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KAM THEORY FOR EQUILIBRIUM STATES IN 1-D
STATISTICAL MECHANICS MODELS
RAFAEL DE LA LLAVE
Abstract. We extend the Lagrangian proof of KAM for twist mappings [SZ89, LM01] to show persistence of quasi-periodic equi- librium solutions in statistical mechanics models. The interactions in the models considered here do not need to be of finite range but they have to decrease sufficiently with the distance. When the interactions are range R, the models admit the dy- namical interpretation of recurrences in (R)^2 R. Note that the small perturbations in the Lagrangian are singular from the dynamical systems point of view since they may increase the dimension of phase space. We show that in these models, given an approximate solution of the equilibrium equation with one Diophantine frequency, which is not too degenerate, there is a true solution nearby. As a conse- quence, we deduce that quasi-periodic solutions of the equilibrium equation with one Diophantine frequency persist under small mod- ifications of the model. The main result can also be used to validate numerical calcula- tions or perturbative expansions. We also show that Lindstedt series can be computed to all orders in these models.
- Introduction It is well known that equilibrium configurations in one-dimensional ferromagnetic models are equivalent to orbits of twist mappings. For example, the equilibrium configurations Frenkel-Kontorova model are equivalent to orbits of the standard map. See [MF94b] as well Sec- tion 1.1 and the references there. In spite of the fact the two problems (orbits of twist maps and equi- librium states of ferromagnetic models), the intuition and the possible natural generalizations are very different for twist mappings and for equilibrium configurations. In this paper, we will consider some mod- els that are very natural from the point of view of statistical mechanics but for which there is no easy equivalent in dynamical systems. The main result of this paper is Theorem 1, which extends the La- grangian proof of the twist mapping theorem in [SZ89, LM01] to some
models in statistical mechanics in which the interactions are not nearest neighbor and indeed need not be finite range. From the point of view of statistical mechanics, this generalization is natural. In the statistical mechanics motivation of the Frenkel-Kontorova models they are used to describe the deposition of a material over a substratum of a differ- ent material. Frenkel-Kontorova models are also natural as description of spin systems over a one-dimensional crystal. In these motivations it is natural to consider extensions of the classical Frenkel-Kontorova models in which all the sites interact, even if the interaction decreases with the distance. Adding a small interaction term for next nearest neighbor interac- tions, even if it seems like a small perturbation from the Lagrangian pic- ture is a very singular perturbation from the point of view of mechan- ics. Adding a next-nearest neighbor to the Frenkel-Kontorova model makes the phase space 4 dimensional, rather than 2 dimensional and some of the terms are very large. We will give more details of this in Section 1.1.4. If one is interested in ground states which are quasi-periodic orbits of the model, it is natural to expect that they persist under small (La- grangian modifications of the model). This is accomplished by KAM theory. Nevertheless, as we have pointed out, from the dynamical sys- tems point of view, the perturbations are not small. Adding a next nearest neighbor interaction, – no matter how small – even changes the dimension of the phase space. In this paper, we will show that quasi-periodic orbits persist under perturbations of the models, provided that some non-degeneracy con- ditions are met. In the case that the interactions are nearest neighbor, this result is the standard twist theorem. When we consider infinite range interactions the equilibrium configurations cannot be considered as orbits of evolution problems, in particular, they cannot be formu- lated as orbits of Hamiltonian systems and the proofs of KAM the- orems based in Hamiltonian formalism do not apply. Hence, we will use the Lagrangian formulation of the KAM theorem. Other KAM theorems which have a Lagrangian formulation but not a Hamiltonian one occur in [Mos88]. In contrast, there are KAM proofs in situa- tions where there is a Hamiltonian formalism but not a Lagrangian one [JdlLZ99, DdlL00, GEHdlL05]. When the variables at each site in the statistical model are one di- mensional, the interaction is of range R and there are some convexity properties in the model – which amount to ferromagnetism or twist
- the equilibrium configurations (critical points of the action) corre- spond to orbits of “monotone recurrences” in R^2 R^ and were considered
to validate the algorithm to produce the solution. One just needs to verify that the equilibrium equation is satisfied with an small error. In the case of numerical computations, reliable upper bounds on the error of the solution are computable (e.g. using interval arithmetic) these type of a-posteriori theorems are the basis of computer assisted proofs. See [Ran87, dlLR91, dlLR90, JdlLZ99, CC95, CC97] for computer as- sisted proofs in Hamiltonian systems and [CdlL05] for computer explo- rations in the systems considered in this paper. From the point of view of numerical explorations it is interesting to remark that the method presented here provides with an algorithm in which the Newton step has a cost of only N log(N ) in the number of discretization points of the torus (this is in common with the algorithms in [SZ89, dlLGJV05]). In contrast, most of the implementations of KAM methods use algo- rithms of order N 3 operations per step. We will present some remarks on numerical algorithms for the quasi-periodic orbits in Section 7.1. Relatedly, as already pointed out in [Mos66b, Mos66a, Zeh75], an “a posteriori” result for analytic functions with good enough quantita- tive bounds implies results for finite differentiable functions. We note that in the papers above, the finitely differentiable results include the assumption that the system is close to an integrable analytic one, but we have removed this assumption here. We hope to come back to this problem. As a technical remark, we note that the proof of KAM theorem we present is not based on the transformation theory that appears in many proofs of KAM theorem, but rather is based on an iterative procedure in which corrections are added. The method is very similar to that of [Mos88, SZ89, LM01]. indeed, some of the identities we use will lead to efficient numerical implementations. We refer to [CdlL05] for a deeper discussion of numerical issues. See also Section 7.1. Besides the KAM theory, we will also discuss some complementary results such as study the existence of Lindstedt series and their con- vergence.
1.1. Models considered. In Section 1.1.1, we discuss twist mappings of the annulus which is the best known and simplest model. For twist mappings, in the statistical mechanics language, the interactions are nearest neighbor and the variables at each site are one dimensional. In Section 1.1.3, we discuss statistical mechanics models of spin chains which include the possibility of long range interactions. The models we consider are models in which the sites are arranged in a one-dimensional lattice Z. That is, a configuration is a mapping from Z :→ R. From the point of view of statistical mechanics, it would have
been natural to consider models defined in higher dimensional lattices or even on trees. These models on higher dimensional lattices do not admit interpretation as dynamical systems and they are not accessible to the methods in this paper. We hope to come back to this problem.
1.1.1. Twist mappings of the annulus. It is well known in Hamiltonian mechanics that orbits {xn}n∈Z ⊆ R of a twist map of the annulus A ≡ T × R can be identified with critical points of the functional given by the formal sum
(1) S({xn}) =
n∈Z
S(xn, xn+1)
where S is the so-called generating function of the map. The standard assumptions for the generating functions of the twist mappings of the annulus are:
(2) ∂x∂yS(x, y) ≤ C < 0
(3) S(x + 1, y + 1) = S(x, y)
The condition (2) is called the twist condition in dynamical systems (it is called ferromagnetism in statistical mechanics) and (3) is implied by the assumption that the xn can be interpreted as angles and that the variational principle has a physical meaning, The meaning of con- dition (3) is that, even if relative phases between two sites may be important for the variational principle, the variational principle of the model is unaffected by to addition of one unit to the states of all the sites. (i.e., if {xn}n∈Z is an orbit, so is {xn + 1}n∈Z.) In some physical interpretations, when the x are angles, it would have been reasonable to assume S(x + 1, y) = S(x, y + 1) = S(x, y) which clearly implies (3). For our purposes the assumption (3) is enough and we will not consider stronger assumptions even if they are natural for the physical models. We recall that {xn} is an equilibrium configuration i.e. a critical point of (1) if and only if
(4) ∂xn S(xn, xn+1) + ∂xn S(xn− 1 , xn) = 0 ∀n ∈ Z
We will refer to the equations (4) as the Euler-Lagrange equations of (1). They are obtained by computing (formally since the sum (1) defining S is only a formal sum) ∂xn S({xn}) by noting that xn only appears in two terms in S({xn}). Note that, even if (1) is just a formal sum, the equilibrium equation (4) is a well defined system of equations. For our studies, the starting point will be the equations (4) and the relation with variational problems serves only motivation. Hence, we will not try to make more precise the well known connection between
The invariance under translation implies that it suffices to consider models of the form:
(7) S({xn}n∈Z) =
L∈N
k∈Z
HL(xk,... , xk+L)
Remark 1. There is a small ambiguity in the definition of the models. For example, the model with Hi = 0, i 6 = 2 and H 2 (x 0 , x 1 , x 2 ) = F (x 0 , x 1 ) + F (x 1 , x 2 ) is equivalent to the model with Hi 6 = 0, i 6 = 1 and H 1 (x 0 , x 1 ) = F (x 0 , x 1 ). This ambiguity will not affect any of our reasonings. We will just state some assumptions on the interaction and draw conclusions from them. So that the results apply for one model provided that there is one of the equivalent interactions that satisfy the assumptions of the results here.
As a point of notation, we observe that it is natural to label the argu- ments of the interaction staring with zero. Hence, write HL(x 0 ,... , xL). We denote ∂iHL = ∂xi HL. Hence, the indices of derivatives run from 0 to L. The analogue of (3) is
(8) HL(x 0 ,... , xL) = HL(x 0 + 1,... , xL + 1)
This indicates that, even if the interactions may depend on the relative phases at different sites, they are independent of a global change of phase by an integer over all the sites. The analogue of (2) that appears in the calculus of variations (e.g. in [MF94b, CdlL98]) is:
∂i∂j HL ≤ 0 i 6 = j ∂ 0 ∂ 1 H 1 ≤ C < 0
which just requires that the two body interaction is strictly ferromag- netic while the many body interactions are not anti-ferromagnetic. In this paper, we will require some weaker assumptions than the assumptions (9) in the calculus of variations. As usual in KAM theory, we will not need positivity assumptions but rather invertibility in a neighborhood of the approximate solution. We will assume that
(10) |det(∂ 0 ∂ 1 H 1 )| > C > 0
and that the longer range interactions are small. See assumption H in Theorem 1 for a precise formulation of the assumption. We will require that the interactions HL decrease fast enough with L and that the nearest neighbor interactions dominate the rest of the interactions.
That is, we will require that
(11)
L> 2
‖HL‖qL^4 < α
The precise definition of the norm used in (11) will be postponed till Section 2.3.2. We just anticipate that the norm is an analytic norm in a neighborhood, hence, it controls derivatives. The constant α will be related to the properties of the nearest neighbor interaction.
1.1.4. An example. A specific model that can serve as motivation is a Frenkel-Kontorova model with a range 2 interaction. The formal variational principle for the model is:
S({xn}) =
n
(xn+1 − xn − a)^2 +
A
(xn+2 − xn − b)^2
ε 4 π^2
(cos(2πxn) − 1)
The equilibrium equations for the Lagrangian (12) are:
(13) (xn+1 − 2 xn + xn− 1 )+A·(xn+2 − 2 xn + xn− 2 )+
ε 2 π
sin(2πxn) = 0
The equation (13) defines a 4-dimensional map for A 6 = 0. In that case, using (13), if we are given xi− 2 , xi− 1 , xi, xi+1, there is one and only one xi+2 so that (13) is satisfied. Hence, associated to (13), there is a mapping that given
Xi = (xi− 2 , xi− 1 , xi, xi+1)
produces Xi+1 = (xi− 1 , xi, xi+1, xi+2). Notice that, when A = 0, the equation (13) reduces to the – two dimensional – standard map. Hence, when A is small, (13) can be considered as a singular perturbation of the standard map. Note that the dimension of the phase space change. On the other hand, it seems quite natural that, for small A, the variational principle is very similar to the solutions of the unperturbed system. As we will see, given the fact that our main result is an a-posteriori result, it will follow that the solutions produced are continuous, indeed, analytic in A. Other generalizations of (12) which are interesting are:
S({xn}) =
n
(xn+1 − xn − a)^2 +
j≥ 2
Aj 2
(xn+j − xn − bj )^2
μ 4 π^2
(cos(2πxn) − 1)
have a variational motivation will not be used explicitly in our study of them, even if some of the identities we will use can be traced back to the variational structure. In the case of finite range interactions, the assumptions (2) show that if we are given xi−R, xi−R+1,... , x 0 ,... , xi+R− 1 there is one and only one xi+R which satisfies the equations (17) for i = R. In other words, given xi−R, xi−R+1,... , x 0 ,... , xi+R− 1 , the equation (2) deter- mines xi−R+1, xi−R+2,... , x 0 ,... , xi+R, Hence, we can think of (17) for interactions of range R as defining a dynamical system in (R)^2 R. Note that, because of the invariance under translation of the interaction, the dynamical system defined above is autonomous. The systems thus defined, have been called “monotone recurrences” in [Ang90] when the second derivative satisfies (2). These monotone recurrences enjoy remarkable dynamical properties such as an Aubry- Mather theory on existence of quasi-periodic orbits with all frequencies. See [Ang90, CdlL98]. In this paper we will consider KAM results for models such as those considered in Section 1.1.3. We will prove that if there is a periodic function u such that Eω[u] defined in (21) is small enough (with respect to other properties of the function and the rotation ω) then, there is another function u∗^ for which E[u∗] = 0. Moreover we can bound u − u∗^ in terms of the original error of the approximation. Since the precise formulation of the result requires definitions of the norms used to measure the error and to formulate some non-degeneracy assump- tions, we defer the formulation of Theorem 1 till we have introduced enough notation and definitions.
- Statement of results We will consider models of the form (7) but with the variables xn being vectors in R as indicated in Section 1.1.
2.1. Plane-like configurations, hull functions. We will be inter- ested in equilibrium configurations {xn}n∈Z ⊂ R that can be written as:
(18) xn = h(nω)
where ω ∈ R and h : R → R satisfies
(19) h(x + e) = h(x) + e ∀e ∈ Z
Notice that because of the periodicity assumption (19) h can be considered a map from the torus T = R/Z to itself. In our applications, we will assume that h is a diffeomorphism of the torus.
We will use the notation h(θ) = θ + u(θ) and often work with the u function which is periodic.
Remark 4. In solid state physics the function h is often referred as “hull” function of the configuration. In dynamical systems, the func- tion h gives a semi-conjugacy between the dynamics and a rotation on a torus. KAM theory always looks for solutions of the form (18).
An obvious consequence of the form (19) is that |xi − ωi| ≤ ‖u‖C 0. Hence, the configurations that can be represented by hull functions are almost linear (usually called “plane-like” in homogenization theory). Our assumption that h is a diffeomorphism of the circle implies that ‖u‖C 0 ≤ 1. The property that a configuration is given by a hull function is equiv- alent to the fact that it satisfies the so-called “Birkhoff property” or “non-intersection property” or “self-conforming property”, which was introduced in [Mat82, ALD83]. This property is very important in variational calculus. Under hypothesis such as the maximum principle (which is implied by (2)) all periodic minimizers satisfy the Birkhoff property.
2.2. Equilibrium equations for hull functions. We note that the Euler-Lagrange equations (17), that express that the configuration is in equilibrium, evaluated on a configuration described by a hull function h = Id +u are just
L
k+j=i;j=0,...,L
∂j HL
h(θ + kω),... , h(θ + (k + j)ω),...
,... , h(θ + (k + L)ω)
L
k+j=i
∂j HL
θ + kω + u(θ + kω),... , θ + (k + j)ω + u(θ + (k + j)ω),...
,... , θ + (k + L)ω + u(θ + (k + L)ω)
the torus. Therefore, if u is a continuous solution of the equilibrium equation (20), so is
(24) uσ = u(θ + σ) + σ
for any choice of σ ∈ T. The fact that (24) is a solution of (20) for all σ can be also checked by substituting directly. It is valid for all ω, including rational ones. In summary, in general, the quasi-periodic solution of the equilibrium equations are not unique and indeed appear in one parameter families. This corresponds to the choice of the origin of time in the of the torus parameterization, which is a symmetry of the problem. The diffeomor- phism of the torus hσ corresponding to uσ is just the translation of the origin in h. That is h(θ + σ) = hσ(θ). The fact that the solutions come in families will play an important role in the study of the equations of equilibrium. Using (24), we see that given one u, we can find a unique σ such that ˜uσ has zero average. An important consequence of the symmetry of the problem under translation and changes of phases is the identity:
(25) Euσ = E[u](θ + σ).
This identity will play an important role in in Section 3.3. Actually, the symmetry under changes of the origin of the phases is true not only for the equilibrium equations but also for the variational principle (22). We have
(26) P[uσ] = P[u].
This identity will also play a role in Section 3.3. Given the importance of the symmetry under shifts in the formulas, it is important for the analysis that the norms we use are also invariant under shifts. The norms we introduce in Section 2.3.1, are indeed in- variant under shifts in the parameterization. The non-degeneracy con- ditions we will consider in Theorem 1 are also invariant under changes of the origin of the phase. Hence, we can assume without loss of gen- erality that the approximate solutions we consider are such that the average of E[u] is zero.
2.3. Formulation of the main result. The main result Theorem 1 below will be of the form that if there is a function which makes E[u] small, and which satisfies non-degeneracy conditions, then close to it, we can find an exact solution close to the approximate one. This solu- tion is unique – up to obvious symmetries — in a small neighborhood of the approximate solution we started with.
This formulation of KAM theorems, as a-posteriori results with- out reference to integrable systems have been common since [Mos66b, Mos66a]. See [dlL01] for a historical review. In order to formulate that periodic functions are small, we will intro- duce appropriate norms in Section 2.3.1 and study some of the elemen- tary properties. In Section 2.3.2, we will describe norms on interactions.
2.3.1. Some families of norms in analytic periodic functions. We will denote by
Tρ^ =
z ∈ C/Z | | Im z| ≤ ρ
We denote by Aρ the Banach space of functions from Tρ^ taking values in CL^ (when L is understood from the context we will omit it from the notation) such that
- They are real for a real argument.
- They are holomorphic in the interior of Tρ^ and continuous on Tρ.
We consider Aρ endowed with the norm
(27) ‖u‖ρ = sup z∈Td,ρ
|u(z)|.
This norm makes Aρ a Banach space. Since the set of L that we will consider will be unbounded, it is important to specify that the norm we will use in CL^ is the supremum of the coordinates. Of course, for finite L all the norms in CL^ are equivalent, but the constants given the equivalence could be unbounded as L grows, so that one needs to pay attention to the choice of norms. We have not optimized the choice of norms in CL. So that it is quite possible that other choices could lead to sharper results. This will also become important when we choose norms in the space of interactions in Section 2.3.2. It is clear that if ⊗ is a bilinear operation of norm 1 (e.g an in- ner product, multiplication or matrices with their operator norms), we have:
(28) ‖u ⊗ v‖ρ ≤ ‖u‖ρ · ‖v‖ρ
We also recall that we have Cauchy estimates for derivatives and for the Fourier coefficients in terms of the family of norms (27). We will write the Fourier series of a function u ∈ Aρ as
u(θ) =
k∈Z
uk e^2 πikθ
It is interesting to compare this with the derivative with respect to the variable θ of the function f˜ η = f ◦ η(θ). We have (^) dθd f η = f ′^ ◦ η(θ). Even if D f˜ and (^) dθd f η are conceptualy very different, they have the same formula. This will play an inportant role later leading to interesting cancellations.
2.3.2. Regularity properties of the interactions. We will assume that the interactions HL are defined on a complex set and analytic there. We will need that the functions HL are analytic in a complex domain large enough to allow their evaluation on the interactions we are con- sidering. The goal of this section is to define norms in the interactions that measure their sizes so that we can state precisely the results. For the spaces that we will consider, it will be quite important to con- sider not only the sizes but also the domains. Even if we will not consider this in this paper, we hope that this will allow to extend the results to finite differentiable interactions using the techniques of [Mos66b, Mos66a, Zeh75] which characterize finite differentiable func- tions by their approximation properties. Thinking about this further extension, the result we will present will pay special attention to the dependence of the smallness conditions and the change required in the conclusions with respect to the domain of the function. In the applications to physical problems, the assumption that the interactions are analytic in a large domain will be often satisfied. Since our result will be formulated for an approximate solution, it is natural to consider domains for the interaction which are defined in the a neighborhood of the range of of a configuration. Given u ∈ Aρ, we consider
DL,u,δ =
(z 0 ,... , zL) ⊂ (C)L+
∃ θ ∈ Td,ρ, |zi − h(θ + iω)| ≤ δ, i = 0,... , L
As usual, we suppress the dependence in ω from the notation unless it can cause confusion. Again, we recall that the norms we use in CL^ are the supremum norms. Since our configurations will be such that they map real values into real values, in some applications it suffices to consider the simpler do- mains
(32) D˜L,δ =
(z 0 ,... , zL) ⊂ (C)L+
∣ |^ Im(zi)| ≤^ δ
Clearly,
(33) DL,u,δ ⊂ D˜L,‖u‖ρ+δ.
Since L will be unbounded, we will need to estimate the dependence in L of several standard results such as Cauchy estimates and the like. The proofs we present are not optimized very seriously. With the choice of supremum norm in CL+1, we have sup θ∈Tρ
u(θ), u(θ+ω),... , u(θ + Lω)
− (˜u(θ), u˜(θ + ω),... , u˜(θ + Lω)| ≤ ||u − u˜||ρ
Therefore, we have
HL((u(θ), u(θ+ω),... , u(θ + Lω)) − HL(˜u(θ), u˜(θ + ω),... , u˜(θ + Lω)| ≤ ||DHL||L∞ ||u − u˜||ρ
On the other hand, we note that the estimate of the norm of a derivative in terms of the partial derivatives does have a dependence on L. |DHL| ≤ (L + 1) max j=0,...L |∂xj HL|
The Cauchy bounds may also have a dependence in L. In Lemma 2, we state the version of Cauchy estimates which we will use (even if we do not know if it is optimal).
Lemma 2. If Ω ⊂ Ω˜ and dist(Ω, CL+1^ − Ω)˜ ≥ δ we have:
(36) ||DHL||Ω ≤ C(L + 1)δ−^1 ||HL||Ω˜
Proof. Given z ∈ Ω we can find circles γi centered in zi with radius δ such that γ = (γ 0 ,... , γL) ⊂ Ω. Cauchy formula gives:
HL(z) =
(2πi)L+
γ 0
dw 0 · · ·
γL
dwL
HL(w) (w 0 − z 0 ) · · · · (wL − zL)
Therefore, given a direction η, DηHL, the directional derivative is:
DηHL(z)
=
(2πi)L+
γ 0
dw 0 · · ·
γL
dwLHL(w)
[ (^) η 0 (w 0 − z 0 )^2 · (w 1 − z 1 ) · · · · (wL − zL)
ηL (w 0 − z 0 ) · (w 1 − z 1 ) · · · · (wL − zL)^2
]
where C is a combinatorial constant that will be made explicit during the proof. H3) Assume that the inverses indicated below exist and have the in- dicated bounds. H3.1) ‖(∂ 0 ∂ 1 H 1 )−^1 (u(θ), u(θ + ω))‖ρ ≤ T. H3.2) Define C 0 , 1 , 1 (θ) = ∂ 0 ∂ 1 H 1 (u(θ), u(θ + ω))h′(θ)h′(θ + ω) (See (56) later for a justification of including the subindices in C) Assume that: (∫
T
C 0 −,^11 , 1
≤ U.
H4) The following bounds measure the non-degeneracy and the accuracy with which the approximate solution solves the problem. H4.a) ‖ Id +u′‖ρ ≤ N+, ‖(Id +u′)−^1 ‖ρ ≤ N−. H4.b) ‖E[u]‖ρ ≤ ε. Assume furthermore that the above upper bounds satisfy the following relations: H4.i) T α < 1 / 2 , U T α < 1 / 4 H4.ii) ‖u‖ρ + ρ ≤ 12 δ H4.iii) ε ≤ ε∗(N+, N−, τ, α, T, U, δ)κ^4 ρ^4 τ^ +A where ε∗^ is a strictly positive function which we will make explicit along the proof. The function ε∗^ makes quantitative the relation between the smallness conditions and the nondegeneracy conditions. A ∈ R+^ is a number which will also be made explicit along the proof. Then, there exists a periodic function u∗^ ∈ Aρ/ 2 such that
(40) E˜ω[u∗] = 0 and
u∗^ = 0. Moreover (41) ‖u − u∗‖ρ/ 2 ≤ Cκ−^2 ρ−^2 τ^ −AT ‖E[u]‖ρ The function u∗^ is the only function in a ball of radius centered at u of radius∫ Cκ−^2 ρ−^2 τ^ −AT ‖E[u]‖ρ satisfying (40) and the normalization. u∗^ = 0.
The most important hypothesis of Theorem 1 is H4.iii) which re- quires that the test function satisfies the equilibrium equation with an
error with is sufficiently small with respect to the other quantities that measure the degeneracy of the problem. The hypothesis H3.1 is the twist condition for the nearest neighbor interaction. The condition H3.2 defines some quantity related to the twist. The quantity in H3.2 appears in H4.i. The importance of H4.i will come from the fact that it will allow us to show that the non- degeneracy condition in the nearest neighbor interaction allows us to control the non-degeneracy of all the other interactions. Note also that the dependence of the smallness conditions on the domain of analyticity ρ is a power and also that the effect of the an- alyticity domain in the correction (41) is also a power of ρ. It is well known to experts that such a result can be used to prove a finitely differentiable result. We hope to come back to this problem. We also note that such a result implies that there is Lipschitz de- pendence on the Diophantine frequencies. (Just take the solution for a frequency as an approximate solution for a nearby frequency). Indeed, by combining a result such as the one above with the Lindstedt series, one gets that the dependence on the frequencies is smooth in the sense of Whitney. See [Van02, dlLV00]. Theorem 1 implies the result of persistence of solutions for quasi- integrable systems. In our case, the integrable systems are the linear systems:
(42) H L^0 (x 0 ,... , xL) = AL
|x 0 − xL|^2.
For systems of the form (42), given ω ∈ R, xn = ωn is a solution, which corresponds to u = 0. If we consider a system HL = H L^0 + μFL. If the perturbation satisfies ‖F ‖L,δL^2 < ∞, for |μ| sufficiently small, we can consider u = 0 as an approximate solution of the system. Note that, to verify the hypothesis of Theorem 1, ε – the error – is bounded by μ and the non-degeneracy conditions remain uniformly bounded as μ approaches zero. Hence, we can obtain the existence of solutions with corresponding frequencies. A more detailed discussion of results for quasi-integrable systems happens in Section 5.1.
- Description of the proof of Theorem 1 The proof is based on an iterative procedure very similar to that of [Mos88, SZ89, LM01]. Given a function u so that ‖E[u]‖ρ is sufficiently small compared with other properties of the function, the iterative procedure constructs another function ˜u defined in a smaller domain and such that
(43) ‖E[˜u]‖ρ′^ ≤ CT A(τ, N+, N−, T, U )κ−^2 (ρ − ρ′)−^2 τ^ −^1 (‖E[u]‖ρ)^2
