Klein-Gordonization: mapping superintegrable quantum mechanics to resonant spacetimes

TL;DR

This paper introduces Klein-Gordonization, a method linking superintegrable quantum systems with resonant spacetimes, enabling new insights into wave dynamics and conserved quantities in curved spacetime geometries.

Contribution

It generalizes the Higgs oscillator-AdS correspondence to a broader class of superintegrable systems, providing a systematic way to construct spacetimes with resonant spectra from quantum models.

Findings

Constructed spacetimes with evenly spaced resonant spectra.

Applied method to Rosochatius systems, generating new resonant spacetimes.

Connected the procedure to the Yamabe problem in differential geometry.

Abstract

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.