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Cet article est consacré à l’étude des mesures limites associées à la solution de l’équation de Helmholtz avec un terme source se concentrant en un point. Le potentiel est supposé $C^{\infty }$ et l’opérateur non-captif. La solution de l’équation de Schrödinger semi-classique s’écrit alors micro-localement comme somme finie de distributions lagrangiennes. Sous une hypothèse géométrique, qui généralise l’hypothèse du viriel, on en déduit que la mesure limite existe et qu’elle vérifie des propriétés standard....

Dans cet exposé, on décrit un travail effectué sous la direction de J. Sjöstrand. On prouve des majorations et des minorations du nombre de résonances d’un opérateur de Schrödinger semi-classique $P=-h^2\Delta +V\left(x\right)$ dans des petits disques centrés en $E_0\>0$, une valeur critique de $p(x,\xi )={\xi }^2+V\left(x\right)$.

We obtain some microlocal estimates of the resonant states associated to a resonance $z_0$ of an $h$-differential operator. More precisely, we show that the normalized resonant states are $𝒪(\sqrt{|\mathrm{Im}{z}_0|/h}$ $+h^{\infty })$ outside the set of trapped trajectories and are $𝒪\left(h^{\infty }\right)$ in the incoming area of the phase space. As an application, we show that the residue of the scattering amplitude of a Schrödinger operator is small in some directions under an estimate of the norm of the spectral projector. Finally we prove such bound...

Let $P$ be a long range metric perturbation of the Euclidean Laplacian on ${ℝ}^d$, $d\ge 2$. We prove local energy decay for the solutions of the wave, Klein-Gordon and Schrödinger equations associated to $P$. The problem is decomposed in a low and high frequency analysis. For the high energy part, we assume a non trapping condition. For low (resp. high) frequencies we obtain a general result about the local energy decay for the group $e^{itf\left(P\right)}$ where $f$ has a suitable development at zero (resp. infinity).

We consider the 3D Schrödinger operator $H=H_0+V$ where $H_0={(-i\nabla -A)}^2-b$, $A$ is a magnetic potential generating a constant magneticfield of strength $b\>0$, and $V$ is a short-range electric potential which decays superexponentially with respect to the variable along the magnetic field. We show that the resolvent of $H$ admits a meromorphic extension from the upper half plane to an appropriate Riemann surface $ℳ$, and define the resonances of $H$ as the poles of this meromorphic extension. We study their distribution near any fixed...

In this note we describe recent results on semiclassical random walk associated to a probability density which may also concentrate as the semiclassical parameter goes to zero. The main result gives a spectral asymptotics of the close to $1$ eigenvalues. This problem was studied in [] and relies on a general factorization result for pseudo-differential operators. In this note we just sketch the proof of this second theorem. At the end of the note, using the factorization, we give a new proof of the...

We study the spectral projection associated to a barrier-top resonance for the semiclassical Schrödinger operator. First, we prove a resolvent estimate for complex energies close to such a resonance. Using that estimate and an explicit representation of the resonant states, we show that the spectral projection has a semiclassical expansion in integer powers of $h$, and compute its leading term. We use this result to compute the residue of the scattering amplitude at such a resonance. Eventually, we...

2000 Mathematics Subject Classification: 35P25, 81U20, 35S30, 47A10, 35B38. We study the microlocal structure of the resolvent of the semiclassical Schrödinger operator with short range potential at an energy which is a unique non-degenerate global maximum of the potential. We prove that it is a semiclassical Fourier integral operator quantizing the incoming and outgoing Lagrangian submanifolds associated to the fixed hyperbolic point. We then discuss two applications of this result...