Estimates on the number of scattering poles near the real axis for strictly convex obstacles

Johannes Sjöstrand; Maciej Zworski

Displaying similar documents to “Estimates on the number of scattering poles near the real axis for strictly convex obstacles”

Resonances for strictly convex obstacles

Johannes Sjöstrand (1997-1998)

Séminaire Équations aux dérivées partielles

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On considère le problème de Dirichlet à l’éxtérieur d’un obstacle strictement convexe borné à bord $C^{\infty}$ . Sous une hypothèse sur la variation de la courbure, on obtient à un facteur $1 + o (1)$ près, le nombre de résonances de module $\leq r$ , associées à la première racine de la fonction d’Airy.

Resonances for transparent obstacles

Georgi Popov, Georgi Vodev (1999)

Journées équations aux dérivées partielles

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This paper is concerned with the distribution of the resonances near the real axis for the transmission problem for a strictly convex bounded obstacle $𝒪$ in $ℝ^{n}$ , $n \geq 2$ , with a smooth boundary. We consider two distinct cases. If the speed of propagation in the interior of the body is strictly less than that in the exterior, we obtain an infinite sequence of resonances tending rapidly to the real axis. These resonances are associated with a quasimode for the transmission problem the frequency...

The resolvent for Laplace-type operators on asymptotically conic spaces

Andrew Hassell, András Vasy (2001)

Annales de l’institut Fourier

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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 $H = Δ + P$ , where $Δ$ 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...

On the distribution of scattering poles for perturbations of the Laplacian

Georgi Vodev (1992)

Annales de l'institut Fourier

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We consider selfadjoint positively definite operators of the form $- Δ + P$ (not necessarily elliptic) in $ℝ^{n}$ , $n \geq 3$ , odd, where $P$ is a second-order differential operator with coefficients of compact supports. We show that the number of the scattering poles outside a conic neighbourhood of the real axis admits the same estimates as in the elliptic case. More precisely, if ${λ_{j}} (Im λ_{j} \geq 0_{)}$ are the scattering poles associated to the operator $- Δ + P$ repeated according to multiplicity, it is proved that for any $ϵ > 0$ there exists...

Asymptotic behaviour of the scattering phase for non-trapping obstacles

Veselin Petkov, Georgi Popov (1982)

Annales de l'institut Fourier

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Let $S (λ)$ be the scattering matrix related to the wave equation in the exterior of a non-trapping obstacle $𝒪 \subset R^{n}$ , $n \geq 3$ with Dirichlet or Neumann boundary conditions on $\partial 𝒪$ . The function $s (λ)$ , called scattering phase, is determined from the equality $e^{- 2 π i s (λ)} = \det S (λ)$ . We show that $s (λ)$ has an asymptotic expansion $s (λ) \sim \sum_{j = 0}^{\infty} c_{j} λ^{n - j}$ as $λ \to + \infty$ and we compute the first three coefficients. Our result proves the conjecture of Majda and Ralston for non-trapping obstacles.

Scattering by two convex bodies

Mitsuru Ikawa (1991-1992)

Séminaire Équations aux dérivées partielles (Polytechnique)

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