  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.
