Quantum integrable systems
Let P be a Markov kernel on a measurable space E with countably generated σ-algebra, let w:E→[1, +∞[ such that Pw≤Cw with C≥0, and let be the space of measurable functions onE satisfying ‖f‖w=sup{w(x)−1|f(x)|, x∈E}<+∞. We prove that Pis quasi-compact on if and only if, for all , contains a subsequence converging in toΠf=∑di=1μi(f)vi, where the vi’s are non-negative bounded measurable functions on E and the μi’s are probability distributions on E. In particular, when the space of...
In these notes, we will describe recent work on globally solving quasilinear wave equations in the presence of trapped rays, on Kerr-de Sitter space, and obtaining the asymptotic behavior of solutions. For the associated linear problem without trapping, one would consider a global, non-elliptic, Fredholm framework; in the presence of trapping the same framework is available for spaces of growing functions only. In order to solve the quasilinear problem we thus combine these frameworks with the normally...
A lattice model with exponential interaction, was proposed and integrated by M. Toda in the 1960s; it was then extensively studied as one of the completely integrable (differential-difference) equations by algebro-geometric methods, which produced both quasi-periodic solutions in terms of theta functions of hyperelliptic curves and periodic solutions defined on suitable Jacobians by the Lax-pair method. In this work, we revisit Toda’s original approach to give solutions of the Toda lattice in terms...
We overview recent existence results and techniques about KAM theory for PDEs.
We prove the existence of quasi-periodic solutions for Schrödinger equations with a multiplicative potential on , finitely differentiable nonlinearities, and tangential frequencies constrained along a pre-assigned direction. The solutions have only Sobolev regularity both in time and space. If the nonlinearity and the potential are then the solutions are . The proofs are based on an improved Nash-Moser iterative scheme, which assumes the weakest tame estimates for the inverse linearized operators...
Soit un difféomorphisme Morse-Smale d’une surface fermée. À une courbe instable de comportement 1 par rapport à un attracteur de correspond une courbe fermée sur un des tores (Bassin. Cette remarque nous permettra de définir de nouveaux invariants de conjugaison de . Nous en déduisons aussi un moyen d’écrire explicitement une puissance de comme le produit du temps 1 d’un champ de vecteurs Morse-Smale topologique par des isotopies à support des disques et des twists de Dehn de supports...
This paper deals with the Hausdorff dimension of the Julia set of quadratic polynomials. It is divided in two parts. The first aims to compute good numerical approximations of the dimension for hyperbolic points. For such points, Ruelle’s thermodynamical formalism applies, hence computing the dimension amounts to computing the zero point of a pressure function. It is this pressure function that we approximate by a Monte-Carlo process combined with a shift method that considerably decreases the computational...