On the distribution of scattering poles for perturbations of the Laplacian

Georgi Vodev

Displaying similar documents to “On the distribution of scattering poles for perturbations of the Laplacian”

On the real analyticity of the scattering operator for the Hartree equation

Changxing Miao, Haigen Wu, Junyong Zhang (2009)

Annales Polonici Mathematici

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We study the real analyticity of the scattering operator for the Hartree equation $i \partial_{t} u = - Δ u + u (V * | u | ²)$ . To this end, we exploit interior and exterior cut-off in time and space, together with a compactness argument to overcome difficulties which arise from absence of good properties as for the Klein-Gordon equation, such as the finite speed of propagation and ideal time decay estimate. Additionally, the method in this paper allows us to simplify the proof of analyticity of the scattering operator for the...

Recovering Asymptotics at Infinity of Perturbations of Stratified Media

Tanya Christiansen, Mark S. Joshi (2000)

Journées équations aux dérivées partielles

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We consider perturbations of a stratified medium $ℝ_{x}^{n - 1} \times ℝ_{y}$ , where the operator studied is $c^{2} (x, y) Δ$ . The function $c$ is a perturbation of $c_{0} (y)$ , which is constant for sufficiently large $| y |$ 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...

Sharp bounds for the number of the scattering poles

Georgi Vodev (1992)

Journées équations aux dérivées partielles

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Poisson formulæ for resonances.

Maciej Zworski (1996-1997)

Séminaire Équations aux dérivées partielles

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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

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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.

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.

Stability of the inverse problem in potential scattering at fixed energy

Plamen Stefanov (1990)

Annales de l'institut Fourier

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We prove an estimate of the kind $∥ q_{1} - q_{2} ∥_{L^{\infty}} \leq C ϕ (∥ A_{q_{1}} - A_{q_{2}} ∥_{R, 3 / 2 - 1 / 2})$ , where $A_{q_{i}} (ω, θ)$ , $i = 1, 2$ is the scattering amplitude related to the compactly supported potential $q_{i} (x)$ at a fixed energy level $k =$ const., $ϕ (t) = (- ln t)^{- δ}$ , $0 < δ < 1$ and $∥ \cdot ∥_{R, 3 / 2 - 1 / 2}$ is a suitably defined norm.