Pierre Lochak

Stability of Hamiltonian Systems over Exponentially Long Times: The Near-Linear Case

We shall be interested here in the classical problem of the stability of the action variables of a nearly integrable hamiltonian system, namely one governed by the following Hamiltonian:

The universal Ptolemy-Teichmuller groupoid

H\left( {p,q} \right) = h\left( p \right) + \varepsilon f\left( {p,q} \right)\;\;\operatorname{with} \;\left( {p,q} \right)\; \in {R^n} \times {T^{n\;}},\;T = R/Z

The Quantum Adiabatic Theorem

(1.1) ,where (p, q) are action-angle variables of the integrable Hamiltonian h. Let us first recall informally the basic result of the theory, due to N.N. Nekhoroshev ([5], [6]; see also [7], [8]): Assume H is defined and analytic over some domain D of phase space; assume moreover that the unperturbed Hamiltonian h is convex, i.e. the hessian matrix ∇2 h(p) is sign definite; then:

Necessary versus sufficient conditions for exact solubility of statistical models on lattices

We derive a series of identities which generalize and simplify the results obtained for adiabatically modulated solitons in the case of perturbed specific integrable equations. It stresses the importance of the variational properties of the solitons, which make an adiabatic theorem plausible. A precise conjecture is made and its validity discussed from different points of view.

Tores invariants à torsion évanescente dans les systèmes hamiltoniens proches de l'intégrable

We define the universal Ptolemy-Teichmüller groupoid, a generalization of Penner’s universal Ptolemy groupoid, on which the Grothendieck-Teichmüller group – and thus also the absolute Galois group – acts naturally as automorphism group. The essential new ingredient added to the definition of the universal Ptolemy groupoid is the profinite local group of pure ribbon braids of each tesselation.

Des oscillateurs à l'espace des modules des courbes

Abstract In this Note we introduce a certain subgroup Г of the Grothendieck-Teichmuller group GT, obtained by adding two new relations to the definition of GT. We show that Г gives an automorphism group of the profinite completions of certain surface mapping class groups with geometric compatibility conditions, and that the absolute Galois group (ℚ/ℚ) is embedded into Г.

Adiabatic Theorems in One Dimension

This chapter is devoted to the quantum adiabatic theorem, and to its connection with the classical cases examined in the preceding chapters. In this section, we present the proof of the quantum theorem in a slightly informal way, without emphasizing the regularity hypotheses nor the domains of definition of the operators involved. The first proof of the theorem appears in an article by M. Born and V. Fock ([Bor]) but it is incomplete in several respects, for example, they consider only discrete spectra, which rarely occur in quantum mechanics. Here we follow the much later proof due to T. Kato ([Kat]).

N Frequency Systems; Neistadt’s Result Based on Anosov’s Method

It is shown that under rather mild conditions the triangle relation represents a necessary condition for the existence of commuting transfer matrices of arbitrary size. The cases of spin models and vertex models are treated separately.