Steklov Institute of Mathematics at St.Petersburg

PREPRINT 07/2003

INVERSE PROBLEMS ON A CLASS OF GRAPHS (TREES) BY THE BC-METHOD

This preprint was accepted June 23, 2003

ABSTRACT:
Planar graph consisting of strings of variable densities is considered.
The spectrum of the Dirichlet problem on the graph and the values of
derivatives of the (normalized) eigenfunctions at the boundary vertices
form the spectral data. We show that the graph without cycles (tree)
and the densities of its edges are determined by the spectral data
uniquely up a natural isometry in the plane. In the
framework of the appoach (the BC-method) we establish a boundary
controllability of the dynamical system gouverned by the wave
equation on the tree and exploit this property for recovering the tree
from its spectral data.