A unitary invariant of semi-bounded operator in reconstruction of manifolds

Abstract

With a densely defined symmetric semi-bounded operator of nonzero defect indexes $L_0$ in a separable Hilbert space $\mathcal{H}$ we associate a topological space ${\Omega }_{L_0}$ ({\it wave spectrum}) constructed from the reachable sets of a dynamical system governed by the equation $u_{tt}+(L_0{)}^{\ast }u=0$. Wave spectra of unitary equivalent operators are homeomorphic. In inverse problems, one needs to recover a Riemannian manifold $\Omega$ via dynamical or spectral boundary data. We show that for a generic class of manifolds, $\Omega$ is isometric to the wave spectrum ${\Omega }_{L_0}$ of the minimal Laplacian $L_0=-\Delta {|}_{C_0^{\infty }(\Omega \backslash \partial \Omega )}$ acting in $\mathcal{H}=L_2(\Omega )$, whereas $L_0$ is determined by the inverse data up to unitary equivalence. Hence, the manifold can be recovered (up to isometry) by the scheme `data $\Rightarrow L_0\Rightarrow {\Omega }_{L_0}\stackrel{\mathrm{isom}}{=}\Omega$'. The wave spectrum is relevant to a wide class of dynamical systems, which describe the finite speed wave propagation processes. The paper elucidates the operator background of the boundary control method (Belishev`1986), which is an approach to inverse problems based on their relations to control theory.

This paper is an amended version of the article arXiv:1004.1646v1 [math.FA] 9 Apr 2010