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Propriétés génériques des rayons refléchissants et applications aux problèmes spectraux. (Generic properties of reflected rays and applications to the spectral problem). (French) Zbl 0597.35092
Sémin. Bony-Sjöstrand-Meyer, Équations Dériv. Partielles 1984-1985, Exp. No. 12, 15 p. (1985).
Let \(-\Delta_ D\) be the self-adjoint realization of -\(\Delta\) under Dirichlet conditions in a domain \(\Omega \subset {\mathbb{R}}^ n\) with \(C^{\infty}\)- boundary \(\partial \Omega\). Associated with \(\Delta_ D\), in the case that \(\Omega\) is bounded, is the tempered distribution \[ (1)\quad \sigma_ D(t)=tr \cos \sqrt{-\Delta_ Dt}, \] while in the case that \(\Omega\) is unbounded, following Lax-Phillips, the distribution \[ (2)\quad \mu_ D(t)=\sum_{j}e^{i\mu_ jt}\in D'({\mathbb{R}}^+). \] In (2), the \(\mu_ j\) are the poles in the half-space Im z\(>0\) of the diffusion matrix S(\(\lambda)\). Let \({\mathcal L}_{\Omega}\) be the set of generalized periodic geodesics and \({\mathcal L}'_{\Omega}\) the set of reflected, periodic geodesics. In this brief survey article the author describes results obtained jointly with L. Stojanov, discussing generic properties of distributions of the form (1) and (2) with respect to domains whose boundaries are images of \(C^{\infty}\) functions f. It is shown that if \(T_{\gamma}\) is the period of a geodesic \(T_{\gamma}\) is of period of a geodesic \(\gamma\), in case (1), then generically \[ (R)\quad T_{\gamma}/T_{\delta}\not\in Q,\quad \forall \gamma,\delta \in {\mathcal L}'_{\Omega}\quad and\quad (S)\quad T_{\gamma},\quad \gamma \in {\mathcal L}'_{\Omega}, \] is isolated and periods of (S) are reflected geodesics allows the authors to verify the conjecture of Weyl on the asymptotic development of \(N(\lambda)=\#\{\lambda^ 2_ j\leq \lambda^ 2\}\). For strictly convex domains in \({\mathbb{R}}^ 2\), generic results are obtained on the characterization of sing supp \(\sigma\) \({}_ D(t)\) and sing supp \(\mu\) \({}_ D(t)\).
Reviewer: M.Thompson
MSC:
35P25 Scattering theory for PDEs
35P20 Asymptotic distributions of eigenvalues in context of PDEs
58J40 Pseudodifferential and Fourier integral operators on manifolds
58J47 Propagation of singularities; initial value problems on manifolds
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