Inverse boundary spectral problem for Riemannian polyhedra

The object under consideration is an admissible Riemannian polyhedron M with a piece-wise smooth boundary δM. This is a finite n-dimensional simplicial complex equipped with a family of Riemannian metrics smooth inside each simplex. We introduce an anisotropic Dirichlet Laplace operator in a weak sense for the admissible Riemannian polyhedron and define a set of boundary spectral data Γ, {λκ, δυφκ|Γ}∞κ=1 on an open part Γ ∩ δM, where λκ are the eigenvalues on Γ and δυφκ|Γ are the traces of normal derivatives of eigenfunctions of the Laplacian. The main result of the work is: if two admissible Riemannian polyhedra M and M have open diffeomorphic parts of the boundaries Γ ∩ δM and Γ ∩ δM such that the set of boundary spectral data on Γ coincides with the set of boundary spectral data on Γ, then there is one-to-one correspondence between M and M as simplicial complexes and they are also isometric as metric spaces. A new technique was developed to tackle the problem. That technique incorporated two methods: BC-method generalized and adjusted for the admissible Riemannian polyhedra and the technique of Gaussian beams extended for anisotropic piecewise smooth media.