Resonances for transparent obstacles

Georgi Popov; Georgi Vodev

Displaying similar documents to “Resonances for transparent obstacles”

Resonances for strictly convex obstacles

Johannes Sjöstrand (1997-1998)

Séminaire Équations aux dérivées partielles

Similarity:

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.

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

Johannes Sjöstrand, Maciej Zworski (1993)

Annales de l'institut Fourier

Similarity:

For the Dirichlet Laplacian in the exterior of a strictly convex obstacle, we show that the number of scattering poles of modulus $\leq r$ in a small angle $θ$ near the real axis, can be estimated by Const $θ^{3 / 2} r^{n}$ for $r$ sufficiently large depending on $θ$ . Here $n$ is the dimension.

Propagation of singularities in many-body scattering in the presence of bound states

András Vasy (1999)

Journées équations aux dérivées partielles

Similarity:

In these lecture notes we describe the propagation of singularities of tempered distributional solutions $u \in 𝒮^{'}$ of $(H - λ) u = 0$ , where $H$ is a many-body hamiltonian $H = Δ + V$ , $Δ \geq 0$ , $V = \sum_{a} V_{a}$ , and $λ$ 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...

The trace of the generalized harmonic oscillator

Jared Wunsch (1999)

Annales de l'institut Fourier

Similarity:

We study a geometric generalization of the time-dependent Schrödinger equation for the harmonic oscillator $(D_{t} + \frac{1}{2} Δ + V) ψ = 0 (0.1)$ where $Δ$ 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 $\frac{1}{2} ω^{2} x^{- 2}$ , with $x$ a boundary defining function for $M$ (e.g. $x = 1 / r$ in the compactified Euclidean...

The resolvent for Laplace-type operators on asymptotically conic spaces

Andrew Hassell, András Vasy (2001)

Annales de l’institut Fourier

Similarity:

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...

Boundary regularity and compactness for overdetermined problems

Ivan Blank, Henrik Shahgholian (2003)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Similarity:

Let $D$ be either the unit ball $B_{1} (0)$ or the half ball $B_{1}^{+} (0),$ let $f$ be a strictly positive and continuous function, and let $u$ and $Ω \subset D$ solve the following overdetermined problem: $Δ u (x) = χ_{_{Ω}} (x) f (x) in D, 0 \in \partial Ω, u = | \nabla u | = 0 in Ω^{c},$ where $χ_{_{Ω}}$ denotes the characteristic function of $Ω,$ $Ω^{c}$ denotes the set $D ∖ Ω,$ and the equation is satisfied in the sense of distributions. When $D = B_{1}^{+} (0),$ then we impose in addition that $u (x) \equiv 0 on {(x^{'}, x_{n}) | x_{n} = 0} .$ We show that a fairly mild thickness assumption on $Ω^{c}$ will ensure enough compactness on...