Results 1 - 10
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82
Methods for the analysis of the Lindstedt series for KAM tori and renormalizability in classical mechanics -- A review with some applications
, 1995
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A renormalization group for Hamiltonians, with applications to KAM theory. Ergodic Theory Dynam
- Systems
, 1999
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Hyperbolic Low-Dimensional Invariant Tori and Summations of Divergent Series
- Comm. Math. Phys
, 2001
"... We consider a class of a priori stable quasi-integrable analytic Hamiltonian systems and study the regularity of low-dimensional hyperbolic invariant tori as functions of the perturbation parameter. We show that, under natural nonresonance conditions, such tori exist and can be identified through th ..."
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Cited by 26 (19 self)
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We consider a class of a priori stable quasi-integrable analytic Hamiltonian systems and study the regularity of low-dimensional hyperbolic invariant tori as functions of the perturbation parameter. We show that, under natural nonresonance conditions, such tori exist and can be identified through the maxima or minima of a suitable potential. They are analytic inside a disc centered at the origin and deprived of a region around the positive or negative real axis with a quadratic cusp at the origin. The invariant tori admit an asymptotic series at the origin with Taylor coe#cients that grow at most as a power of a factorial and a remainder that to any order N is bounded by the (N + 1)-st power of the argument times a power of N !. We show the existence of a summation criterion of the (generically divergent) series, in powers of the perturbation size, that represent the parametric equations of the tori by following the renormalization group methods for the resummations of perturbative series in quantum field theory. 1.
KAM Theorem and Quantum Field Theory
, 1998
"... We give a new proof of the KAM theorem for analytic Hamiltonians. The proof is inspired by a quantum field theory formulation of the problem and is based on a renormalization group argument treating the small denominators inductively scale by scale. The crucial cancellations of resonances are show ..."
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Cited by 26 (1 self)
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We give a new proof of the KAM theorem for analytic Hamiltonians. The proof is inspired by a quantum field theory formulation of the problem and is based on a renormalization group argument treating the small denominators inductively scale by scale. The crucial cancellations of resonances are shown to follow from the Ward identities expressing the translation invariance of the corresponding field theory.
Quasi-Periodic Solutions for Two-Level Systems
"... We consider the Schörodinger equation for a class of two-level atoms in a quasi-periodic external field in the case in which the spacing 2 epsilon between the two unperturbed energy levels is small. We prove the existence of quasi-periodic solutions for a Cantor set E of values of epsilon around the ..."
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Cited by 24 (18 self)
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We consider the Schörodinger equation for a class of two-level atoms in a quasi-periodic external field in the case in which the spacing 2 epsilon between the two unperturbed energy levels is small. We prove the existence of quasi-periodic solutions for a Cantor set E of values of epsilon around the origin which is of positive Lebesgue measure: such solutions can be obtained from the formal power series by a suitable resummation procedure. The set E can be characterized by requesting infinitely many Diophantine conditions of Mel'nikov type.
Scaling Properties For The Radius Of Convergence Of Lindstedt Series: Generalized Standard Maps
- J. Math. Pures Appl
, 2000
"... . For a class of symplectic two-dimensional maps which generalize the standard map by allowing more general nonlinear terms, the radius of convergence of the Lindstedt series describing the homotopically non-trivial invariant curves is proved to satisfy a scaling law as the complexied rotation numbe ..."
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Cited by 22 (13 self)
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. For a class of symplectic two-dimensional maps which generalize the standard map by allowing more general nonlinear terms, the radius of convergence of the Lindstedt series describing the homotopically non-trivial invariant curves is proved to satisfy a scaling law as the complexied rotation number tends to a rational value non-tangentially to the real axis, thus generalizing previous results of the authors. The function conjugating the dynamics to rotations by ! possesses a limit which is explicitly computed and related to hyperelliptic functions in the case of nonlinear terms which are trigonometric polynomials. The case of the standard map is shown to be non-generic. 1. Introduction In this paper we generalize the results of [1] considering maps of the kind T ";f : 8 > < > : x 0 = x + y + "f(x); y 0 = y + "f(x); (1.1) where f(x) is a 2-periodic function of x, analytic in a strip S = fj Im(x)j < g of width 2 around the real x axis. The nonlinear term f(x) can be expande...
Kolmogorov theorem and classical perturbation theory
- J. of App. Math. and Phys. (ZAMP
, 1997
"... Abstract. We reconsider the original proof of Kolmogorov’s theorem in the light of classical perturbation methods based on expansions in some parameter. This produces quasiperiodic solutions on invariant tori in the form of power series in a small parameter, that we prove to be absolutely convergent ..."
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Cited by 21 (16 self)
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Abstract. We reconsider the original proof of Kolmogorov’s theorem in the light of classical perturbation methods based on expansions in some parameter. This produces quasiperiodic solutions on invariant tori in the form of power series in a small parameter, that we prove to be absolutely convergent.
