2017/11/30 by Oleg Evnin, Hovhannes Demirchian, Armen Nersessian · 1 citation
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Classical mechanics #De Sitter universe #Equations of motion #Geometry #Klein–Gordon equation #Massless particle #Mathematical physics #Mathematics #Nonlinear Photonic Systems #Nonlinear system #Physics #Quantum #Quantum Mechanics and Non-Hermitian Physics #Quantum mechanics #Scalar (mathematics) #Spacetime #Universe #gr-qc #hep-th #math-ph #math.DG #math.MP #nlin.SI
paper · pdf · doi:10.1103/physrevd.97.025014
published as Phys. Rev. D 97, 025014 (2018) · v2: minor improvements, to appear in PRD
arxiv created 2018/01/15 · openalex publication_date 2018/01/24 · openalex created_date 2018/01/26 · arxiv updated 2018/01/31 · openalex updated_date 2026/08/05
We describe a procedure naturally associating relativistic Klein-Gordon equations in static curved spacetimes to nonrelativistic 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.