Resummations of self energy graphs and KAM theory G. Gentile (Roma3), G.G. (I.H.E.S.) Communications in Mathematical Physics, 227, , 2002.
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1 Resummations of self energy graphs and KAM theory G. Gentile (Roma3), G.G. (I.H.E.S.) Communications in Mathematical Physics, 227, ,
2 Hamiltonian for a rotator system H = 1 2 (A2 + B 2 )+εf(α, β) A = (A 1,...,A r ), B = (B 1,...,B n r ) α = (a 1,...,a r ), β = (β 1,...,β n r ) with f an even trigonometric polynomial of degree N. Unperturbed motions A = ω, B = 0, ω ν > 1 C ν τ α = α 0 + ωt, β = β 0 r = n maximal tori or KAM tori Are there motions of the same type in the perturbed system?? i.e. A = ω + H(ψ), B = K(ψ) α = ψ + h(ψ), β = β 0 + k(ψ) with ψ ψ + ωt is a solution? It must be (iff): (ω ψ ) 2 h(ψ) = ε α f(ψ + h(ψ), β 0 + k(ψ)) (ω ψ ) 2 k(ψ) = ε β f(ψ + h(ψ), β 0 + k(ψ)) and β 0 must be anextremum forthe α averagef(β) = f(α, β) dα (2π) r of f: (ω ψ ) 2 k 1 (ψ) = β f(ψ, β 0 ) 0 = β f(β 0 ) 1
3 Case of maximal tori: r=n Given a tree ϑ with p nodes v 0,...,v p 1 and a root r Associate with each node v a node momentum Z r for f 0. The root momentum will be a unit vector n; and define a line current ν(v) def = w<v ν w and assume 0 Note that 0 < N ω > c > 0 (Diophantine ω) 1 v 5 r n λ 0 ν = ν λ0 v 0 0 v 1 v 3 v 6 v 7 v 2 v 4 v 8 v 9 v 10 Fig. 1: A tree ϑ with mv 0 = 2,mv 1 = 2,mv 2 = 3,mv 3 = 2,mv 4 = 2 and k = 12, and some decorations. Only two mode label and one momentum are explicitly written; the number labels, distinguishing the branches, are not shown. The arrows represent the partial ordering on the tree. v 11 Define the value of a tree ϑ with distinguishable branches by Val(ϑ) = 1 p! ( linesλ=(v v) ϑ )( (ω ν(v)) 2 v ϑ f ) counting tree up to pivoting equivalence. Linstedt, Newcomb, Poincaré: h (p) ν n = ϑ root current= ν 2 Val(ϑ)
4 Siegel Bryuno Pöschel bound Definescaleofλ = (v v)toben = 0, 1, 2,...if2 n 1 < ω ν(v) 2 n N 2 2 2n if 2 n 1 < ω ν(v) 2 n (ω ν(v)) 2 Val(ϑ) 1 p! N2p F p 0 n= 2 2nN n N n def = nombre des lignes d échelle n Si ν(v) ν(w) pour tout v > w alors N n est petit N n an2 n/τ p for some a > 0. Il faut consommer 2 n/τ N 1 noeuds avec moment N pour atteindre une ligne v v telle que ω ν(v) 2 n, i.e. ν(v) = O(2 n/τ ). Pour en trouver une autre de la meme echelle il en faut encore autant: donc N n = O(p2 n/τ N). Val(ϑ) 1 0 p! p!4p N 2p F p ( 2 2naN2n/τ p ) = ϑ avec p noeds n= = 1 p! p!4p N 2p F p (2 2aN 0 n= n2n/τ ) p = B p Simple self energy graphs R ν out ν in 3
5 R ν out ν in This is a self-energy subgraph if the entering line and the exiting line have the samecurrent ν with scale nand allinternal lines havescale m n+3 and their number is < a2 n/τ, i.e. not too large, and w R ν w = 0 and all lines in the subgraph have different currents (i.e. no self energy sub-sub graphs! simple ). Resummation of simple self energy subgraphs The contribution to the value of a self-energy subgraph R inserted in the line v v on which is ν out (ω ν) 2 ( λ=(w w) R 1 (ω ν) 2 M R (ν) (ω ν) 2 ν w ν w (ω ν(λ)) 2 ) ν in (ω ν) 2 We define M(ν) = R ε R M R (ν). One can insert m = 0,1,2,... self energy subgraphs inside each line of a tree which has no such subgraphs m=0 = 1 (ω ν) 2 ( M(ν) (ω ν) 2 ) m = 1 (ω ν) 2 M(ν) convergent by the Siegel Bryuno Pöschel boun.. 4
6 Cancellations Not useful because (ω ν) 2 M(ν) may vanish!! However (theorem) M(ν) = (ω ν) 2 m 1 ε(ν) andthepropagatoris(ω ν) 2 (1+m 1 def ε (ν)) 1 = (ω ν) 2 G (1) (ν) i.e. we have eliminated the self energy subgraphs without inner self energy subgraphs. Elimination of overlapping self energy graphs Define m 2 ε(ν) in the same way : consider all trees with at most simple self energy graphs and define their values as in the previous case but using the new propagators. Iterate: one checks that G (k) ε (ν) converges to a limit G ( ) ε (ν). The equation of the invariant torus is then obtained by considering all tree graphs without self energies and evaluating them with the propagator (ω ν) 2 (1+m ε (ν)) 1 def = (ω ν) 2 G ( ) (ν) which by the Siegel Bryuno Pöschel bound has no convergence problems and in fact it yields an effective computational algorithm to evaluate the LNP series. Low dimensional tori If f(α, β) = ν,µ eiν α+iµ β f ν,µ the Feynman rules change in a minor way. Namely after resummation of the self energy graphs (defined in the same way) the propagator is the a matrix (n n as before) which has the form 5
7 α β α (r r) (r (n r)) = β (r r) ((n r) (n r)) ( ( (ω ν) 2 (1+O(ε 2 )) i(ω ν)bε+o(ε 2 ) = i(ω ν)bε+o(ε 2 ) (ω ν) 2 ε β f(β 0 )+O(ε 2 ) where the α α elements take into account the cancellations. ) ) 1 However the β β elements can vanish on or close to the infinite set (ω ν) 2 ε β f(β 0 ) = 0. If ε > 0 and β 0 is a maximum there is no 0 eigenvalue and in fact the eigenvalues are bounded below by (ω ν) 2. Hence we are back to the maximal case. The convergence occurs in the domain D γ (γ > 0) of complex ε where (ω ν) 2 ε β f(β 0 ) γ ω ν 2. This has the form: complex ε plane O Analyticity domain D 0 for the lower dimension invariant torus. The cusp at the origin is a second order cusp. The figure corresponds to the hyperbolic case. 6
8 complex ε plane The domain D 0 of Figure 1 can be further extended? the conjecture above asks whether the extended analyticity domain could possibly be represented (close to the origin) as here: the domain reaches the real axis at cusp points which are in Iε 0 and correspond, in the complex ε-plane, to the elliptic tori which are the analytic continuations of the hyperbolic tori. The analytic continuation would be continuous thorough the real axis at the points of Iε 0. The cusps would be at least quadratic. 7
9 Some references Low dim tori S.M. Graff: On the conservation for hyperbolic invariant tori for Hamiltonian systems, J. Differential Equations 15 (1974), V.K. Mel nikov: On some cases of conservation of conditionally periodic motions under a small change of the Hamiltonian function, Soviet Math. Dokl. 6 (1965), ; A family of conditionally periodic solutions of a Hamiltonian systems, Soviet Math. Dokl. 9 (1968), J. Moser: Convergent series expansions for quasi periodic motions, Math. Ann. 169 (1967), Eliasson, L.H.: Absolutely convergent series expansions for quasi-periodic motions, Math. Phys. Electron. J. 2(1996), Paper 4, < mpej.unige.ch>. A. Jorba, R. Llave, M. Zou: Lindstedt series for lower dimensional tori, in Hamiltonian systems with more than two degrees of freedom (S Agaró, 1995), , NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. Vol. 533, Ed. C. Simó, Kluwer Academic Publishers, Siegel-Bryuno-Pöschel bound A.D. Bryuno: Analytic form of differential equations. I, II. (Russian), Trudy Moskov. Mat. Obšč. 25 (1971), ; ibid. 26 (1972), J. Pöschel: Invariant manifolds of complex analytic mappings near fixed points, in Critical Phenomena, Random Systems, Gauge Theories, Les Houches, Session XLIII (1984), Vol. II, , Ed. K. Osterwalder & R. Stora, North Holland, Amsterdam, Graph methods L. Chierchia, C. Falcolini: Compensations in small divisor problems, Comm. Math. Phys. 175 (1996), G. Gallavotti: Twistless KAM tori, Comm. Math. Phys. 164 nd (1994), 8
10 no. 1, Twistless KAM tori, quasiflat homoclinic intersections, and other cancellations in the perturbation series of certain completely integrable Hamiltonian systems. A review, Rev. Math. Phys. 6 (1994), no. 3, More recent J. Bricmont, A. Kupiainen, A. Schenkel: Renormalization group for the Melnikov problem for PDE s, Comm. Math. Phys. 221 (2001), no. 1, J. Xu, J. You: Persistence of Lower Dimensional Tori Under the First Melnikov s Non-resonance Condition, Nanjing University Preprint, 1 21, 2001, J. Math. Pures Appl., in press. Xiaoping Yuan: Construction of quasi-periodic breathers via KAM techniques Comm. Math. Phys., in press. 9
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Resummations of self-energy graphs and KAM theory G. Gentile (Roma3), G.G. (The.H.E.S.) Communications in Mathematical Physics, 227, 421 460, 2002. 1 Hamiltonian for a rotators system H = 1 2 (A2 + B 2
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