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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...

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,...

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 $\parallel \tilde{\chi }{R}_h(E+i0)\tilde{\chi }\parallel =𝒪\left(a\left(h\right)h^{-1}\right)$.

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...

Let $X$ be a compact manifold with boundary, and $g$ a scattering metric on $X$, which may be either of short range or “gravitational” long range type. Thus, $g$ gives $X$ the geometric structure of a complete manifold with an asymptotically conic end. Let $H$ be an operator of the form $H=\Delta +P$, where $\Delta$ is the Laplacian with respect to $g$ and $P$ is a self-adjoint first order scattering differential operator with coefficients vanishing at $\partial X$ and satisfying a “gravitational” condition. We define a symbol calculus for...

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.