1 Divergent series summation in Hamilton Jacobi equation (resonances) G. Gentile, G.G. α = ε α f(α)

Transcription

1 Divergent series summation in Hamilton Jacobi equation (resonances) Eq. Motion: G. Gentile, G.G. α = ε α f(α) α α 2 α l Representation of phase space in terms of l rotators.

2 2 α = (α,...,α l ) T l f(α) analytic or f(α) = ν Z l f ν e iν α, f ν 0 if ν > N Equations of motions α = ε α f(α)

3 3 Resonance of order s with frequencies ω 0 R r (unperturbed) motions with rotation ω = (ω 0,0) R r R s, l = r +s ω 0 ν > C ν τ, 0 ν Zr α def = (γ,β) T r T s, t (γ +ω 0 t,β) γ= fast angles, β= slow angles

4 4 Hamilton-Jacobi: Find h(ψ) def = ( ) g(ψ), ψ T k(ψ) r, β 0 T s with g(ψ),k(ψ) R r R s so that α = ε α f(α), Bα (γ,β), is solved by γ = ψ +g(ψ), β = β 0 +k(ψ), ψ ψ +ω 0 t This means ( ) (ω 0 ψ ) 2 g(ψ) = ε k(ψ) α f ( ψ +g(ψ),β 0 +k(ψ) ) Resonance dimensionality drop from l to r β f(β 0 ) = 0, Let f(β) def = dγ (2π) f(γ,β). Condition det 2 r ββ f(β 0) 0

5 5 Proposition: power series solution (elementary) ν 5 u root θ ν ν 0 ν ν 2 ν 3 ν 4 Notation: f(γ,β) def = ν Z r f ν(β)e iγ ν, ν 6 ν 7 ν 8 ν 0 ν j f ν (β 0 ) def = iν j f ν (β 0 ), j f ν (β 0 ) def = βj f ν (β 0 ), J f ν (β 0 ) def = j0,...,j p f ν (β 0 ), J = (j 0,...,j p ) trivial node v def = one entering line and 0 harmonic label (ν v = 0). ν 0 ν i i 0 j 0 j Let M 0;i0 j 0 def = ( ) ε 2 f(β 0 )

6 6 () To node v attach a harmonic ν v Z r (2) To line λ v v attach current ν(λ) = w v ν w (3) and two component labels j λ,j λ j j v v j 0 j j2 Component labels around v v j p v J v = (j 0,...,j p ) and Jv f νv (β 0 ) are defined. (4) Value : Val(θ) = k! ( ε Jv f νv (β 0 ) )( ) g jλ j λ v ( ) linesλ def δ g ij = ij 0 0, or g (ω ν(λ)) 2 ij = 0 (ε β 2f(β 0)) if ν(λ) = 0 h ν θ Val(θ) : no trivial node with 0 incoming current Estimate: h (k) ν bb k ε /2 kk! 2τ!! Results:

7 7 Theorem: The tree series can be rearranged to yield a convergent series representation of h, hence its existence, in ε > 0, hyperbolic case ( 2 ββ f(β 0) < 0) analyticity region common to all ε > 0 estimate k! 2τ at ε = 0 4 E ( ε 0,0] has open dense complement but 0 is a (Lebesgue) density point

8 8 Fig.4: ε E; E dense at 0. The figure illustrates the analyticity domain D(ε) associated with a single ε E, represented by the dot, ( elliptic case ). The cusp at the origin is quadratic. ε-plane 4 Need k! 2 for Borel summability (r = 2?). Question: is there uniqueness? Are others results the same? (Delshams, Llave, Zhou l = 3,r = 2, Treshev ε > 0 only)

9 9 Inserting a trivial node with 0 harmonic ( ν 0) get ν 0 ν i i 0 j 0 j δ ij (ω ν) 2 δ ii 0 (ω ν) 2 ( M0;i0 j 0 δ j0 j M 0;i0 j 0 = ( ε 2 i 0 j 0 f 0 (β 0 ) Let M 0;i0 j 0 def = ( ) ε 2 f(β 0 ) ) (ω ν) propagator modification 2 ), f 0 (β 0 ) f(β 0 )

10 0 Can form chains of trivial nodes (large values k! 2τ ) i i 0 j 0 i j j 6. δ ij (ω ν) 2 (ω ν) 2 ( M0 (ω ν) 2 ) k ij Simplify : NO trivial nodes; price : (ω ν) 2 (ω ν) 2 k=0 BUT z = M 0 (ω ν) 2 <? NO! ( M0 (ω ν) 2 ) k (ω ν) 2 M 0

11 so we are using k=0 zk = z, z, e.g. 2 k = = k=0 dont mind! Eventually: Check that the h is solution must be done If accepted ( ) k (ω ν) 2 (ω ν) 2 k=0 M0 (ω ν) 2 (ω ν) 2 M 0 M 0 = ( ) ε β 2f(β 0) gives ε > 0 easier than ε < 0. For ε < 0 expect to exclude ε s.t ω ν = ± εµ j.

12 2 PROBLEM: there are LOTS of other chains! They cause values k! η, η > 0 IDEA: eliminate them by resummation : left with convergent series KEY: Siegel s theorem Given a tree θ let N n be the number of lines of scale n: i.e. s.t. 2 n < C ω ν 2 n+, n, < C ω ν, n = 0 n = 0,,... IF no pair lines λ < λ with ν(λ ) = ν(λ) with only lower scale intermediates THEN N n 4N2 n/τ k

13 3 N n 4N2 n/τ k v (ω 0 ν) 2 C2k 2 2nNn C 2k( 2 2n(4N2 n/τ ) ) k n=0 n=0 Trivial bound (ε > 0): J vf νv (β 0 ) v v N Jv F k N 2k F k number of harmonics ν: (2N +) k, number of trees k k Convergence: ε < ( N 2 C 2 (2N +) l 3 F 2 8N n n2 n/τ )

14 4 Multiscale analysis: Organize the lines of θ into clusters Definition: A cluster of scale n is a maximal connected set of lines of θ with scales p n and with one line at least of scale n. 7 Self energy clusters ν in = ν out ν out ν in 8

15 5 Eliminate self energy clusters by resummations Necessary multiscale analysis to avoid overlapping divergences First identify the self energy clusters of scale [0] i.e. with x def = Cω ν with x 2 and resum all chains giant tree ν ν ν νgiant tree Key remark: each s.e. cluster does not contain s.e. clusters Summing over the contents of each s.e. cluster convergent sum by Siegel s lemma. 9

16 6 Summing over s.e. clusters of scale [0] leads to modify propagators of lines on scale [ ] giant tree ν ν ν νgiant tree 9 g [ ] (x) = x 2 M 0 x 2 M 0 (M x 2 ) n M 0 n=0 which becomes x 2 M 0 M and graphs simplify with no s.e. subgraphs of scale [0]. Iterate! at every step only graphs with no s.e. subgraphs have to be considered; convergent additions made on propagators

17 7 In the hyperbolic case no real new problems arise. In the elliptic the situation is very different. As the scale decreases the scale 2 n ε is reached and x 2 M 0 M... M n can vanish (a) More values of ε excluded (b)the successive scales must be measured by the size of x 2 M 0 M...; analysis becomes delicate:

18 8 DIFFICULTY: even in the hyperbolic case it is necessary to check that the renormalized propagators have the same size as the bare ones Siegel s lemma applies only to graphs in which the propagators size is of the order of x 2 (ω ν) 2. Not automatic but checked via the cancellation mechanism of the KAM theory: this time the cancellations are only partial but still enough.

19 9 OPEN PROBLEM: uniqueness (and relation with alternative existence results) complex εplane

20 20 O. Costin, G. Gentile, A. Giuliani, GG For r = 2 possible τ =. In this case Borel summability for ε > 0. Uniqueness, within the framework of multiscale analysis, is proved. F admits a Taylor exp. at 0 F(η) k=2 F kη k then F B (p) k=2 F k p k (k )!. If series for F converges (i.e. analytic at 0) then F(η) = + 0 e p/η F B (p)dp

21 2 Definition: η F(η) be (i) analytic in a disk tangent at 0 with admits asymptotic Taylor series at 0 (ii) F B (p) real analytic for p > 0, and growing at most exp. as p +, (iii) it can be expressed, for η > 0 small enough, as F(η) = + 0 e p/η F B (p)dp. Theorem: If η = ε, h(ψ) is Borel summable in η for ε > 0. The h constructed is the unique possible within the class of Borel summable functions. BUT this does not exclude existence of other less regular quasi periodic motions.

22 22 Regularize h(ψ): h (N) defined as h with sum to trees with only lines of scale n N. BT of h (N) (ψ,η) is entire function that can be written for p real and positive as (h (N) ) B (ψ,p) = L h (N) (ψ,p) = η N +i η N i e pz h (N) ( ψ, z where η N is the convergence radius of h (N) (ψ,η). Key: ω 0 ν 2 εm 0 is the paradigm of a Borel summable function. Inductively: h (N) (ψ, z ) is analytic for z > 2η 0. Integral can be shifted to a contour on the vertical line with abscissa ρ > 2η 0, N-independent. ) dz 2πi,

23 23 References G.Gentile, GG: Hyperbolic low-dimensional invariant tori and summations of divergent series, Communications in Mathematical Physics, 227, , 2002, doi:0.007/s G.Gentile, GG: Degenerate elliptic resonances, Communications in Mathematical Physics, 257, , 2005, doi: 0.007/s O.Costin, A.Giuliani, G.Gentile, GG: Borel summability and Lindstedt series, Communications in Mathematical Physics, 269, 75-93, 2007, doi:0.007/s A. Jorba, R. de la Llave, M. Zou: Lindstedt series for lower-dimensional tori, in Hamiltonian systems with three or more degrees of freedom, NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, Editors A. Delshams, C. Simó, 533, 5 67, 999, Kluwer Acad. Publ.

24 24 Precise formulation in HJ (fixed ω (C,τ)-Diophant.) F(A,α) def = 2( A + α Φ(A,α) ) 2 +εf(α) A n n A { and ρ n,ξ such that in S ρn (A n) (T l ) ξ Φ n (A α Φ n n n,α) H(α), A Φ n h(α) n αφ 2 n H n (α), α,a 2 Φ n H n (α) 2 (A + H(α)) 2 +εf(α) = E = α indep. A F(A n,α) n ω ψ = α+ h(α) α = ψ +h(ψ) ψ(t) = ψ +ωt is solution