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


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