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Periods of multiple reflecting geodesics and inverse spectral results. (English) Zbl 0652.35027
Let $$\Delta_{\Omega}$$ be the self-adjoint operator defined by the Dirichlet problem in a bounded domain $$\Omega \subset {\mathbb{R}}^ n$$. Consider the distribution $$\sigma(t)=tr \cos \sqrt{\Delta_{\Omega}}t\in S'({\mathbb{R}})$$, denote by $${\mathcal L}_{\Omega}$$ the set of all periodic generalized geodesics in $${\bar \Omega}$$ and by $$T_{\gamma}$$ the periods (lengths) of $$\gamma\in {\mathcal L}_{\Omega}$$. For $$\Omega$$ with $$C^{\infty}$$ boundary $$\partial \Omega =X$$ denote by $$C^{\infty}_{emb}(X,{\mathbb{R}}^ n)$$ the subspace of all $$C^{\infty}$$ embeddings in the space of all $$C^{\infty}$$ maps with the Whitney topology. A set $$R$$ will be called residual if it is a countable intersection of open dense sets and $$\Omega_ f$$ is the bounded domain with $$\partial \Omega_ f=f(x)$$, $$f\in C^{\infty}_{emb}(X,{\mathbb{R}}^ n).$$
Theorem 1.1. If $$\Omega \subset {\mathbb{R}}^ 2$$ is strictly convex then there exists a neighbourhood W of Id in $$C^{\infty}_{emb}(x,{\mathbb{R}}^ 2)$$ and a residual set $$R\subset W$$ such that for Dirichlet laplacian $$\Delta_ f$$ in $$\Omega_ f$$ $$\sin g \sup p \sigma_ f(t)=\cup_{\gamma \in {\mathcal L}_ f}\{- T_{\gamma}\}\cup \{0\}\cup_{\gamma \in {\mathcal L}_ f}\{T_{\gamma}\}.$$ Moreover for every reflecting $$\gamma\in {\mathcal L}_ f$$, $$f\in R$$ we can recover the spectrum of Poincaré map related to $$\gamma$$, from the spectrum of $$\Delta_ f$$. The same results are true for the Neumann and Robin laplacians in $$\Omega_ f.$$
This theorem extends the so called Poisson relation for the manifolds with boundary proved by R. B. Melrose and J. Sjöstrand [Commun. Pure Appl. Math. 35, 129-168 (1982; Zbl 0546.35083)]. Similar results are proved for unbounded domains when the Poisson relation includes the poles of Lax-Phillips scattering matrix.
Theorem 1.4. If $$\Omega \subset {\mathbb{R}}^ n$$ is a smooth bounded domain then there exists a residual set $$R\subset C^{\infty}_{emb}(X,{\mathbb{R}}^ n)$$ such that all periodic geodesics with reflection points on $$f(X)$$ are at most countable, $$\forall f\in {\mathbb{R}}.$$
Combining this theorem with results by V. Ya. Ivrij [Funkts. Anal. Prilozh. 15, No.1, 74-75 (1981; Zbl 0455.35093)] authors establish two- term asymptotics of the Weyl counting function $$N(\lambda)$$ for domains $$\Omega_ f$$ from Theorem 1.4.
The paper contains also a variety of other results on the structure of the geodesics in the generic situation, interrelation between the spectrum of laplacian and the geodesics lengths spectrum, etc.
Reviewer: J.Frehse

##### MSC:
 35J25 Boundary value problems for second-order elliptic equations 35R30 Inverse problems for PDEs
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