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Journal of Operator Theory

Volume 69, Issue 2, Spring 2013  pp. 299-326.

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

Authors M.I. Belishev
Author institution: St-Petersburg Department of the Steklov Mathematical Institute, St-Petersburg State University, Saint-Petersburg, Russia

Summary:  Let $L_0$ be a densely defined symmetric semi-bounded operator of non-zero defect indexes in a separable Hilbert space ${\mathcal H}$. With $L_0$ 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 {\it isometric} to the wave spectrum $\Omega_{L_0}$ of the minimal Laplacian $L_0=-\Delta|_{C^\infty_0(\Omega\backslash \partial \Omega)}$ acting in ${\mathcal H}=L_2(\Omega)$. In the mean time, $L_0$ is determined by the inverse data up to unitary equivalence. Hence, the manifold can be recovered by the scheme "data $\Rightarrow L_0 \Rightarrow \Omega_{L_0} \overset{\mathrm{isom}}{=} \Omega$".

Keywords:  symmetric semi-bounded operator, lattice with inflation, evolutionary dynamical system, wave spectrum, reconstruction of manifolds

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