An invertible linearization map for the quartic oscillator
Abstract
The set of world lines for the nonrelativistic quartic oscillator satisfying Newton's equation of motion for all space and time in 11 dimensions with no constraints other than the "spring" restoring force is shown to be equivalent (11onto) to the corresponding set for the harmonic oscillator. This is established via an energy preserving invertible linearization map which consists of an explicit nonlinear algebraic deformation of coordinates and a nonlinear deformation of time coordinates involving a quadrature. In the context stated, the map also explicitly solves Newton's equation for the quartic oscillator for arbitrary initial data on the real line. This map is extended to all attractive potentials given by even powers of the space coordinate. It thus provides classes of new solutions to the initial value problem for all these potentials.
 Publication:

Journal of Mathematical Physics
 Pub Date:
 December 2010
 DOI:
 10.1063/1.3527070
 arXiv:
 arXiv:1204.0765
 Bibcode:
 2010JMP....51l2904A
 Keywords:

 03.65.Ge;
 02.30.f;
 03.65.Fd;
 Solutions of wave equations: bound states;
 Function theory analysis;
 Algebraic methods;
 Mathematical Physics;
 Physics  Classical Physics
 EPrint:
 Journal of Mathematical Physics 51, 122904 (2010)