Inverse resonance scattering for rotationally symmetric manifolds

We consider the Laplacian on a rotationally symmetric manifolds M. We assume that M is a cylindrical manifold with warped product. We show that the Laplacian has infinite number of eigenvalues for a specific rotation radius, and has not eigenvalues for another specific rotation radius.
We describe the resonances. Moreover, we show that the resonances and eigenvalues determine the rotation radius uniquely.
It is a joint result with Hiroshi Isozaki, Japan.
The talk will be in Russian.