NANOSYSTEMS: PHYSICS, CHEMISTRY, MATHEMATICS, 2013, 4 (4), P. 484–489
C*-ALGEBRAS IN RECONSTRUCTION OF MANIFOLDS
M. I. Belishev – St. Petersburg Department of the Steklov Mathematical Institute, St. Petersburg State University, St. Petersburg, Russia; belishev@pdmi.ras.ru
We deal with two dynamical systems associated with a Riemannian manifold with boundary. The first one is a system governed by the scalar wave equation, the second is governed by Maxwells equations. Both of the systems are controlled from the boundary. The inverse problems are to recover the manifold via the relevant measurements at the boundary (inverse data). We show that that the inverse data determine a C*-algebras, whose (topologized) spectra are identical to the manifold. By this, to recover the manifold is to determine a proper algebra from the inverse data, find its spectrum, and provide the spectrum with a Riemannian structure. This paper develops an algebraic version of the boundary control method (M.I.Belishev’1986), which is an approach to inverse problems based on their relations to control theory.
Keywords: inverse problems on manifolds, C*-algebras, boundary control method.
PACS 02.30.Jr, 02.30.Yy