  We discuss inverse resonance scattering for the Laplacian on a rotationally
symmetric manifold $M = (0,\infty) \times Y$ whose rotation radius is constant
outside some compact interval. The Laplacian on $M$ is unitarily equivalent to
a direct sum of one-dimensional Schr\"odinger operators with compactly
supported potentials on the half-line. We prove
  o Asymptotics of counting function of resonances at large radius
  o Inverse problem: The rotation radius is uniquely determined by its
eigenvalues and resonances. Moreover, there exists an algorithm to recover the
rotation radius from its eigenvalues and resonances.
  The proof is based on some non-linear real analytic isomorphism between two
Hilbert spaces.

Spectral Theory;Differential Geometry