Resolvent estimates for normally hyperbolic trapped sets

We give pole free strips and estimates for resolvents of semiclassical operators which, on the level of the classical flow, have normally hyperbolic smooth trapped sets of codimension two in phase space. Such trapped sets are structurally stable and our motivation comes partly from considering the wave equation for Kerr black holes and their perturbations, whose trapped sets have precisely this structure. We give applications including local smoothing effects with epsilon derivative loss for the Schr\"odinger propagator as well as local energy decay results for the wave equation.Comment: Further changes to erratum correcting small problems with Section 3.5 and Lemma 4.1; this now also corrects hypotheses, explicitly requiring trapped set to be symplectic. Erratum follows references in this versio

Correlations for pairs of periodic trajectories for open billiards

In this paper we prove two asymptotic estimates for pairs of closed trajectories for open billiards similar to those established by Pollicott and Sharp for closed geodesics on negatively curved compact surfaces. The first of these estimates holds for general open billiards in any dimension. The more intricate second estimate is established for open billiards satisfying the so called Dolgopyat type estimates. This class of billiards includes all open billiards in the plane and open billiards in $\R^N, N \geq 3$ satisfying some additional conditions

Bounds on the growth of high Sobolev norms of solutions to 2D Hartree Equations

In this paper, we consider Hartree-type equations on the two-dimensional torus and on the plane. We prove polynomial bounds on the growth of high Sobolev norms of solutions to these equations. The proofs of our results are based on the adaptation to two dimensions of the techniques we previously used to study analogous problems on $S^1$, and on $\mathbb{R}$.Comment: 38 page

Nondispersive solutions to the L2-critical half-wave equation

We consider the focusing $L^2$-critical half-wave equation in one space dimension $$i \partial_t u = D u - |u|^2 u,$$ where $D$ denotes the first-order fractional derivative. Standard arguments show that there is a critical threshold $M_* > 0$ such that all $H^{1/2}$ solutions with $\| u \|_{L^2} < M_*$ extend globally in time, while solutions with $\| u \|_{L^2} \geq M_*$ may develop singularities in finite time. In this paper, we first prove the existence of a family of traveling waves with subcritical arbitrarily small mass. We then give a second example of nondispersive dynamics and show the existence of finite-time blowup solutions with minimal mass $\| u_0 \|_{L^2} = M_*$. More precisely, we construct a family of minimal mass blowup solutions that are parametrized by the energy $E_0 >0$ and the linear momentum $P_0 \in \R$. In particular, our main result (and its proof) can be seen as a model scenario of minimal mass blowup for $L^2$-critical nonlinear PDE with nonlocal dispersion.Comment: 51 page

L^{2}-restriction bounds for eigenfunctions along curves in the quantum completely integrable case

We show that for a quantum completely integrable system in two dimensions,the $L^{2}$-normalized joint eigenfunctions of the commuting semiclassical pseudodifferential operators satisfy restriction bounds ofthe form $\int_{\gamma} |\phi_{j}^{\hbar}|^2 ds = {\mathcal O}(|\log \hbar|)$ for generic curves $\gamma$ on the surface. We also prove that the maximal restriction bounds of Burq-Gerard-Tzvetkov are always attained for certain exceptional subsequences of eigenfunctions.Comment: Correct some typos and added some more detail in section

Is there exist a hadron spin-flip contribution in the Coulomb-hadron interference at small transfer momenta and high energies

The analysing power $A_N$ is examined in the range of the Coulomb-hadron interferenceon on the basis of the experimental data from p_L = 6 GeV/c up to 200 GeV/c taking account of a phenomenological analysis at p_L = 6 GeV/c and a dynamic high-energy spin model. The results are compared with the new RHIC data at p_L = 100 GeV/c. The new experimental data obtained at RHIC indicate the small contributions of the hadrons spin-flip amplitude.Comment: 18 pages, 8 figures, Table Essentially revision. Figures and Table are added, fit with new final data are made. Conclusion is not chang

The subconvexity problem for \GL_{2}

Generalizing and unifying prior results, we solve the subconvexity problem for the $L$-functions of \GL_{1} and \GL_{2} automorphic representations over a fixed number field, uniformly in all aspects. A novel feature of the present method is the softness of our arguments; this is largely due to a consistent use of canonically normalized period relations, such as those supplied by the work of Waldspurger and Ichino--Ikeda.Comment: Almost final version to appear in Publ. Math IHES. References updated

Concerning the Wave equation on Asymptotically Euclidean Manifolds

We obtain KSS, Strichartz and certain weighted Strichartz estimate for the wave equation on $(\R^d, \mathfrak{g})$, $d \geq 3$, when metric $\mathfrak{g}$ is non-trapping and approaches the Euclidean metric like $x ^{- \rho}$ with $\rho>0$. Using the KSS estimate, we prove almost global existence for quadratically semilinear wave equations with small initial data for $\rho> 1$ and $d=3$. Also, we establish the Strauss conjecture when the metric is radial with $\rho>0$ for $d= 3$.Comment: Final version. To appear in Journal d'Analyse Mathematiqu

Global well-posedness for the KP-I equation on the background of a non localized solution

We prove that the Cauchy problem for the KP-I equation is globally well-posed for initial data which are localized perturbations (of arbitrary size) of a non-localized (i.e. not decaying in all directions) traveling wave solution (e.g. the KdV line solitary wave or the Zaitsev solitary waves which are localized in $x$ and $y$ periodic or conversely)

New developments in threshold pion photo- and electroproduction

Photoproduction of neutral and charged pions off nucleons and deuterium has been precisely calculated in baryon chiral perturbation theory. I review the predictions with the accurate data that have become available over the last few years. Some progress in the description of neutral pion electroproduction off protons is also discussed.Comment: 17 pp, LaTeX, uses epsf and lamuphys.sty, invited plenary talk, Workshop on ``Chiral Dynamics: Theory and Experiment'', Mainz, September 1997, to appear in the proceeding