Inverse resonance scattering on rotationally symmetric manifolds

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