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

TL;DR

This paper introduces a topological invariant called the wave spectrum for semi-bounded operators, enabling the reconstruction of Riemannian manifolds from boundary data in inverse problems, and elucidates its operator-theoretic background.

Contribution

It establishes that the wave spectrum of a minimal Laplacian uniquely determines the manifold up to isometry, linking inverse boundary data to geometric reconstruction.

Findings

Wave spectrum is homeomorphic for unitarily equivalent operators.

Manifolds can be reconstructed from inverse data via the wave spectrum.

The approach extends the boundary control method to a broader class of systems.

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.