Resonances, Divergent series and R.G.: (G. Gentile, G.G.) α = ε α f(α) α 2. 0 ν Z r

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1 B Resonances, Divergent series and R.G.: (G. Gentile, G.G.) Eq. Motion: α = ε α f(α) A A 2 α α 2 α l A l Representation of phase space in terms of l rotators: α=(α,...,α l ) T l Potential: f(α) = ν Z l f ν e iν α, f ν 0 if ν > N Motions: α def = (γ,β) T r T s, t (γ +ω 0 t,β) γ= fast angles, β= slow angles Resonance of order s Independence: ω 0 ν > C ν τ, 0 ν Z r

2 2 L Find a resonance copy, i.e. h (g(ψ),k(ψ)) R r R s s.t. α (γ,β), and γ = ψ +g(ψ), β = β 0 +k(ψ), ψ ψ +ω 0 t solves α = ε α f(α), i.e. ( ) g(ψ) (ω 0 ψ ) 2 = ε α f ( ψ +g(ψ),β 0 +k(ψ) ) k(ψ) If f(β) def = dγ (2π) r f(γ,β) β f(β 0 ) = 0 (0 average force) There is a power series solution

3 A A 2 B 3 α α 2 α l A l Representation of phase space in terms of l rotators: α=(α,...,α l ) T l Notation: f(γ,β) def = ν Z r f ν (β)e iγ ν. Rules: () 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 j0 j j 2 Component labels around v v j p v J v = (j 0,...,j p ) and Jv f νv (β 0 ) are defined.

4 4 L () 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 j0 j j 2 Component labels around v (4) Value : Val(θ) = k! g ij def = ( v v j p ε Jv f νv (β 0 ) )( ( δ ij 0 0 (ω ν(λ)), or g 2 ij = 0 (ε 2 β f(β 0)) g jλ j λ linesλ ) ) if ν(λ) = 0 h ν θval(θ) : no trivial node with 0 incoming current Estimate: h (k) ν bb k ε k k! 2τ!! Results:

5 B 5 If det 2 β f(β 0) 0 The tree series can be rearranged to yield a convergent series representation of h = (g(ψ), k(ψ), hence its existence, in j j v v j0 Component labels around v v j j 2 j p 3 E ( ε 0,0] with open dense complement; 0 is a density point. Fig.4: ε E; E dense at 0. The figure illustrates the analyticity domain D(ε) associated with ε E, for ε < 0 (elliptic case), assuming 2 ββ f(β 0) <0 ε-plane 4 Need k! 2 for Borel sum: at 0 derivatives grow as k! 2τ.

6 6 L The k! arise because of chains in graphs (self-energy insertions): i i 0 j 0 i j j Inserting a single trivial node with 0 harmonic ( ν 0): 6 ν 0 ν i i 0 j 0 j def Let M 0;i0 j 0 = ( ) ε 2 f(β 0 ) graph value modified trivially : propagator modification δ ij (ω ν) 2 (ω ν) 2 ( M0 (ω ν) 2 ) k Simplify : NO trivial nodes if (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

7 B 7 using k=0 zk = z, z, e.g. k=0 2k = = If accepted (ω ν) ( ) k 2 (ω ν) 2 k=0 M0 (ω ν) 2 (ω ν) 2 M 0 M 0 0 (β 0 =maximum) gives ε > 0 easier than ε < 0. For ε < 0 expect to exclude ε s.t ω ν = ± εµ j. PROBLEM: LOTS of other chains! They cause values k! η, η > 0 IDEA: eliminate them all by resummation (RG needed). 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 = 0,,... IF no pair lines λ < λ with ν(λ ) = ν(λ) with only lower scale intermediates THEN N n 4N2 n/τ k

8 8 L If there were no chains : N n 4N2 n/τ k v (ω 0 ν) 2 C2k 2 2nN n C 2k( 2 2n(4N2 n/τ ) ) k n=0 Product of couplings: v J vf ν v (β 0 ) v N Jv F k N 2k F k, number of harmonics ν: (2N +) k, n. of trees k k n=0 Convergence: ε < ( N 2 C 2 (2N +) l 3 F 2 8N n n2 n/τ) Multiscale analysis (RG): Organize the lines of θ into clusters

9 B 9 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. ν 0 ν i i 0 j 0 j Self energy clusters : line in and line out with ν in = ν out 7 ν out ν in 8

10 0 L Eliminate self energy clusters by resummations Necessary multiscale 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 ν giant 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

11 B Summing over s.e. clusters of scale [0] leads to modify propagators of lines on scale [ ] giant tree ν ν ν ν g [ ] (x) = x 2 M 0 x 2 M 0 n=0 (M eliminated also s.e. subgraphs of scale [0]. giant tree x 2 M 0 ) n x 2 M 0 M Iterate! at every step only graphs with no s.e. subgraphs have to be considered; convergent additions made on propagators. Full resummation of self energy in the hyperbolic case. In the elliptic the situation is very different. 9

12 2 L 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: 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.

13 B 3 Case det 2 β f(β 0) = 0? If s = and c = k 0+ f(β 0 ) 0 same but analyticity in η = ε k 0 (+ A. Giuliani) Borel Summability? yes if r = 2(perhaps: w. progress +O.Costin,A.Giuliani) OPEN PROBLEM: uniqueness (and relation with alternative existence results) complex ε plane 0