Non-existence of a Gelfand-Levitan Coordinate System for the Wave Equation
, by Bert Jacobs
The Gelfand-Levitan inversion procedure can be extended to multidimensional problems when the scattering potential is both local in one of the spatial variables and is frequency independent. Unfortunately, there is, in general, no coordinate transformation of the spatial variables which converts the pressure wave equation into a Schrodinger equation of the desired form. Thus, the procedure is probably not applicable to the pressure wave equation when the propogation medium is laterally and vertically heterogenous.
STANFORD