In this talk we describe the propagation of ${𝒞}^{\infty }$ and Sobolev singularities for the wave equation on ${𝒞}^{\infty }$ manifolds with corners $M$ equipped with a Riemannian metric $g$. That is, for $X=M\times {ℝ}_t$, $P=D_t^2-{\Delta }_M$, and $u\in H_{\text{loc}}^1\left(X\right)$ solving $Pu=0$ with homogeneous Dirichlet or Neumann boundary conditions, we show that ${WF}_{\text{b}}\left(u\right)$ is a union of maximally extended generalized broken bicharacteristics. This result is a ${𝒞}^{\infty }$ counterpart of Lebeau’s results for the propagation of analytic singularities on real analytic manifolds with appropriately stratified boundary,...

In these lecture notes we describe the propagation of singularities of tempered distributional solutions $u\in {𝒮}^{\text{'}}$ of $(H-\lambda )u=0$, where $H$ is a many-body hamiltonian $H=\Delta +V$, $\Delta \ge 0$, $V={\sum }_a{V}_a$, and $\lambda$ is not a threshold of $H$, under the assumption that the inter-particle (e.g. two-body) interactions $V_a$ are real-valued polyhomogeneous symbols of order $-1$ (e.g. Coulomb-type with the singularity at the origin removed). Here the term “singularity” provides a microlocal description of the lack of decay at infinity. Our result is then that the...

Motivated by the study of resolvent estimates in the presence of trapping, we prove a semiclassical propagation theorem in a neighborhood of a compact invariant subset of the bicharacteristic flow which is isolated in a suitable sense. Examples include a global trapped set and a single isolated periodic trajectory. This is applied to obtain microlocal resolvent estimates with no loss compared to the nontrapping setting.

We study the “hyperboloidal Cauchy problem” for linear and semi-linear wave equations on Minkowski space-time, with initial data in weighted Sobolev spaces allowing singular behavior at the boundary, or with polyhomogeneous initial data. Specifically, we consider nonlinear symmetric hyperbolic systems of a form which includes scalar fields with a $\lambda {\phi }^p$ nonlinearity, as well as wave maps, with initial data given on a hyperboloid; several of the results proved apply to general space-times admitting conformal...

We extend our recent results on propagation of semiclassical resolvent estimates through trapped sets when a priori polynomial resolvent bounds hold. Previously we obtained non-trapping estimates in trapping situations when the resolvent was sandwiched between cutoffs $\chi$ microlocally supported away from the trapping: $\parallel \chi R_h(E+i0)\chi \parallel =𝒪\left(h^{-1}\right)$, a microlocal version of a result of Burq and Cardoso-Vodev. We now allow one of the two cutoffs, $\tilde{\chi }$, to be supported at the trapped set, giving $\parallel \chi R_h(E+i0)\tilde{\chi }\parallel =𝒪\left(\sqrt{a\left(h\right)}{h}^{-1}\right)$ when the a priori bound is $\parallel \tilde{\chi }{R}_h(E+i0)\tilde{\chi }\parallel =𝒪\left(a\left(h\right)h^{-1}\right)$.

We study a geometric generalization of the time-dependent Schrödinger equation for the harmonic oscillator$$\left(D_t+\frac{1}{2}\Delta +V\right)\psi =0(0.1)$$where $\Delta$ is the Laplace-Beltrami operator with respect to a “scattering metric” on a compact manifold $M$ with boundary (the class of scattering metrics is a generalization of asymptotically Euclidean metrics on ${ℝ}^n$, radially compactified to the ball) and $V$ is a perturbation of $\frac{1}{2}{\omega }^2{x}^{-2}$, with $x$ a boundary defining function for $M$ (e.g. $x=1/r$ in the compactified Euclidean case). Using the quadratic-scattering...

Kashiwara et Schapira ont proposé une condition de régularité appelée ($\mu )$ sur un couple de sous-variétés $X,Y$ d’une variété $C^{2}M:(T_Y^{*}M\widehat{+}{T}_X^{*}M)\cap (T^{*}M){|}_Y\subseteq T_Y^{*}M$, où $\widehat{+}$ est une somme géométrique naturelle dans l’analyse microlocale. Nous démontrons que la $(\mu$)-régularité est équivalente à la $\left(w\right)$-régularité de Verdier, répondant ainsi à une question de Kashiwara.