1). Therefore the problem of existence of response solutions , i.e. quasi-periodic solutions to (1.1) with frequency vector ω , can be seen as a problem of persistence oflower-dimensional (or resonant) tori, more precisely of d -dimensional tori for a system with d + 1degrees of freedom. In the case (1.3) the unperturbed (i.e. with ε = 0) Hamiltonian is isochronousin all but one angle variables. The existence of d -dimensional tori in systems with d +1 degrees offreedom, without imposing any non-degeneracy condition on the perturbation except analyticity,was first studied by Cheng [4]. He proved that, for convex unperturbed Hamiltonians, thereexists at least one d -dimensional torus continuing a d -dimensional submanifold of the d + 1unperturbed resonant torus on which the flow is quasi-periodic with frequency vector ω ∈ R d satisfying the standard Diophantine condition | ω · ν | ≥ γ | ν | − τ for all ν ∈ Z d \ { } , and forsome γ > τ > d − · denotes the standard scalar product in R d and | ν | = | ν | = | ν | + . . . + | ν d | ).We prove a result of the same kind for the equation (1.1), that is the existence of at leastone response solution for ε small enough – see Theorem 2.2 in Section 2. Even if the system(1.3) can be seen as a simplified model for the problem of lower-dimensional tori, we think thatour result can be of interest by its own. First of all, Cheng’s result does not directly apply, sinceboth the convexity property he requires is obviously not satisfied by the Hamiltonian (1.3) andwe allow a weaker Diophantine condition on the frequency vector. Moreover, just because of itssimplicity, the model is particularly suited to point out the main issues of the proof, avoiding allaspects that would add only technical intricacies without shedding further light on the problem.Finally, our method is completely different: it is based on the analysis and resummation of theperturbation series through renormalisation group techniques, and not on an iteration scheme `ala KAM. In particular a crucial role in the proof will be played by remarkable identities betweenclasses of diagrams. By exploiting the analogy of the method with the techniques of quantumfield theory, one can see the solution as the one-point Schwinger function of a suitable Euclideanfield theory – this has been explicitly shown in the case of KAM tori [11] –; then the identitiesbetween diagrams can be imagined as due to a suitable Ward identity that follows from thesymmetries of the field theory – again, this has been checked for the KAM theorem [2], and weleave it as a conjecture in our case. 2
Results
Consider equation (1.1) and take the solution for the unperturbed system given by β ( t ) = β . Wewant to study whether for some value of β such a solution can be continued under perturbation. Hypothesis 1. ω satisfies the Bryuno condition B ( ω ) < ∞ , where B ( ω ) := ∞ X m =0 m log 1 α m ( ω ) , α m ( ω ) := inf < | ν |≤ m | ω · ν | . Write f ( α , β ) = X ν ∈ Z d f ν ( β ) e i ν · α , F ( α , β ) = X ν ∈ Z d F ν ( β ) e i ν · α . (2.1) Hypothesis 2. β ∗ is a zero of order n for F ( β ) with n odd. Assume also ε∂ n β F ( β ∗ ) < forfixed ε = 0 . Eventually we shall want to get rid of Hypothesis 2: however, we shall first assume it tosimplify the analysis, and at the end we shall show how to remove it.We look for a solution to (1.1) of the form β ( t ) = β + b ( t ), with b ( t ) = X ν ∈ Z d ∗ e i ν · ω t b ν (2.2)where Z d ∗ = Z d \ { } . In Fourier space (1.1) becomes( ω · ν ) b ν = ε [ F ( ω t, β )] ν , ν = , (2.3a)[ F ( ω t, β )] = 0 , (2.3b)where [ F ( ψ , β )] ν = X r ≥ X ν + ... + ν r = νν ∈ Z d ν i ∈ Z d ∗ , i =1 ,...,r r ! ∂ rβ F ν ( β ) r Y i =1 b ν i . Our first result will be the following.
Theorem 2.1.
Consider the equation (1.1) and assume Hypotheses 1 and 2. If ε is smallenough, there exists at least one quasi-periodic solution β ( t ) to (1.1) with frequency vector ω ,such that β ( t ) → β ∗ as ε → . The proof will be carried out through Sections 3 to 5. First, after introducing the basicnotations in Section 3, we shall show in Section 4 that, under the assumption that furtherconditions are satisfied, for ε small enough and arbitrary β there exists a solution β ( t ) = β + b ( t ; ε, β ) , (2.4)to (2.3a), depending on ε, β , with b ( t ) = b ( t ; ε, β ) a zero-average function. For such a solutiondefine G ( ε, β ) := [ F ( ω t, β ( t ))] , (2.5)3nd consider the implicit function equation G ( ε, β ) = 0 . (2.6)Then we shall prove in Section 5 that one can fix β = β ( ε ) in such a way that (2.6) holds andthe conditions mentioned above are also satisfied. Hence for that β ( ε ) the function (2.4) is asolution of the whole system (2.3).Next, we shall see how to remove Hypothesis 2 in order to prove the existence of a responsesolution without any assumption on the forcing function, so as to obtain the following result,which is the main result of the paper. Theorem 2.2.
Consider the equation (1.1) and assume Hypothesis 1. There exists ε > suchthat for all ε with | ε | < ε there is at least one quasi-periodic solution to (1.1) with frequencyvector ω . Note that if F ( β ) does not identically vanish, then Theorem 2.2 follows immediately fromTheorem 2.1. Indeed, the function f ( β ) is analytic and periodic, hence, if it is not identicallyconstant, it has at least one maximum point β ′ and one minimum point β ′′ , where ∂ n ′ +1 β f ( β ′ ) < ∂ n ′′ +1 β f ( β ′′ ) >
0, for some n ′ and n ′′ both odd. Let ε be fixed small enough, say | ε | < ε for a suitable ε : choose β ∗ = β ′ if ε > β ∗ = β ′′ if ε <
0. Then Hypothesis 2 issatisfied, and we can apply Theorem 2.1 to deduce the existence of a quasi-periodic solutionwith frequency vector ω . However, the function f ( β ) can be identically constant, and hence F ( β ) can vanish identically, so that some further work will be needed to prove Theorem 2.2:this will be performed in Section 6. We want to study whether it is possible to express the function b ( t ; ε, β ) appearing in (2.4) asa convergent series. Let us start by writing formally b ( t ; ε, β ) = X k ≥ ε k b ( k ) ( t ; β ) = X k ≥ ε k X ν ∈ Z d ∗ e i ν · ω t b ( k ) ν ( β ) . (3.1)If we define recursively for k ≥ b ( k ) ν ( β ) = 1( ω · ν ) [ F ( ω t, β )] ( k − ν , (3.2)where [ F ( ω t, β )] (0) ν = F ν ( β ) and, for k ≥ F ( ω t, β )] ( k ) ν = X s ≥ X ν + ... + ν s = νν ∈ Z d ν i ∈ Z d ∗ , i =1 ,...,s s ! ∂ sβ F ν ( β ) X k + ... + k s = k,k i ≥ s Y i =1 b ( k i ) ν i ( β ) , (3.3)the series (3.1) turns out to be a formal solution of (2.3a): the coefficients b ( k ) ν ( β ) are welldefined for all k ≥ ν ∈ Z d ∗ – by Hypothesis 1 – and solve (2.3a) order by order – as itis straightforward to check. 4rite also, again formally, G ( ε, β ) = X k ≥ ε k G ( k ) ( β ) , (3.4)with G (0) ( β ) = F ( β ) and, for k ≥ G ( k ) ( β ) = X s ≥ X ν + ... + ν s = ν ∈ Z d ν i ∈ Z d ∗ , i =1 ,...,s s ! ∂ sβ F ν ( β ) X k + ... + k s = k,k i ≥ s Y i =1 b ( k i ) ν i ( β ) . (3.5)Of course, Hypothesis 1 yields that the formal series (3.4) is well-defined too.Unfortunately the power series (3.1) and (3.4) may not be convergent (as far as we know).However we shall see how to construct two series (convergent if β is suitably chosen) whoseformal expansion coincide with (3.1) and (3.4). As we shall see, this leads to express the responsesolution as a series of contributions each of which can be graphically represented as a suitablediagram.A graph is a set of points and lines connecting them. A tree θ is a graph with no cycle, suchthat all the lines are oriented toward a unique point ( root ) which has only one incident line ℓ θ ( root line ). All the points in a tree except the root are called nodes . The orientation of thelines in a tree induces a partial ordering relation ( (cid:22) ) between the nodes and the lines: we canimagine that each line carries an arrow pointing toward the root. Given two nodes v and w , weshall write w ≺ v every time v is along the path (of lines) which connects w to the root.We denote by N ( θ ) and L ( θ ) the sets of nodes and lines in θ respectively. Since a line ℓ ∈ L ( θ ) is uniquely identified with the node v which it leaves, we may write ℓ = ℓ v . We write ℓ w ≺ ℓ v if w ≺ v , and w ≺ ℓ = ℓ v if w (cid:22) v ; if ℓ and ℓ ′ are two comparable lines, i.e. ℓ ′ ≺ ℓ , wedenote by P ( ℓ, ℓ ′ ) the (unique) path of lines connecting ℓ ′ to ℓ , with ℓ and ℓ ′ not included (inparticular P ( ℓ, ℓ ′ ) = ∅ if ℓ ′ enters the node ℓ exits).With each node v ∈ N ( θ ) we associate a mode label ν v ∈ Z d and we denote by s v thenumber of lines entering v . With each line ℓ we associate a momentum ν ℓ ∈ Z d ∗ , except for theroot line which can have either zero momentum or not, i.e. ν ℓ θ ∈ Z d . Finally, we associate witheach line ℓ also a scale label such that n ℓ = − ν ℓ = , while n ℓ ∈ Z + if ν ℓ = . Note thatone can have n ℓ = − ℓ is the root line of θ .We force the following conservation law ν ℓ = X w ∈ N ( θ ) w ≺ ℓ ν w . (3.6)In the following we shall call trees tout court the trees with labels, and we shall use the term unlabelled tree for the trees without labels.We shall say that two trees are equivalent if they can be transformed into each other bycontinuously deforming the lines in such a way that these do not cross each other and also labelsmatch. This provides an equivalence relation on the set of the trees – as it is easy to check.From now on we shall call trees tout court such equivalence classes.Given a tree θ we call order of θ the number k ( θ ) = | N ( θ ) | = | L ( θ ) | (for any finite set S we denote by | S | its cardinality) and total momentum of θ the momentum associated with ℓ θ .We shall denote by Θ k, ν the set of trees with order k and total momentum ν . More generally,5f T is a subgraph of θ (i.e. a set of nodes N ( T ) ⊆ N ( θ ) connected by lines L ( T ) ⊆ L ( θ )), wecall order of T the number k ( T ) = | N ( T ) | . We say that a line enters T if it connects a node v / ∈ N ( T ) to a node w ∈ N ( T ), and we say that a line exits T if it connects a node v ∈ N ( T )to a node w / ∈ N ( T ). Of course, if a line ℓ enters or exits T , then ℓ / ∈ L ( T ) Remark 3.1.
One has X v ∈ N ( θ ) s v = k ( θ ) − cluster T on scale n is a maximal subgraph of a tree θ such that all the lines have scales n ′ ≤ n and there is at least a line with scale n . The lines entering the cluster T and the linecoming out from it (unique if existing at all) are called the external lines of T .A self-energy cluster is a cluster T such that (i) T has only one entering line ℓ ′ T and oneexiting line ℓ T , (ii) one has ν ℓ T = ν ℓ ′ T and hence X v ∈ N ( T ) ν v = . (3.7)For any self-energy cluster T , set P T = P ( ℓ T , ℓ ′ T ). More generally, if T is a subgraph of θ with only one entering line ℓ ′ and one exiting line ℓ , we can set P T = P ( ℓ, ℓ ′ ). We shall say thata self-energy cluster is on scale −
1, if N ( T ) = { v } with of course ν v = (so that P T = ∅ ).A left-fake cluster T on scale n is a connected subgraph of a tree θ with only one enteringline ℓ ′ T and one exiting line ℓ T such that (i) all the lines in T have scale ≤ n and there is in T atleast a line on scale n , (ii) ℓ ′ T is on scale n + 1 and ℓ T is on scale n , and (iii) one has ν ℓ T = ν ℓ ′ T .Analogously a right-fake cluster T on scale n is a connected subgraph of a tree θ with only oneentering line ℓ ′ T and one exiting line ℓ T such that (i) all the lines in T have scale ≤ n and thereis in T at least a line on scale n , (ii) ℓ ′ T is on scale n and ℓ T is on scale n + 1, and (iii) onehas ν ℓ T = ν ℓ ′ T . Roughly speaking, a left-fake (respectively right-fake) cluster T fails to be aself-energy cluster only because the exiting (respectively the entering) line is on scale equal tothe scale of T . Remark 3.2.
Given a self-energy cluster T , the momenta of the lines in P T depend on ν ℓ ′ T because of the conservation law (3.6). More precisely, for all ℓ ∈ P T one has ν ℓ = ν ℓ + ν ℓ ′ T with ν ℓ = X w ∈ N ( T ) w ≺ ℓ ν w , (3.8)while all the other labels in T do not depend on ν ℓ ′ T . Clearly, this holds also for left-fake andright-fake clusters.We shall say that two self-energy clusters T , T have the same structure if forcing ν ℓ ′ T = ν ℓ ′ T one has T = T . Of course this provides an equivalence relation on the set of all self-energyclusters. The same consideration apply for left-fake and right-fake clusters. From now on weshall call self-energy, left-fake and right-fake clusters tout court such equivalence classes.A renormalised tree is a tree in which no self-energy clusters appear; analogously a renor-malised subgraph is a subgraph of a tree θ which does not contains any self-energy cluster.Denote by Θ R k, ν the set of renormalised trees with order k and total momentum ν , by R n the setof renormalised self-energy clusters on scale n , and by LF n and RF n the sets of (renormalised)left-fake and right-fake clusters on scale n respectively.6or any θ ∈ Θ R k, ν we associate with each node v ∈ N ( θ ) a node factor F v ( β ) := 1 s v ! ∂ s v β F ν v ( β ) . (3.9)We associate with each line ℓ ∈ L ( θ ) with n ℓ ≥
0, a dressed propagator G n ℓ ( ω · ν ℓ ; ε, β ) (prop-agator tout court in the following) defined recursively as follows.Let us introduce the sequences { m n , p n } n ≥ , with m = 0 and, for all n ≥ m n +1 = m n + p n + 1, where p n := max { q ∈ Z + : α m n ( ω ) < α m n + q ( ω ) } . Then the subsequence { α m n ( ω ) } n ≥ of { α m ( ω ) } m ≥ is decreasing. Let χ be a C ∞ non-increasing function such that χ ( x ) = ( , | x | ≤ / , , | x | ≥ . (3.10)Set χ − ( x ) = 1 and χ n ( x ) = χ (4 x/α m n ( ω )) for n ≥
0. Set also ψ ( x ) = 1 − χ ( x ), ψ n ( x ) = ψ (4 x/α m n ( ω )), and Ψ n ( x ) = χ n − ( x ) ψ n ( x ), for n ≥
0; see Figure 1. xα α α m α m α m ( x ) Ψ ( x ) Ψ ( x ) Figure 1: Graphs of some of the C ∞ functions Ψ n ( x ) partitioning the unity in R \ { } ; here α m = α m ( ω ).The function χ ( x ) = χ (4 x/α ) is given by the sum of all functions Ψ n ( x ) for n ≥ Lemma 3.3.
For all x = 0 and for all p ≥ one has ψ p ( x ) + X n ≥ p +1 Ψ n ( x ) = 1 . Proof.
For fixed x = 0 let N = N ( x ) := min { n : χ n ( x ) = 0 } and note that max { n : ψ n ( x ) =0 } ≤ N −
1. Then if p ≤ N − ψ p ( x ) + X n ≥ p +1 Ψ n ( x ) = ψ N − ( x ) + χ N − ( x ) = 1 , while if p ≥ N one has ψ p ( x ) + X n ≥ p +1 Ψ n ( x ) = ψ p ( x ) = 1 . Remark 3.4.
Lemma 3.3 implies P n ≥ Ψ n ( x ) = 1 for all x = 0. Hence { Ψ n } n ≥ is a partitionof unity in R \ { } . 7efine, for n ≥ G n ( x ; ε, β ) := Ψ n ( x ) (cid:0) x − M n − ( x ; ε, β ) (cid:1) − , (3.11)with formally, M n − ( x ; ε, β ) := n − X q = − χ q ( x ) M q ( x ; ε, β ) , M q ( x ; ε, β ) := X T ∈ R q ε k ( T ) V T ( x ; ε, β ) , (3.12)where V T ( x ; ε, β ) is the renormalised value of T , V T ( x ; ε, β ) := Y v ∈ N ( T ) F v ( β ) Y ℓ ∈ L ( T ) G n ℓ ( ω · ν ℓ ; ε, β ) . (3.13)Here and henceforth, the sums and the products over empty sets have to be considered as zeroand 1 respectively. Note that V T depends on ε because the propagators do, and on x = ω · ν ℓ ′ T only through the propagators associated with the lines ℓ ∈ P T (see Remark 3.2). Remark 3.5.
One has | R − | = 1 , so that M − ( x ; ε, β ) = M − ( x ; ε, β ) = ε∂ β F ( β ).Set M = {M n ( x ; ε, β ) } n ≥− . We call self-energies the quantities M n ( x ; ε, β ). Remark 3.6.
One has ∂ β G n ( x ; ε, β ) = G n ( x ; ε, β ) (cid:0) x − M n − ( x ; ε, β ) (cid:1) − ∂ β M n − ( x ; ε, β ) . Set also G − (0; ε, β ) = 1 (so that we can associate a propagator also with the root line of θ ∈ Θ R k, ). For any subgraph S of any θ ∈ Θ R k, ν define the renormalised value of S as V ( S ; ε, β ) := Y v ∈ N ( S ) F v ( β ) Y ℓ ∈ L ( S ) G n ℓ ( ω · ν ℓ ; ε, β ) . (3.14)Finally set b [ k ] ν ( ε, β ) := X θ ∈ Θ R k, ν V ( θ ; ε, β ) , (3.15)and G [ k ] ( ε, β ) := X θ ∈ Θ R k +1 , V ( θ ; ε, β ) , (3.16)and define formally b R ( t ; ε, β ) := X k ≥ ε k X ν ∈ Z d ∗ e i ν · ω t b [ k ] ν ( ε, β ) , (3.17)and G R ( ε, β ) := X k ≥ ε k G [ k ] ( ε, β ) . (3.18)The series (3.17) and (3.18) will be called the resummed series . The term “resummed” comesfrom the fact that if we formally expand (3.17) and (3.18) in powers of ε , we obtain (3.1) and(3.4) respectively, as it is easy to check. 8 emark 3.7. If T is a renormalised left-fake (respectively right-fake) cluster, we can (andshall) write V ( T ; ε, β ) = V T ( ω · ν ℓ ′ T ; ε, β ) since the propagators of the lines in P T depend on ω · ν ℓ ′ T . In particular one has X T ∈ LF n ε k ( T ) V T ( x ; ε, β ) = X T ∈ RF n ε k ( T ) V T ( x ; ε, β ) = M n ( x ; ε, β ) . Remark 3.8.
Given a renormalised tree θ such that V ( θ ; ε, β ) = 0, for any line ℓ ∈ L ( θ )(except possibly the root line) one has Ψ n ℓ ( ω · ν ℓ ) = 0, and hence α m nℓ ( ω )8 < | ω · ν ℓ | < α m nℓ − ( ω )4 , (3.19)where α m − ( ω ) has to be interpreted as + ∞ . The same considerations apply to any subgraphof θ and to renormalised self-energy clusters. Moreover, by the definition of { α m n ( ω ) } n ≥ , thenumber of scales which can be associated with a line ℓ in such a way that the propagator doesnot vanishes is at most 2; see Figure 1.For θ ∈ Θ R k, ν , let N n ( θ ) be the number of lines on scale ≥ n in θ , and set K ( θ ) := X v ∈ N ( θ ) | ν v | . (3.20)More generally, for any renormalised subgraph T of any tree θ call N n ( T ) the number of lineson scale ≥ n in T , and set K ( T ) := X v ∈ N ( T ) | ν v | . (3.21) Lemma 3.9.
For any θ ∈ Θ R k, ν such that V ( θ ; ε, β ) = 0 one has N n ( θ ) ≤ − ( m n − K ( θ ) , forall n ≥ .Proof. First of all we note that if N n ( θ ) ≥
1, then K ( θ ) ≥ m n − . Indeed, if a line ℓ has scale n ℓ ≥ n , then | ω · ν ℓ | ≤ α m n − ( ω ) < α m n − + p n − ( ω ) = 12 α m n − ( ω ) < α m n − ( ω ) , and hence, by definition of α m ( ω ), one has K ( θ ) ≥ | ν ℓ | ≥ m n − . Now we prove the bound N n ( θ ) ≤ max { − ( m n − K ( θ ) − , } by induction on the order.If the root line of θ has scale n ℓ θ < n then the bound follows by the inductive hypothesis.If n ℓ θ ≥ n , call ℓ , . . . , ℓ r the lines with scale ≥ n closest to ℓ θ (that is such that n ℓ ′ < n for alllines ℓ ′ ∈ P ( ℓ θ , ℓ i ), i = 1 , . . . , r ); see Figure 2. If r = 0 then N n ( θ ) = 1 and | ν | ≥ m n − , so thatthe bound follows. If r ≥ r = 1,then ℓ is the only entering line of a cluster T which is not a self-energy cluster as θ ∈ Θ R k, ν , andhence ν ℓ = ν . But then | ω · ( ν − ν ℓ ) | ≤ | ω · ν | + | ω · ν ℓ | ≤ α m n − ( ω ) < α m n − + p n − ( ω ) = α m n − ( ω ) , as both ℓ θ and ℓ are on scale ≥ n , so that one has K ( T ) ≥ | ν − ν ℓ | ≥ m n − . Now, call θ thesubtree of θ with root line ℓ . Then one has N n ( θ ) = 1 + N n ( θ ) ≤ { − ( m n − K ( θ ) − , } , θ ≥ n < n ≥ n ℓ θ ≥ nℓ θ ≥ nℓ r θ r Figure 2: Construction used in the proof of Lemma 3.9 when n ℓ θ ≥ n . so that N n ( θ ) ≤ − ( m n − ( K ( θ ) − K ( T )) ≤ − ( m n − K ( θ ) − , again by induction. Lemma 3.10.
For any T ∈ R n such that V T ( x ; ε, β ) = 0 , one has N p ( T ) ≤ − ( m p − K ( T ) ,for all ≤ p ≤ n .Proof. We first prove that for all n ≥ T ∈ R n , one has K ( T ) ≥ m n − . In fact if T ∈ R n then T contains at least a line on scale n . If there is ℓ ∈ L ( T ) \ P T with n ℓ = n , then | ω · ν ℓ | < α m n − ( ω ) < α m n − ( ω ) , and hence K ( T ) ≥ | ν ℓ | > m n − . Otherwise, let ℓ ∈ P T be the line on scale n which isclosest to ℓ ′ T . Call e T the subgraph (actually the cluster) consisting of all lines and nodes of T preceding ℓ ; see Figure 3. Then ν ℓ = ν ℓ ′ T , otherwise e T would be a self-energy cluster. Therefore K ( T ) > | ν ℓ − ν ℓ ′ T | > m n − as both ℓ, ℓ ′ T are on scale ≥ n . ℓ ℓ ′ T e TT = Figure 3: Construction used to prove K ( T ) ≥ m n − when there is a line ℓ ∈ P T on scale n . Given a tree θ , call C ( n, p ) the set of renormalised subgraphs T of θ with only one entering line ℓ ′ T and one exiting line ℓ T both on scale ≥ p , such that L ( T ) = ∅ and n ℓ ≤ n for any ℓ ∈ L ( T ).Note that R n ⊂ C ( n, p ) for all n, p ≥ K ( T ) ≥ m q − ,with q = min { n, p } , for all T ∈ C ( n, p ). We prove that N p ( T ) ≤ max { K ( T )2 − ( m p − − , } for all 0 ≤ p ≤ n and all T ∈ C ( n, p ). The proof is by induction on the order. Call N ( P T )the set of nodes in T connected by lines in P T . If all lines in P T are on scale < p , then N p ( T ) = N p ( θ ) + . . . + N p ( θ r ) if θ , . . . , θ r are the subtrees with root line entering a node in N ( P T ), and hence the bound follows from (the proof of) Lemma 3.9. If there exists a line ℓ ∈ P T on scale ≥ p , call T and T the subgraphs of T such that L ( T ) = { ℓ } ∪ L ( T ) ∪ L ( T ), and notethat if L ( T ) , L ( T ) = ∅ , then T , T ∈ C ( n, p ); see Figure 4.10 = ≥ p T ≥ pℓ T ≥ p Figure 4: Construction used to prove Lemma 3.10.
Hence, by the inductive hypothesis one has N p ( T ) = 1 + N p ( T ) + N p ( T ) ≤ { − ( m p − K ( T ) − , } + max { − ( m p − K ( T ) − , } . If both N p ( T ) , N p ( T ) are zero the bound trivially follows as K ( T ) ≥ m p − , while if both arenon-zero one has N p ( T ) ≤ − ( m p − ( K ( T ) + K ( T )) − − ( m p − K ( T ) − . Finally if only one is zero, say N p ( T ) = 0 and N p ( T ) = 0, N p ( T ) ≤ − ( m p − K ( T ) = 2 − ( m p − K ( T ) − − ( m p − K ( T ) . On the other hand, either T ∈ C ( n, p ) or it is constituted by only one node v with ν v = , sothat K ( T ) > m p − in both cases. The same argument can be used in the case N p ( T ) = 0 and N p ( T ) = 0. To prove that the resummed series (3.17) converges, we first make the assumption that thepropagators G n ℓ ( x ; ε, β ) are bounded essentially as 1 /x : we shall see that in that case theconvergence of the series can be easily proved. Then, in Section 5, we shall check that theassumption is justified. Definition 4.1.
We shall say that M satisfies property 1 if one has Ψ n +1 ( x ) | x − M n ( x ; ε, β ) | ≥ Ψ n +1 ( x ) x / , for all n ≥ − . Lemma 4.2.
Assume M to satisfy property 1. Then the series (3.17) and (3.18) with thecoefficients given by (3.15) and (3.16) respectively, converge for ε small enough.Proof. Let θ ∈ Θ R k, ν . The analyticity of f , hence of F , implies that there exist positive constants F , F , ξ such that for all v ∈ N ( θ ) one has |F v ( β ) | = 1 s v ! | ∂ s v β F ν v ( β ) | ≤ F F s v e − ξ | ν v | . (4.1)11oreover property 1 implies |G n ( x ; ε, β ) | ≤ α m n ( ω ) − for all n ≥
0, and hence by Lemma 3.9one can bound Y ℓ ∈ L ( θ ) |G n ℓ ( ω · ν ℓ ; ε, β ) | ≤ Y n ≥ (cid:18) α m n ( ω ) (cid:19) N n ( θ ) ≤ α m n ( ω ) ! k Y n ≥ n +1 (cid:18) α m n ( ω ) (cid:19) N n ( θ ) ≤ α m n ( ω ) ! k Y n ≥ n +1 / α m n ( ω ) ! − ( mn − K ( θ ) ≤ α m n ( ω ) ! k exp K ( θ ) X n ≥ n +1 m n log 2 / α m n ( ω ) ≤ D k ( n )exp( ξ ( n ) K ( θ )) , with D ( n ) = 2 α m n ( ω ) , ξ ( n ) = 8 X n ≥ n +1 m n log 2 / α m n ( ω ) . Then, by Hypothesis 1, one can choose n such that ξ ( n ) ≤ ξ/
2. The sum over the other labelsis bounded by a constant to the power k , and hence one can bound X θ ∈ Θ R k, ν | V ( θ ; ε, β ) | ≤ C C k e − ξ | ν | / , for some constants C , C , and this is enough to prove the assertion. Lemma 4.3.
Assume M to satisfy property 1. Then for ε small enough the function (3.17),with the coefficients given by (3.15), solves the equation (2.3a).Proof. We shall prove that, the function b R defined in (3.17) satisfies the equation of motion(2.3a), i.e. we shall check that b R = εgF ( ω t, β + b R ), where g is the pseudo-differential operatorwith kernel g ( ω · ν ) = 1 / ( ω · ν ) . We can write the Fourier coefficients of b R as b R ν = X n ≥ b [ n ] ν , b [ n ] ν = X k ≥ ε k X θ ∈ Θ R k, ν ( n ) V ( θ ; ε, β ) , (4.2)where Θ R k, ν ( n ) is the subset of Θ R k, ν such that n ℓ θ = n .Using Remark 3.4 and Lemma 4.2, in Fourier space one can write g ( ω · ν )[ εF ( ω t, β + b R )] ν = g ( ω · ν ) X n ≥ Ψ n ( ω · ν )[ εF ( ω t, β + b R )] ν = g ( ω · ν ) X n ≥ Ψ n ( ω · ν )( G n ( ω · ν ; ε, β )) − G n ( ω · ν ; ε, β )[ εF ( ω t, β + b R )] ν = g ( ω · ν ) X n ≥ (cid:0) ( ω · ν ) − M n − ( ω · ν ; ε, β ) (cid:1) G n ( ω · ν ; ε, β )[ εF ( ω t, β + b R )] ν = g ( ω · ν ) X n ≥ (cid:0) ( ω · ν ) − M n − ( ω · ν ; ε, β ) (cid:1) X k ≥ ε k X θ ∈ Θ R k, ν ( n ) V ( θ ; ε, β ) , R k, ν ( n ) differs from Θ R k, ν ( n ) as it contains also trees θ which have one self-energy clusterwith exiting line ℓ θ . If we separate the trees containing such self-energy cluster from the others,we obtain g ( ω · ν )[ εF ( ω t, β + b R )] ν = g ( ω · ν ) X n ≥ (cid:0) ( ω · ν ) − M n − ( ω · ν ; ε, β ) (cid:1) b [ n ] ν + g ( ω · ν ) X n ≥ Ψ n ( ω · ν ) X p ≥ n n − X q = − M q ( ω · ν ; ε, β ) b [ p ] ν + g ( ω · ν ) X n ≥ Ψ n ( ω · ν ) n − X p =0 p − X q = − M q ( ω · ν ; ε, β ) b [ p ] ν = g ( ω · ν ) X n ≥ (cid:0) ( ω · ν ) − M n − ( ω · ν ; ε, β ) (cid:1) b [ n ] ν + g ( ω · ν ) X p ≥ p − X q = − M q ( ω · ν ; ε, β ) X n ≥ q +1 Ψ n ( ω · ν ) b [ p ] ν = g ( ω · ν ) X n ≥ (cid:0) ( ω · ν ) − M n − ( ω · ν ; ε, β ) (cid:1) b [ n ] ν + g ( ω · ν ) X n ≥ n − X q = − M q ( ω · ν ; ε, β ) χ q ( ω · ν ) b [ n ] ν = g ( ω · ν ) X n ≥ (cid:0) ( ω · ν ) − M n − ( ω · ν ; ε, β ) (cid:1) b [ n ] ν + g ( ω · ν ) X n ≥ M n − ( ω · ν ; ε, β ) b [ n ] ν = X n ≥ b [ n ] ν = b R ν , so that the proof is complete. Definition 4.4.
We shall say that M satisfies property 2- p if one has Ψ n +1 ( x ) | x − M n ( x ; ε, β ) | ≥ Ψ n +1 ( x ) x / , for all − ≤ n < p . Lemma 4.5.
Assume M to satisfy property 2- p . Then for any ≤ n ≤ p the self-energies arewell defined and one has | M n ( x ; ε, β ) | ≤ ε K e − K mn , (4.3a) | ∂ jx M n ( x ; ε, β ) | ≤ ε C j e − C j mn , j = 1 , , (4.3b) for suitable constants K , K , C , C , C and C .Proof. Property 2- p implies |G n ( x ; ε, β ) | ≤ α m n ( ω ) − for all 0 ≤ n ≤ p . Then, using alsoLemma 3.10 and the fact that any self-energy cluster in R n has at least two nodes for any n ≥ | M n ( x ; ε, β ) | ≤ X T ∈ R n | ε | k ( T ) | V T ( x ; ε, β ) | ≤ X k ≥ | ε | k C k e − K mn ,
13o that (4.3a) is proved for ε small enough. Now we prove (4.3b) by induction on n . For n = 0the bound is obvious. Assume then (4.3b) to hold for all n ′ < n . For any T ∈ R n such that V T ( x ; ε, β ) = 0 one has ∂ x V T ( x ; ε, β ) = X ℓ ∈P T Y v ∈ N ( T ) F v ( β ) ∂ x G n ℓ ( x ℓ ; ε, β ) Y ℓ ′ ∈ L ( T ) \{ ℓ } G n ℓ ′ ( ω · ν ℓ ′ ; ε, β ) , where x ℓ = ω · ν ℓ = x + ω · ν ℓ and ∂ x G n ℓ ( x ℓ ; ε, β ) = ddx G n ℓ ( ω · ν ℓ + x ; ε, β )= ∂ x Ψ n ℓ ( x ℓ ) x ℓ − M n ℓ − ( x ℓ ; ε, β ) − Ψ n ℓ ( x ℓ ) (2 x ℓ − ∂ x M n ℓ − ( x ℓ ; ε, β )) (cid:0) x ℓ − M n ℓ − ( x ℓ ; ε, β ) (cid:1) . One has | ∂ x Ψ n ℓ ( x ℓ ) | ≤ | ∂ x χ n ℓ − ( x ℓ ) | + | ∂ x ψ n ℓ ( x ℓ ) | ≤ B α m nℓ ( ω ) , for some constant B and, by (4.3a), the inductive hypothesis and Hypothesis 1, | ∂ x M n ℓ − ( x ℓ ; ε, β ) | ≤ n ℓ − X q =0 | ( ∂ x χ q ( x ℓ )) M q ( x ℓ ; ε, β ) | + n ℓ − X q =0 | ∂ x M q ( x ℓ ; ε, β ) |≤ ε B K X q ≥ α m q ( ω ) e − K mq + ε C X q ≥ e − C mq ≤ ε B , for some constant B . Hence, at the cost of replacing the bound for the propagators with e Cα m nℓ ( ω ) − for some constant e C , one can rely upon Lemma 3.10 to obtain (4.3b) for j = 1.For j = 2 one can reason analogously. Lemma 4.6.
Assume M to satisfy property 2- p . Then one has M n ( x ; ε, β ) = M n (0; ε, β ) + O ( ε x ) for all ≤ n ≤ p .Proof. We shall prove that M n ( x ; ε, β ) = M n ( − x ; ε, β ), by induction on n ≥ −
1. For n = − M − does not depend on x . Assume now M q ( x ; ε, β ) = M q ( − x ; ε, β ) for all q < n . This implies G q ( x ; ε, β ) = G q ( − x ; ε, β ) for q ≤ n . Let T ∈ R n andconsider the self-energy cluster T obtained from T by taking ℓ T as the entering line and ℓ ′ T asthe exiting line (i.e. ℓ ′ T = ℓ T and ℓ T = ℓ ′ T ) and by taking ν ℓ ′ T = − ν ℓ ′ T . Hence the momenta ofthe lines belonging to P T change signs, while all the other momenta do not change: therefore allpropagators are left unchanged. Hence M n ( x ; ε, β ) = M n ( − x ; ε, β ), so that ∂ x M n (0; ε, β ) = 0for all n ≤ p , and, by Lemma 4.5, this is enough to prove the assertion. Lemma 4.7.
Assume M to satisfy property 1. Then the function G R ( ε, β ) and the self-energies M n ( x ; ε, β ) are C ∞ in both ε and β .Proof. It follows from the explicit expressions for G R ( ε, β ) and M n ( x ; ε, β ).Define formally M ∞ ( x ; ε, β ) = lim n →∞ M n ( x ; ε, β ) , (4.4)14nd note that if M satisfies property 1, then M ∞ ( x ; ε, β ) is well defined and moreover it is C ∞ in both ε and β .The following result plays a crucial role. The proof is deferred to Appendix A. Lemma 4.8.
Assume M to satisfy property 1. Then one has ε∂ β G R ( ε, β ) = M ∞ (0; ε, β ) . Remark 4.9.
If we take the formal power expansions of both G R ( ε, β ) and M ∞ (0; ε, β ), weobtain tree expansions where self-energy clusters are allowed; see Section 6 for further details.Then the identity ε∂ β G R ( ε, β ) = M ∞ (0; ε, β ) is easily found to be satisfied to any perturba-tion order. However, without any resummation procedure, we are no longer able to prove theconvergence of the series, so that the identity becomes a meaningless “ ∞ = ∞ ”. Remark 4.10.
The identity ε∂ β G R ( ε, β ) = M ∞ (0; ε, β ), in Lemma 4.8, can be seen asan identity between classes of diagrams. In turn, in light of a possible quantum field formula-tion of the problem, this can be thought as a consequence of some deep Ward identity of thecorresponding field theory. Ward identities play a crucial role in quantum field theory. Theanalogy between KAM theory and quantum field theory has been widely stressed in the litera-ture [11, 2, 6]; in particular the cancellations which assure the convergence of the perturbationseries for maximal KAM tori are deeply related to a Ward identity, as shown in [2], which canbe seen as a remarkable identity between classes of graphs. In the case studied in this paper,we have a similar situation, made fiddlier by the fact that we have to deal with nonconvergentseries to be resummed, and it is well known that identities which are trivial on a formal levelcan turn out to be difficult to prove rigorously [19]. However, we expect a Ward identity to holdalso in our case, so as to imply that ε∂ β G R ( ε, β ) = M ∞ (0; ε, β ). It would be interesting toconfirm the expectation and to determine the Ward identity explicitly. Lemma 4.11.
Assume M to satisfy property 1. Then the implicit function equation G R ( ε, β ) =0 admits a solution β = β ( ε ) , such that β (0) = β ∗ . Moreover in a suitable half-neighbourhoodof ε = 0 , one has ε∂ β G R ( ε, β ( ε )) ≤ .Proof. Property 1 allows us to write G R ( ε, β ) = F ( β ) + O ( ε ), so that by Hypothesis 2 onehas ∂ n β G R (0 , β ∗ ) = 0. Then there exist two half-neighbourhood V − , V + of β = β ∗ such that G R (0 , β ) > β ∈ V + and G R (0 , β ) < β ∈ V − . Hence, by continuity, for all β ∈ V + there exists a neighbourhood U + ( β ) of ε = 0 such that G R ( ε, β ) > ε ∈ U + ( β ) and,for the same reason, for all β ∈ V − there exists a neighbourhood U − ( β ) of ε = 0 such that G R ( ε, β ) < ε ∈ U − ( β ). Therefore, again by continuity, there exists a continuouscurve β = β ( ε ) defined in a suitable neighbourhood U = ( − ε, ε ) such that β (0) = β ∗ and G R ( ε, β ( ε )) ≡
0. Moreover, if ∂ n β G R (0 , β ∗ ) >
0, then V + , V − are of the form ( β ∗ , v + ) and( v − , β ∗ ) respectively, and therefore ∂ β G R ( c, β ( c )) ≥ c ∈ U . If on the contrary ∂ n β G R (0 , β ∗ ) <
0, one has V + = ( v + , β ∗ ) and V − = ( β ∗ , v − ), and then ∂ β G R ( c, β ( c )) ≤ c ∈ U . Hence the assertion follows in both cases, again by Hypothesis 2. Remark 4.12. If M satisfies property 1, one has G R ( ε, β ) = [ F ( ω t, β + b R ( t ; ε, β ))] , and hence, if β = β ( ε ) is the solution referred to in Lemma 4.11, by Lemma 4.3 the function β ( t ; ε ) = β ( ε ) + b R ( t ; ε, β ( ε )) , solves the equation of motion (1.1). 15 emark 4.13. In Lemma 4.11 we widely used that the variable β is one-dimensional. All theother results in this paper could be quite easily extended to higher dimension. Remark 4.14.
The results of this section are not sufficient to prove Theorem 2.1 because wehave assumed – without proving – that property 1 is satisfied. In Section 5 we shall show that,thanks to the symmetry property of Lemma 4.6 and the identity of Lemma 4.8, property 1 issatisfied along a suitable continuous curve β = β ( ε ) such that G R ( ε, β ( ε )) = 0. In this section we shall remove the assumption that the self-energies satisfy property 1 of Def-inition 4.1 – see Remark 4.14. For all n ≥
0, define the C ∞ non-increasing functions ξ n suchthat ξ n ( x ) = ( , x ≤ α m n +1 ( ω ) / , , x ≥ α m n +1 ( ω ) / , (5.1)and set ξ − ( x ) = 1. Define recursively, for all n ≥
0, the propagators G n ( x ; ε, β ) = Ψ n ( x ) (cid:0) x − M n − ( x ; ε, β ) ξ n − ( M n − (0; ε, β )) (cid:1) − , (5.2)with M − ( x ; ε, β ) = ε∂ β F ( β ), and for n ≥ M n ( x ; ε, β ) = M n − ( x ; ε, β ) + χ n ( x ) M n ( x ; ε, β ) , (5.3)where we have set M n ( x ; ε, β ) = X T ∈ R n ε k ( T ) V T ( x ; ε, β ) , (5.4)with V T ( x ; ε, β ) = Y v ∈ N ( T ) F v ( β ) Y ℓ ∈ L ( T ) G n ℓ ( ω · ν ℓ ; ε, β ) , (5.5)and x = ω · ν ℓ ′ T .Set also M = {M n ( x ; ε, β ) } n ≥− , and M ξ = {M n ( x ; ε, β ) ξ n ( M n (0; ε, β )) } n ≥− . Lemma 5.1. M ξ satisfies property 1.Proof. We shall prove that M ξ satisfies property 2- p for all p ≥
0, by induction on p . Property2-0 is trivially satisfied for ε small enough. Assume M ξ to satisfy property 2- p . Then we canrepeat (almost word by word) the proofs of Lemmas 4.5 and 4.6 so as to obtain M p ( x ; ε, β ) = M p (0; ε, β ) + O ( ε x ) , hence, by the definition of the function ξ p , M ξ satisfies property 2-( p + 1), and thence theassertion follows.Set V ( θ ; ε, β ) = Y v ∈ N ( θ ) F v ( β ) Y ℓ ∈ L ( θ ) G n ℓ ( ω · ν ℓ ; ε, β ) , (5.6)16nd b [ k ] ν ( ε, β ) = X θ ∈ Θ R k, ν V ( θ ; ε, β ) , (5.7)and define b ( t, ε, β ) = X k ≥ ε k b [ k ] ( ε, β ) = X k ≥ ε k X ν ∈ Z d ∗ e i ν · ω t b [ k ] ν ( ε, β ) . (5.8)Note that, by (the proof of) Lemma 4.2 the series (5.8) converges.Define also M ∞ ( x ; ε, β ) := lim n →∞ M n ( x ; ε, β ) , (5.9)and note that, by Lemma 5.1 the limit in (5.9) is well defined and it is C ∞ in both ε and β .Introduce the C ∞ functions G ( ε, β ) such that M ∞ (0; ε, β ) = ε∂ β G ( ε, β ) and G (0 , β ∗ ) = 0,and for any such function consider the implicit function equation G ( ε, β ) = 0 . (5.10) Lemma 5.2.
The implicit function equation (5.10) admits a solution β = β ( ε ) such that β (0) = β ∗ . Moreover in a suitable half-neighbourhood of ε = 0 , one has ε∂ β G ( ε, β ( ε )) ≤ .Proof. By construction, all the functions G ( ε, β ) are smooth and of the form G ( ε, β ) = F ( β ) + O ( ε ). Then the result follows straightforward from (the proof of) Lemma 4.11. Lemma 5.3.
Let β = β ( ε ) be the solution referred to in Lemma 5.2. Then one has ξ n ( M n (0; ε, β ( ε ))) ≡ for all n ≥ , in a suitable half-neighbourhood of ε = 0 .Proof. If β = β ( ε ), by Lemma 5.2 in a suitable half-neighbourhood of ε = 0 one has M ∞ (0; ε, β ( ε )) = ε∂ β G ( ε, β ( ε )) ≤
0. Hence, as the bound (4.3a) holds also for M n ( x ; ε, β ),one has M n (0; ε, β ( ε )) ≤ M n (0; ε, β ( ε )) − M ∞ (0; ε, β ( ε )) ≤ X p ≥ n +1 | M p (0; ε, β ( ε )) |≤ K ε e − K mn ≤ α m n +1 , (5.11)so that the assertion follows by the definition of ξ n . Lemma 5.4.
For β = β ( ε ) , one has M = M = M ξ , and hence one can choose G ( ε, β ) suchthat G R ( ε, β ( ε )) = G ( ε, β ( ε )) = 0 . In particular β ( t ; ε ) = β ( ε ) + b R ( t ; ε, β ( ε )) defined in(3.17) solves the equation of motion (1.1).Proof. It follows from the results above. If F ( β ) vanishes identically, let us come back to the formal expansion (3.4) of G ( ε, β ), where G (0) ( β ) = F ( β ) ≡ k ∈ N such that all functions G ( k ) ( β ) are identically zero for0 ≤ k ≤ k −
1, while G ( k ) ( β ) is not identically vanishing. Then we can write G ( ε, β ) = ε k (cid:16) G ( k ) ( β ) + G ( >k ) ( ε, β ) (cid:17) , (6.1)with G ( >k ) ( ε, β ) = O ( ε ), and we can solve the equation of motion up to order k without fixingthe parameter β .Any primitive function g ( k ) ( β ) of G ( k ) ( β ) is therefore analytic and periodic: since it isnot identically constant, it admits at least one maximum ¯ β ′ and one minimum ¯ β ′′ , so that onecan assume the following Hypothesis 3. β ∗ is a zero of order ¯ n for G ( k ) ( β ) with ¯ n odd, and ε k +1 ∂ ¯ n β G ( k ) ( β ∗ ) < . Indeed, if k is even one can choose β ∗ = ¯ β ′ for ε >
0, and β ∗ = ¯ β ′′ for ε <
0; if k is odd wehave to fix β ∗ = ¯ β ′ : in both cases Hypothesis 3 is satisfied.Then one can adapt the proof in the previous sections to cover this case. Namely, as theformal expansion of G R coincide with that of G , one sets G R ( ε, β ) =: ε k G ∗ ( ε, β ) , and hence, if M satisfies property 1, M ∞ (0; ε, β ) = ε k +1 ∂ β G ∗ ( ε, β ) . (6.2)On the other hand, Hypothesis 3 and Lemma 4.11 guarantee the existence of a continuous curve β ( ε ) such that β (0) = β ∗ , G ∗ ( ε, β ( ε )) ≡ k is even then ε k +1 ∂ β G ∗ ( ε, β ( ε )) ≤ ε = 0, while if k is odd and β ∗ is a maximum for g ( k ) , then ∂ β G ∗ ( ε, β ( ε )) ≤ ε = 0. Then one can reason as in Section 5 toobtain the result.Finally, assume G ( k ) ( β ) ≡ k ≥
0. We shall see that no resummation is necessaryin that case: this situation is reminiscent of the “null-renormalisation” case considered in [16]when studying the stability problem for Hill’s equation with a quasi-periodic perturbation.We define trees and clusters according to the definitions previously done. On the other hand,we slight change the definition of self-energy clusters. Namely, a cluster T on scale n ≥ ℓ ′ T and one exiting line ℓ T , and with ν ℓ T = ν ℓ ′ T , is called a self-energycluster if n + 2 ≤ n T := min { n ℓ T , n ℓ ′ T } . The definition of self-energy cluster does not changefor the self-energy cluster on scale −
1. We denote by Θ k, ν the set of trees with order k andmomentum ν as in Section 3, and by S kn the set of (non-renormalised) self-energy clusters withorder k and scale n ; note that self-energy clusters are allowed both in Θ k, ν and in S kn .For any subgraph S of any tree θ ∈ Θ k, ν , and for any T ∈ S kn , define the (non-renormalised)value of S and T as in (3.14) and (3.13) respectively, but with the (undressed) propagatorsdefined as G n ℓ ( ω · ν ℓ ) := Ψ n ℓ ( ω · ν ℓ ) ω · ν ℓ , n ℓ ≥ , , n ℓ = − . (6.3)Note that now the values of trees and self-energy clusters do not depend on ε , and they dependon β only through the node factors. From now on we do not write explicitly the dependence18n β to lighten the notations. For all k ≥
1, define b ( k ) ν := X θ ∈ Θ k, ν V ( θ ) , (6.4a) G ( k − := X θ ∈ Θ k, V ( θ ) , (6.4b) M ( k ) n ( x ) := X T ∈ S kn V T ( x ) , n ≥ − M ( k ) n ( x ) := n X p =0 M ( k ) p ( x ) , n ≥ − M ( k ) ∞ ( x ) := lim n →∞ M ( k ) n ( x ) . (6.4e)The coefficients (6.4a) and (6.4b) coincide with (3.2) and (3.5) respectively as it is easy to check;in particular, for all k ≥ X θ ∈ Θ k, V ( θ ) ≡ , by assumption. Remark 6.1.
One has S k − = S n = ∅ for k ≥ n ≥
0. On the other hand | S − | = 1 and V T ( x ) = ∂ β F ≡ T is the self-energy cluster in S − ; see Remark 3.5. Hence M (1) n ( x ) = M (1) n ( x ) = M (1) ∞ ( x ) = M ( k ) − = M ( k ) − ≡ n ≥ − k ≥ θ with V ( θ ) = 0, we shall say that a line ℓ ∈ L ( θ ) is resonant if it is the exitingline of a self-energy cluster T , otherwise we shall say that ℓ is non-resonant . For any subgraph T of any tree θ ∈ Θ k, ν , denote by N ∗ n ( T ) the number of non-resonant lines on scale ≥ n in T ,and set K ( T ) as in (3.21). Then we can prove the analogous of Lemmas 3.9 and 3.10, namelythe following results. Lemma 6.2.
For any θ ∈ Θ k, ν such that V ( θ ) = 0 one has N ∗ n ( θ ) ≤ − ( m n − K ( θ ) , for all n ≥ . Lemma 6.3.
For any T ∈ S kn such that V T ( x ) = 0 one has N ∗ p ( T ) ≤ − ( m p − K ( T ) , for all ≤ p ≤ n . We omit the proofs of the two results above as it would be essentially a repetition of thosefor Lemmas 3.9 and 3.10, respectively. Note that, since self-energy clusters are now allowed, forthe proof of Lemma 6.3 one needs that the momenta of the lines in P T are different from thoseof the external lines: this explains the new definition of self-energy clusters.In light of Lemmas 6.2 and 6.3, although one has the ‘good bound’ 1 /x for the propagators,one cannot prove the convergence of the power series (3.1) as done in Lemma 4.2, because wedo not have any bound for the number of resonant lines, which in principle can accumulate ‘toomuch’. In fact, we need a gain factor proportional to ( ω · ν ℓ ) for each resonant line ℓ . Lemma 6.4.
For all n ≥ and for all k ≥ one has ∂ x M ( k ) n (0) = 0 , and hence ∂ x M ( k ) n (0) = 0 for all k ≥ . roof. As the propagators are trivially even in the momenta, one can repeat (almost word byword) the proof of Lemma 4.6 so as to obtain the result.
Lemma 6.5.
One has M ( k ) ∞ (0) ≡ for all k ≥ .Proof. One has (see also Remark 4.9) ∂ β G ( k − ≡ M ( k ) ∞ (0) so that the assertion follows. Lemma 6.6.
For all k ≥ one has |M ( k ) n ( x ) | Ψ n +2 ( x ) ≤ C k x Ψ n +2 ( x ) , (6.5) for some positive constant C .Proof. First of all note that (6.5) is trivially satisfied if Ψ n +2 ( x ) = 0. Assume then α m n +2 ( ω )8 < | x | < α m n +1 ( ω )4 . (6.6)Note also that the bound (6.5) provides the gain factor which is needed for the resonant lines.This can be seen as follows.Let θ ∈ Θ k, ν and let S be any subgraph of θ . For any ℓ ∈ L ( S ) set A ℓ ( S, x ℓ ) := (cid:16) Y v ∈ N ( S ) v ℓ F v (cid:17)(cid:16) Y ℓ ′ ∈ L ( S ) ℓ ′ ℓ G n ℓ ′ ( x ℓ ′ ) (cid:17) , (6.7)and B ℓ ( S ) := (cid:16) Y v ∈ N ( S ) v ≺ ℓ F v (cid:17)(cid:16) Y ℓ ′ ∈ L ( S ) ℓ ′ ≺ ℓ G n ℓ ′ ( x ℓ ′ ) (cid:17) , (6.8)where x ℓ = ω · ν ℓ . If ℓ ∈ L ( S ) is a resonant line exiting a self-energy cluster T ∈ S kn (with ofcourse n ≤ n T − ≤ n ℓ −
2) and also ℓ ′ T ∈ L ( S ) we can write V ( S ) = A ℓ ( S, x ℓ ) G n ℓ ( x ℓ ) V T ( x ℓ ) G n ℓ ′ T ( x ℓ ) B ℓ ′ T ( S ) , (6.9)where we have used x ℓ = x ℓ ′ T . But then, if we sum over all S ′ which can be obtained from S byreplacing T with any self-energy cluster T ′ ∈ S kn ′ for any n ′ ≤ n T − A ℓ ( S, x ℓ ) G n ℓ ( x ℓ ) M ( k ) n T − ( x ℓ ) G n ℓ ′ T ( x ℓ ) B ℓ ′ T ( S ) , (6.10)and hence, by (6.5), we obtain the gain factor which is needed.We shall prove the bound (6.11) by induction on k . For k = 1 (6.5) is trivially satisfied.Assume (6.5) to hold for all k ′ < k . By Lemma 6.4 we can write M ( k ) n ( x ) = M ( k ) n (0) + x Z dt (1 − t ) ∂ M ( k ) n ( tx ) , (6.11)where ∂ denotes the second derivative of M ( k ) n with respect to its argument. Then we shallprove |M ( k ) n (0) | ≤ A k α m n +2 ( ω )64 , (6.12a) | ∂ M ( k ) n ( x ) | ≤ A k , (6.12b)20or suitable constants A , A . Note that Lemma 6.3 and the inductive hypothesis yield | M ( k ) n ( x ) | ≤ B k e − B mn , (6.13a) | ∂ M ( k ) n ( x ) | ≤ D k e − D mn , (6.13b)for some positive constants B , B , D and D . But then |M ( k ) n (0) | = |M ( k ) n (0) − M ( k ) ∞ (0) | ≤ X p ≥ n +1 | M ( k ) p (0) | ≤ B k e − B mn , (6.14)so that (6.12a) follows if A is suitably chosen. Moreover | ∂ M ( k ) n ( x ) | ≤ n X p =0 | ∂ M ( k ) p ( x ) | ≤ DD k (6.15)for some constant D . Hence the assertion follows. Remark 6.7.
We have obtained the convergence of the power series (3.1) and (3.4) for any β and any ε small enough. Hence, in this case, the response solution turns out to be analytic inboth ε, β . Remark 6.8.
Note that the problem under study has analogies with the problem consideredin [15]. In that case, the resummation adds to the small divisor i ω · ν a quantity − ε ( ω · ν ) + M n ( ω · ν ; ε ), and one can prove that M n ( x, ε ) is smooth in x and it is real at x = 0, so thatthe dressed propagator is proportional to 1 / (i ω · ν − ε ( ω · ν ) + M n ( ω · ν ; ε )), and hence canbe bounded essentially as the undressed one. In the present case, both the small divisor ( ω · ν ) and the correction are real, but they turn out to have the same sign (for a suitable choice of β ∗ ), so that once more the dressed propagator can be bounded as the undressed one. A Proof of Lemma 4.8
First of all, for any renormalised tree θ set ∂ v V ( θ ; ε, β ) := ∂ β F v ( β ) Y w ∈ N ( θ ) \{ v } F w ( β ) Y ℓ ∈ L ( θ ) G n ℓ ( ω · ν ℓ ; ε, β ) (A.1)and ∂ ℓ V ( θ ; ε, β ) := ∂ β G n ℓ ( x ℓ ; ε, β ) Y v ∈ N ( θ ) F v ( β ) Y λ ∈ L ( θ ) \{ ℓ } G n λ ( x λ ; ε, β ) = A ℓ ( θ, x ℓ ; ε, β ) ∂ β G n ℓ ( x ℓ ; ε, β ) B ℓ ( θ ; ε, β ) , (A.2)where x ℓ := ω · ν ℓ , ∂ β G n ℓ ( x ℓ ; ε, β ) is written according to Remark 3.6, A ℓ ( θ, x ℓ ; ε, β ) := (cid:16) Y v ∈ N ( θ ) v ℓ F v ( β ) (cid:17)(cid:16) Y ℓ ′ ∈ L ( θ ) ℓ ′ ℓ G n ℓ ′ ( x ℓ ′ ; ε, β ) (cid:17) , (A.3)21nd B ℓ ( θ ; ε, β ) := (cid:16) Y v ∈ N ( θ ) v ≺ ℓ F v ( β ) (cid:17)(cid:16) Y ℓ ′ ∈ L ( θ ) ℓ ′ ≺ ℓ G n ℓ ′ ( x ℓ ′ ; ε, β ) (cid:17) , (A.4)see also (6.7) and (6.8). Let us define in the analogous way ∂ v V T ( x ; ε, β ) and ∂ ℓ V T ( x ; ε, β )for any self-energy cluster T , and let us write ∂ β V ( θ ; ε, β ) = ∂ N V ( θ ; ε, β ) + ∂ L V ( θ ; ε, β ) , (A.5)where ∂ N V ( θ ; ε, β ) := X v ∈ N ( θ ) ∂ v V ( θ ; ε, β ) , (A.6)and ∂ L V ( θ ; ε, β ) := X ℓ ∈ L ( θ ) ∂ ℓ V ( θ ; ε, β ) . (A.7)Let us also write ∂ β V T ( x ; ε, β ) = ∂ N V T ( x ; ε, β ) + ∂ L V T ( x ; ε, β ) , (A.8)for any T ∈ R n , n ≥
0, where the derivatives ∂ N and ∂ L are defined analogously with theprevious cases (A.6) and (A.7), with N ( T ) and L ( T ) replacing N ( θ ) and L ( θ ), respectively, sothat we can split ∂ β M n ( x ; ε, β ) = ∂ N M n ( x ; ε, β ) + ∂ L M n ( x ; ε, β ) ,∂ β M n ( x ; ε, β ) = ∂ N M n ( x ; ε, β ) + ∂ L M n ( x ; ε, β ) , (A.9)again with obvious meaning of the symbols. Remark A.1.
We can interpret the derivative ∂ v as all the possible ways to attach an extraline (carrying a momentum ) to the node v , so that X k ≥ ε k +1 X θ ∈ Θ R k +1 , ∂ N V ( θ ; ε, β ) , produces contributions to M ∞ (0; ε, β ).Given any θ ∈ Θ R k, we have to study the derivative (A.5). The terms (A.6) produce imme-diately contributions to M ∞ (0; ε, β ) by Remark A.1. Thus, we have to study the derivatives ∂ ℓ V ( θ ; ε, β ) appearing in the sum (A.7). Here and henceforth, we shall not write any longerexplicitly the dependence on ε and β of both propagators and self-energies, in order not tooverwhelm the notation.For any θ ∈ Θ R k, such that V ( θ ; ε, β ) = 0 and for any line ℓ ∈ L ( θ ), either there is only onescale n such that Ψ n ( x ℓ ) = 0 (and in that case Ψ n ( x ℓ ) = 1 and Ψ n ′ ( x ℓ ) = 0 for all n ′ = n ) orthere exists only one n ≥ n ( x ℓ )Ψ n +1 ( x ℓ ) = 0. If Ψ n ( x ℓ ) = 1 one has ∂ ℓ V ( θ ; ε, β ) = A ℓ ( θ, x ℓ ) Ψ n ( x ℓ ) x ℓ − M n − ( x ℓ ) ∂ β M n − ( x ℓ ) 1 x ℓ − M n − ( x ℓ ) B ℓ ( θ )= A ℓ ( θ, x ℓ ) Ψ n ( x ℓ ) x ℓ − M n − ( x ℓ ) ∂ β M n − ( x ℓ ) Ψ n ( x ℓ ) x ℓ − M n − ( x ℓ ) B ℓ ( θ )= A ℓ ( θ, x ℓ ) G n ( x ℓ ) ∂ β M n − ( x ℓ ) G n ( x ℓ ) B ℓ ( θ ) , (A.10)where (here and henceforth) we shorten A ℓ ( θ, x ℓ ) = A ℓ ( θ, x ℓ ; ε, β ) and B ℓ ( θ ) = B ℓ ( θ ; ε, β ).22 emark A.2. Note that if we split ∂ β = ∂ N + ∂ L in (A.10), the term with ∂ N M n − ( x ℓ ) is acontribution to M ∞ (0).If there is only one n ≥ n ( x ℓ )Ψ n +1 ( x ℓ ) = 0, then Ψ n ( x ℓ ) + Ψ n +1 ( x ℓ ) = 1and χ q ( x ℓ ) = 1 for all q = − , . . . , n −
1, so that ψ n +1 ( x ℓ ) = 1 and hence Ψ n +1 ( x ℓ ) = χ n ( x ℓ ).Moreover it can happen only (see Remark 3.7) n ℓ = n or n ℓ = n + 1. Consider first the case n ℓ = n + 1. One has ∂ ℓ V ( θ ; ε, β ) = A ℓ ( θ, x ℓ ) G n +1 ( x ℓ ) ∂ β M n ( x ℓ ) 1 x ℓ − M n ( x ℓ ) B ℓ ( θ )= A ℓ ( θ, x ℓ ) G n +1 ( x ℓ ) ∂ β M n − ( x ℓ ) Ψ n ( x ℓ ) + Ψ n +1 ( x ℓ ) x ℓ − M n ( x ℓ ) B ℓ ( θ )+ A ℓ ( θ, x ℓ ) G n +1 ( x ℓ ) ∂ β M n ( x ℓ ) χ n ( x ℓ ) x ℓ − M n ( x ℓ ) B ℓ ( θ )= A ℓ ( θ, x ℓ ) G n +1 ( x ℓ ) n X q = − ∂ β M q ( x ℓ ) G n +1 ( x ℓ ) B ℓ ( θ )+ A ℓ ( θ, x ℓ ) G n +1 ( x ℓ ) n − X q = − ∂ β M q ( x ℓ ) G n ( x ℓ ) B ℓ ( θ )+ A ℓ ( θ, x ℓ ) G n +1 ( x ℓ ) n − X q = − ∂ β M q ( x ℓ ) G n ( x ℓ ) M n ( x ℓ ) G n +1 ( x ℓ ) B ℓ ( θ ) . (A.11)We can represent graphically the three contributions in (A.11) as in Figure 5: we represent thederivative ∂ β as an arrow pointing toward the graphical representation of the differentiatedquantity; see also Figures 7, 10 and 12. n +1 ≤ n n +1 + n +1 ≤ n − n + n +1 ≤ n − n n n +1 Figure 5: Graphical representation of the derivative ∂ ℓ V ( θ ; ε, β ) according to (A.11). Remark A.3.
Note that the M n ( x ℓ ) appearing in the latter line of (A.11) has to be interpreted(see Remark 3.7) as X T ∈ LF n ε k ( T ) V T ( x ℓ ; ε, β ) . Note also that, again, if we split ∂ β = ∂ N + ∂ L in (A.11), all the terms with ∂ N M q ( x ℓ ) arecontributions to M ∞ (0). 23ow consider the case n ℓ = n . If ℓ is not the exiting line of a left-fake cluster, set ¯ θ = θ ; otherwise, if ℓ is the exitingline of a left-fake cluster T , define – if possible – ¯ θ as the renormalised tree obtained from θ by removing T and ℓ ′ T . In both cases, define – if possible – τ (¯ θ, ℓ ) as the set constituted byall the renormalised trees θ ′ obtained from ¯ θ by inserting a left-fake cluster, together with itsentering line, between ℓ and the node v which ℓ exits; see Figure 6. Here and henceforth, if S is a subgraph with only one entering line ℓ ′ S = ℓ v and one exiting line ℓ S and we “remove” S together with ℓ ′ S , we mean that we also reattach the line ℓ S to the node v .¯ θ = nℓ θ ′ = nℓ n n +1 Figure 6: The renormalised tree ¯ θ and the renormalised trees θ ′ of the set τ (¯ θ, ℓ ) associated with ¯ θ . Remark A.4.
The construction of the set τ (¯ θ, ℓ ) could be impossible if the removal or theinsertion of a left-fake cluster T , together with its entering line ℓ ′ T , produce a self-energy cluster.We shall see later how to deal with these cases.Then one has ∂ ℓ V (¯ θ ; ε, β ) + ∂ ℓ X θ ′ ∈ τ (¯ θ,ℓ ) V ( θ ′ ; ε, β ) = A ℓ (¯ θ, x ℓ ) ∂ β G n ( x ℓ ) (1 + M n ( x ℓ ) G n +1 ( x ℓ )) B ℓ (¯ θ ) , (A.12)where ∂ β G n ( x ℓ ) (1 + M n ( x ℓ ) G n +1 ( x ℓ ))= G n ( x ℓ ) ∂ β M n − ( x ℓ ) G n ( x ℓ )+ G n ( x ℓ ) ∂ β M n − ( x ℓ ) Ψ n +1 ( x ℓ ) x ℓ − M n − ( x ℓ )+ G n ( x ℓ ) ∂ β M n − ( x ℓ ) G n ( x ℓ ) M n ( x ℓ ) G n +1 ( x ℓ )+ G n ( x ℓ ) ∂ β M n − ( x ℓ ) Ψ n +1 ( x ℓ ) x ℓ − M n − ( x ℓ ) M n ( x ℓ ) G n +1 ( x ℓ )= G n ( x ℓ ) ∂ β M n − ( x ℓ ) G n ( x ℓ ) + G n ( x ℓ ) ∂ β M n − ( x ℓ ) G n +1 ( x ℓ ) − G n ( x ℓ ) ∂ β M n − ( x ℓ ) χ n ( x ℓ ) x ℓ − M n − ( x ℓ ) M n ( x ℓ ) G n +1 ( x ℓ )+ G n ( x ℓ ) ∂ β M n − ( x ℓ ) G n ( x ℓ ) M n ( x ℓ ) G n +1 ( x ℓ )+ G n ( x ℓ ) ∂ β M n − ( x ℓ ) Ψ n +1 ( x ℓ ) x ℓ − M n − ( x ℓ ) M n ( x ℓ ) G n +1 ( x ℓ )= G n ( x ℓ ) ∂ β M n − ( x ℓ ) G n ( x ℓ ) + G n ( x ℓ ) ∂ β M n − ( x ℓ ) G n +1 ( x ℓ )+ G n ( x ℓ ) ∂ β M n − ( x ℓ ) G n ( x ℓ ) M n ( x ℓ ) G n +1 ( x ℓ ) , (A.13)so that also in this case, if we split ∂ β = ∂ N + ∂ L , all the terms with ∂ N M n − are contributionsto M ∞ (0) – see Remark A.2. Again, we can represent graphically the three contributionsobtained inserting (A.13) in (A.12): see Figure 7. Assume now that ℓ is not the exiting line of a left-fake cluster, and the insertion of a left-fakecluster, together with its entering line, produces a self-energy cluster. Note that this can happenonly if ℓ is the entering line of a renormalised right-fake cluster T . Let ℓ be the exiting line (on24 ≤ n − n + n ≤ n − n +1+ n ≤ n − n n n +1 Figure 7: Graphical representation of the three contributions in the last two lines of (A.13). scale n + 1) of the renormalised right-fake cluster T , call θ the renormalised tree obtained from θ by removing T and ℓ and call τ ( θ, ℓ ) the set of renormalised trees θ ′ obtained from θ byinserting a right-fake cluster, together with its entering line, before ℓ ; see Figure 8. θ ′ = n +1 ℓ n nℓ θ = ℓn +1 Figure 8: The trees θ ′ of the set τ ( θ, ℓ ) obtained from θ when ℓ ∈ L ( θ ) enters a right-fake cluster. By construction one has V ( θ ; ε, β ) = A ℓ ( θ, x ℓ ) G n +1 ( x ℓ ) B ℓ ( θ ) X θ ′ ∈ τ ( θ,ℓ ) V ( θ ′ ; ε, β ) = A ℓ ( θ, x ℓ ) G n +1 ( x ℓ ) M n ( x ℓ ) G n ( x ℓ ) B ℓ ( θ ) , where we have used that x ℓ = x ¯ ℓ .Consider the contribution to ∂ ℓ V ( θ ; ε, β ) – see (A.11) – given by A ℓ ( θ, x ℓ ) G n +1 ( x ℓ ) ∂ L M n ( x ℓ ) G n +1 ( x ℓ ) B ℓ ( θ ) . (A.14)Call R n ( T ) the subset of R n such that if T ′ ∈ R n ( T ) the exiting line ℓ T ′ exits also the renor-malised right-fake cluster T ; note that the entering line ℓ of T must be also the exiting line ofsome renormalised left-fake cluster T ′′ contained in T ′ ; see Figure 9. T T ′′ T ′ n +1 n n n n +1 ℓℓ T ′ ℓ ′ T ′ Figure 9: A self-energy cluster T ′ ∈ R n ( T ). Define M n ( T, x ℓ ; ε, β o ) = X T ′ ∈ R n ( T ) V T ′ ( x ℓ ; ε, β ) . (A.15)25ence one has ∂ ℓ X θ ′ ∈ τ ( θ,ℓ ) V ( θ ′ ; ε, β ) + A ℓ ( θ, x ℓ ) G n +1 ( x ℓ ) ∂ ℓ X T ∈ RF n M n ( T, x ℓ ) G n +1 ( x ℓ ) B ℓ ( θ )= A ℓ ( θ, x ℓ ) G n +1 ( x ℓ ) M n ( x ℓ ) ∂ β G n ( x ℓ ) (1 + M n ( x ℓ ) G n +1 ( x ℓ )) B ℓ ( θ ) , (A.16)where we have used again that x ℓ = x ℓ . Thus, one can reason as in (A.13), so as to obtain thesum of three contributions, as represented in Figure 10. n +1 n n ≤ n − n + n +1 n n ≤ n − n +1+ n +1 n n ≤ n − n n n +1 Figure 10: Graphical representation of the three contributions arising from (A.16). Finally, consider the case in which ℓ is the exiting line of a renormalised left-fake cluster, T and the removal of T and ℓ ′ T creates a self-energy cluster.Set (for a reason that will become clear later) θ = θ and ℓ = ℓ . Then there is a maximal m ≥ m lines ℓ , . . . , ℓ m and ℓ ′ , . . . ℓ ′ m , with the following properties:(i) ℓ i ∈ P ( ℓ θ , ℓ i − ), for i = 1 , . . . , m ,(ii) n ℓ i = n + i < max { p : Ψ p ( x ℓ i ) = 0 } = n + i + 1, for i = 0 , . . . , m −
1, while n m := n ℓ m = n + m + σ , with σ ∈ { , } ,(iii) ν ℓ i = ν ℓ i − and the lines preceding ℓ i but not ℓ i − are on scale ≤ n + i −
1, for i = 1 , . . . , m ,(iv) ν ℓ ′ i = ν ℓ i , for i = 1 , . . . , m ,(v) if m ≥ ℓ ′ i is the exiting line of a left-fake cluster T i , for i = 1 , . . . , m − ℓ ′ i ≺ ℓ ′ T i − and all the lines preceding ℓ ′ T i − but not ℓ ′ i are on scale ≤ n + i −
1, for i = 1 , . . . , m ,(vii) n ′ m := n ℓ ′ m = n + m + σ ′ with σ ′ ∈ { , } .Note that one cannot have σ = σ ′ = 1, otherwise the subgraph between ℓ m and ℓ ′ m wouldbe a self-energy cluster. Note also that (ii), (iv) and (v) imply n ℓ ′ i = n + i for i = 1 , . . . , m − m ≥
2. Call S i the subgraph between ℓ i +1 and ℓ i , and S ′ i the cluster between ℓ ′ T i and ℓ ′ i +1 for all i = 0 , . . . , m −
1. For i = 1 , . . . , m , call θ i the renormalised tree obtained from θ by removingeverything between ℓ i and the part of θ preceding ℓ ′ i , and note that if m ≥
2, properties (i)–(vii)hold for θ i but with m − i instead of m , for all i = 1 , . . . , m − i = 1 , . . . , m , call R i the self-energy cluster obtained from the subgraph of θ i − between ℓ i and ℓ ′ i , by removing the left-fake cluster T i − together with ℓ ′ T i . Note that L ( R i ) = L ( S i − ) ∪{ ℓ i − } ∪ L ( S ′ i − ) and N ( R i ) = N ( S i − ) ∪ N ( S ′ i − ); see Figure 11.For i = 0 , . . . , m −
1, given ℓ ′ , ℓ ∈ L ( θ i ), with ℓ ′ ≺ ℓ , call P ( i ) ( ℓ, ℓ ′ ) the path of lines in θ i connecting ℓ ′ to ℓ (hence P ( i ) ( ℓ, ℓ ′ ) = P ( ℓ, ℓ ′ ) ∩ L ( θ i )). For any i = 0 , . . . , m − ℓ ∈ P ( i ) ( ℓ i , ℓ ′ m ), let τ ( θ i , ℓ ) be the set of all renormalised trees which can be obtained from θ i by replacing each left-fake cluster preceding ℓ but not ℓ ′ m with all possible left-fake clusters. Setalso τ ( θ m − , ℓ ′ m ) = θ m − . 26 = n +1 ℓ ≤ n S nℓ n T n +1 ℓ ′ T ≤ n S ′ ℓ ′ n +1 θ = n +1 ℓ R nℓ ≤ n ≤ nS ′ S Figure 11: The renormalised trees θ and θ and the self-energy cluster R in case 5 with m = 1 and σ = σ ′ = 0. Note that the set S ′ is a cluster, but not a self-energy cluster. Note that A ℓ m ( θ m , x ℓ m ) G n m ( x ℓ m ) V ( S m − ) = A ℓ m − ( θ m − , x ℓ m − ) , V ( S ′ m − ) G n ′ m ( x ℓ m ) B ℓ m ( θ m ) = B ℓ ′ Tm − ( θ m − ) , (A.17)and one among cases 1–4 holds for ℓ m ∈ L ( θ m ) so that we can consider the contribution to ∂ ℓ m V ( θ m ; ε, β ) (together with other contributions as in 3 and 4 if necessary) given by – see(A.10), (A.11) and (A.13) – A ℓ m ( θ m , x ℓ m ) G n m ( x ℓ m ) ∂ ℓ m − V R m ( x ℓ m ) G n ′ m ( x ℓ m ) B ℓ m ( θ m ) . Then one has A ℓ m ( θ m , x ℓ m ) G n m ( x ℓ m ) ∂ ℓ m − V R m ( x ℓ m ) G n ′ m ( x ℓ m ) B ℓ m ( θ m ) + ∂ ℓ m − X θ ′ ∈ τ ( θ m − ,ℓ m − ) V ( θ ′ ; ε, β )= A ℓ m − ( θ m − , x ℓ m − ) ∂ β G n + m − ( x ℓ m − ) (cid:0) M n + m − ( x ℓ m − ) G n + m ( x ℓ m − ) (cid:1) × B ℓ ′ Tm − ( θ m − ) , (A.18)and hence we obtain, reasoning as in (A.13), A ℓ m − ( θ m − , x ℓ m − ) G n + m − ( x ℓ m − ) ∂ β M n + m − ( x ℓ m − ) G n + m − ( x ℓ m − ) B ℓ ′ Tm − ( θ m − )+ A ℓ m − ( θ m − , x ℓ m − ) G n + m − ( x ℓ m − ) ∂ β M n + m − ( x ℓ m − ) G n + m ( x ℓ m − ) × B ℓ ′ Tm − ( θ m − )+ A ℓ m − ( θ m − , x ℓ m − ) G n + m − ( x ℓ m − ) ∂ β M n + m − ( x ℓ m − ) G n + m − ( x ℓ m − ) × M n + m − ( x ℓ m − ) G n + m ( x ℓ m − ) B ℓ ′ Tm − ( θ m − ) . (A.19)Then, for i = m − , . . . , B ℓ ′ Ti ( τ ( θ i , ℓ ′ i +1 )) := X θ ′ ∈ τ ( θ i ,ℓ ′ i +1 ) B ℓ ′ Ti ( θ ′ )27nd note that A ℓ i ( θ i , x ℓ i ) G n + i ( x ℓ i ) V ( S i − ) = A ℓ i − ( θ i − , x ℓ i − ) , V ( S ′ i − ) G n + i ( x ℓ i ) M n + i ( x ℓ i ) G n + i +1 ( x ℓ i ) B ℓ ′ Ti ( τ ( θ i , ℓ ′ i +1 )) = B ℓ ′ Ti − ( τ ( θ i − , ℓ ′ i )) . (A.20)Consider the contribution A ℓ i ( θ i , x ℓ i ) G n + i ( x ℓ i ) ∂ ℓ i − V R i ( x ℓ i ) G n + i ( x ℓ i ) M n + i ( x ℓ i ) G n + i +1 ( x ℓ i ) B ℓ ′ Ti ( τ ( θ i , ℓ ′ i +1 )) (A.21)obtained at the ( i + 1)-th step of the recursion. By (A.20) one has (see Figure 12) A ℓ i ( θ i , x ℓ i ) G n + i ( x ℓ i ) ∂ ℓ i − V R i ( x ℓ i ) G n + i ( x ℓ i ) M n + i ( x ℓ i ) G n + i +1 ( x ℓ i ) B ℓ ′ Ti ( τ ( θ i , ℓ ′ i +1 ))+ ∂ ℓ i − X θ ′ ∈ τ ( θ i − ,ℓ i − ) V ( θ ′ ; ε, β ) = A ℓ i − ( θ i − , x ℓ i − ) ∂ β G n + i − ( x ℓ i − ) × (cid:0) M n + i − ( x ℓ i − ) G n + i ( x ℓ i − ) (cid:1) B ℓ ′ Ti − ( τ ( θ i − , ℓ ′ i )) , (A.22)which produces, as in (A.19), the contribution A ℓ i − ( θ i − , x ℓ i − ) G n + i − ( x ℓ i − ) ∂ ℓ i − V R i − ( x ℓ i − ) G n + i − ( x ℓ i − ) × M n + i − ( x ℓ i − ) G n + i ( x ℓ i − ) B ℓ ′ Ti − ( τ ( θ i − , ℓ ′ i )) . (A.23) n + iℓ i ≤ n + i − S i − R i ℓ i − ≤ n + i − S ′ i − ℓ ′ i n + i n + i n + i +1 ℓ ′ Ti + n + iℓ i ≤ n + i − S i − ℓ i − n + i − n + iℓ ′ T i − ≤ n + i − S ′ i − n + iℓ ′ i n + i n + i +1 ℓ ′ Ti Figure 12: Graphical representation of the left hand side of (A.22).
Hence we can proceed recursively from θ m up to θ , until we obtain A ℓ ( θ , x ℓ ) G n ( x ℓ ) ∂ β M n − ( x ℓ ) G n ( x ℓ ) B ℓ ′ T ( τ ( θ , ℓ ′ ))+ A ℓ ( θ , x ℓ ) G n ( x ℓ ) ∂ β M n − ( x ℓ − ) G n +1 ( x ℓ ) B ℓ ′ T ( τ ( θ , ℓ ′ ))+ A ℓ ( θ , x ℓ ) G n ( x ℓ ) ∂ β M n − ( x ℓ ) G n ( x ℓ ) M n ( x ℓ ) G n +1 ( x ℓ ) B ℓ ′ T ( τ ( θ , ℓ ′ )) . (A.24)Once again, if we split ∂ β = ∂ N + ∂ L , all the terms with ∂ N M n − are contributions to M ∞ (0). We are left with the derivatives ∂ L M q ( x ; ε, β ), q ≤ n , when the differentiated propagatoris not one of those used along the cases 4 or 5; see for instance (A.16), (A.18) and (A.22).28ne can reason as in the case ∂ L V ( θ ; ε, β ), by studying the derivatives ∂ ℓ V T ( x ℓ ; ε, β ) andproceed iteratively along the lines of cases 1 to 5 above, until only lines on scales 0 are left. Inthat case the derivatives ∂ β G ( x ℓ ; ε, β ) produce derivatives ∂ β M − ( x ; ε, β ) = ε∂ β F ( β ) (seeRemarks 3.5 and 3.6). Therefore, for n = −
1, in the splitting (A.9), there are no terms with thederivatives ∂ ℓ , and the derivatives ∂ v can be interpreted as said in Remark A.1. It is also easyto realize that, by construction, each contribution to M ∞ (0; ε, β ) appears as one term amongthose considered in the discussion above. Hence the assertion follows. Remark A.5.
If we used a sharp scale decomposition instead of the C ∞ one, the proof abovewould be much more easier. More precisely, if we defined the (discontinuous) function χ ( x ) := ( , | x | ≤ , , | x | > , and consequently changed the definitions of ψ , and χ n , ψ n and Ψ n for n ≥
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