But a new approach offers reasonable hope of determining certain types of Riemannian manifolds or bounded domains from their spectra. This work was developed by Guillemin and Francoise and culminated in Guillemin's breakthrough result of 1993 that the quantum analogue of the Birkhoff normal form around an elliptic closed geodesic is a spectral invariant. In all proofs of this kind, one must assume that the length spectrum is simple so that its wave invariants can be separated out in the trace formula. In the case of a surface of revolution, the profile function is already even in a suitable sense so at least in the analytic case one can determine the surface from the spectrum. If an analytic plane domain has an up-down symmetry then it should also be spectrally determined.
Simple type means there are global action-angle variables for the geodesic flow.