A unitary invariant of a 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 ${\cal H}$ we associate a topological space $\Omega_{L_0}$ ({\it wave spectrum}) constructed from the reachable sets of a dynamical system governed by the equation $u_{tt}+(L_0)^*u=0$. 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 $L_0=-\Delta|_{C^\infty_0(\Omega\backslash \partial \Omega)}$ acting in ${\cal H}=L_2(\Omega)$, 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 $\Rightarrow L_0 \Rightarrow \Omega_{L_0} \overset{\rm isom}= \Omega$'. 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.