Sur une variété analytique paracompacte de dimension 2, on considère un opérateur différentiel $P$ à symbole principal $p_m$ analytique vérifiant la condition $\left(𝒫\right)$ de Nirenberg et Treves. En ajoutant une nouvelle variable et en utilisant des estimations a priori de type Carleman, on montre qu’il y a propagation des singularités pour $P$, dans $p_m^{-1}\left(0\right)$, le long des feuilles intégrales du système différentiel engendré par les champs hamiltoniens de Re$p_m$ et Im$p_m$.

Let $P$ be a classical pseudodifferential operator of order $m$ on a paracompact $C^{\infty }$ manifold $X$. Let $p_m$ be the principal symbol and assume that $\Sigma =p_m^{-1}\left(0\right)$ is an involutive $C^{\infty }$ sub-manifold of $T^{*}X\setminus 0$, satisfying a certain transversality condition. We assume that $p_m$ vanishes exactly to order $M$ on $\Sigma$ and that the derivatives of order $M$ satisfy a certain condition, inspired from the Calderòn uniqueness theorem (usually empty when $M=2$). Suppose that a Levi condition is valid for the lower order symbols. If $u\in {𝒟}^{\text{'}}\left(X\right)$, $Pu\in C^{\infty }\left(X\right)$, then $WF\left(u\right)$ is a union...

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,...

This paper is a continuation of Part I of the same title which has appeared at the last issue of this journal.

We study spectral asymptotics and resolvent bounds for non-selfadjoint perturbations of selfadjoint $h$-pseudodifferential operators in dimension 2, assuming that the classical flow of the unperturbed part is completely integrable. Spectral contributions coming from rational invariant Lagrangian tori are analyzed. Estimating the tunnel effect between strongly irrational (Diophantine) and rational tori, we obtain an accurate description of the spectrum in a suitable complex window, provided that the...

In this paper we will give a brief survey of recent regularity results for Fourier integral operators with complex phases. This will include the case of real phase functions. Applications include hyperbolic partial differential equations as well as non-hyperbolic classes of equations. An application to an oblique derivative problem is also given.

We consider perturbations of a stratified medium ${ℝ}_x^{n-1}\times {ℝ}_y$, where the operator studied is $c^2(x,y)\Delta$. The function $c$ is a perturbation of $c_0\left(y\right)$, which is constant for sufficiently large $\left|y\right|$ and satisfies some other conditions. Under certain restrictions on the perturbation $c$, we give results on the Fourier integral operator structure of the scattering matrix. Moreover, we show that we can recover the asymptotic expansion at infinity of $c$ from knowledge of $c_0$ and the singularities of the scattering matrix at fixed energy....