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

$\Rightarrow L_0 \Rightarrow \Omega_{L_0} \overset{\mathrm{isom}}{=} \Omega$ ".

DOI: http://dx.doi.org/10.7900/jot.2010oct22.1925
Keywords: symmetric semi-bounded operator, lattice with inflation, evolutionary dynamical system, wave spectrum, reconstruction of manifolds

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