2012/08/15 by Μ. I. Belishev, Belishev, M. I.
Computer Science · Mathematics · #35R01 #47Axx #51-XX #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Functional Analysis (math.FA) #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.1208.3084
openalex publication_date 2012/08/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
With a densely defined symmetric semi-bounded operator of nonzero defect indexes L0 in a separable Hilbert space \cal H we associate a topological space ΩL0 (\it wave spectrum) constructed from the reachable sets of a dynamical system governed by the equation utt+(L0)^*u=0. Wave spectra of unitary equivalent operators are homeomorphic. In inverse problems, one needs to recover a Riemannian manifold Ω via dynamical or spectral boundary data. We show that for a generic class of manifolds, Ω is isometric to the wave spectrum ΩL0 of the minimal Laplacian L0=-Δ|C^∞0(Ω\backslash ∂ Ω) acting in \cal H=L2(Ω), whereas L0 is determined by the inverse data up to unitary equivalence. Hence, the manifold can be recovered (up to isometry) by the scheme `data ⇒ L0 ⇒ ΩL0 \overset\rm isom= Ω'. 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.