An unitary invariant of semi-bounded operator and its application to inverse problems

TL;DR

The paper introduces the wave spectrum as a unitary invariant of semi-bounded operators and demonstrates its application in reconstructing unknown Riemannian manifolds from boundary data.

Contribution

It establishes that the wave spectrum of the minimal Laplacian uniquely determines the manifold in inverse boundary problems.

Findings

Wave spectrum is isometric for unitarily equivalent operators.

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

The approach connects inverse problems with system and control theory.

Abstract

Let $L_0$ be a closed densely defined symmetric semi-bounded operator with nonzero defect indexes in a separable Hilbert space ${\cal H}$. With $L_0$ we associate a metric space $\Omega_{L_0}$ that is named a {\it wave spectrum} and constructed from trajectories $\{u(t)\}_{t \geq 0}$ of a dynamical system governed by the equation $u_{tt}+(L_0)^*u=0$. The wave spectrum is introduced through a relevant von Neumann operator algebra associated with the system. Wave spectra of unitary equivalent operators are isometric. In inverse problems on {\it unknown} manifolds, 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 ${\cal H}=L_2(\Omega)$, whereas $L_0$ is determined by the inverse data up to unitary equivalence. By this, one can recover the manifold by the scheme "the 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) based on relations of inverse problems to system and control theory.