  We describe a procedure naturally associating relativistic Klein-Gordon
equations in static curved spacetimes to non-relativistic quantum motion on
curved spaces in the presence of a potential. Our procedure is particularly
attractive in application to (typically, superintegrable) problems whose energy
spectrum is given by a quadratic function of the energy level number, since for
such systems the spacetimes one obtains possess evenly spaced, resonant spectra
of frequencies for scalar fields of a certain mass. This construction emerges
as a generalization of the previously studied correspondence between the Higgs
oscillator and Anti-de Sitter spacetime, which has been useful for both
understanding weakly nonlinear dynamics in Anti-de Sitter spacetime and
algebras of conserved quantities of the Higgs oscillator. Our conversion
procedure ("Klein-Gordonization") reduces to a nonlinear elliptic equation
closely reminiscent of the one emerging in relation to the celebrated Yamabe
problem of differential geometry. As an illustration, we explicitly demonstrate
how to apply this procedure to superintegrable Rosochatius systems, resulting
in a large family of spacetimes with resonant spectra for massless wave
equations.
