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Scattering frequencies and Gevrey 3 singularities. (English) Zbl 0723.35058
Scattering frequences (or resonances) \((\mu_ k)\) for an exterior Dirichlet-Laplacian \(\Delta =\Delta_{\Omega}\) \((\Omega ={\mathbb{R}}^ n\setminus Q\) the complement of a compact obstacle) can be described in many different ways: (i) Poles of the analytic continuation of the Green function \((z^ 2-\Delta)^{-1}\) in the non-physical sheet; (ii) Asymptotic eigenmodes of the wave-equation (1) \(u_{tt}-\Delta u=0\), \(u|_{t=0}=f\), \(u_ t|_{t=0}=g\), with compactly supported initial data; any such u can be expanded into the series u(x,t)\(\simeq \sum e^{\mu_ kt}\psi_ k(x)\), as \(t\to \infty\), where \(\psi =\psi_ k\) solves the reduced equation \((\Delta -\mu^ 2_ k)\psi =0\), subject to the Sommerfeld radiation condition at \(\infty\); (iii) Eigenvalues of the Lax-Phillips “incoming-outgoing” semigroup \(Z(t)=e^{tB}\). An interesting problem related to \(\{\mu_ k\}\) is the behavior of Re \(\mu\) \({}_ k\) as \(| \mu_ k| \to \infty\). P. D. Lax and R. S. Phillips [Arch. Ration. Mech. Anal. 40, 268-280 (1971; Zbl 0216.130)] showed that (2) Re \(\mu\) \({}_ k\leq -C \log | \mu_ k|\) for all nontrapping smooth obstacles, and linked (2) to propagation of singularities of the wave-equation, i.e., \(C^{\infty}\)-smoothness of u(x,t). G. Lebeau [Commun. Partial Differ. Equations 9, No.15, 1437-1494 (1984; Zbl 0559.35019)] improved the latter and showed that for analytic nontrapping obstacles Q, the solution u(x,t) belongs to the quasianalytic Gevrey class 3, i.e., \(\partial^ ku=O(k!^ 3)\). This yields the estimate (3) Re \(\mu\) \({}_{kj}\leq C| \mu_{kj}|^{-1/3}\). The authors prove that estimate (3) is optimal, i.e., (4) Re \(\mu\) \({}_{k_ j}\geq -C| \mu_{k_ j}|^{-1/3}\) holds for certain subsequences of \(\{\mu_ k\}\). Furthermore, they link the constant C in the RHS of (4) and a subsequence \(\{\mu_{k_ j}\}\) to a stable closed geodesic \(\gamma\subset \partial \Omega\), namely \[ C=C_{\gamma}=const \cos (\pi /6)/(2^{1/3}T)\int^{T}_{0}\rho^{2/3}(s)ds, \] where \(T=``length\) of \(\gamma\) ”, s is the arclength parameter along \(\gamma\), and \(\rho\) (s) is the curvature of \(\gamma\).

35P25 Scattering theory for PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35L05 Wave equation
Full Text: DOI EuDML
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