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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 \(\gamma\).

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