Liouvillian Propagators, Riccati Equation and Differential Galois Theory
Abstract
In this paper a Galoisian approach to build propagators through Riccati equations is presented. The main result corresponds to the relationship between the Galois integrability of the linear Schr\"odinger equation and the virtual solvability of the differential Galois group of its associated characteristic equation. As main application of this approach we solve the Ince's differential equation through Hamiltonian Algebrization procedure and Kovacic Algorithm to find the propagator for a generalized harmonic oscillator that has applications describing the process of degenerate parametric amplification in quantum optics and the description of the light propagation in a nonlinear anisotropic waveguide. Toy models of propagators inspired by integrable Riccati equations and integrable characteristic equations are also presented.
Keywords
Cite
@article{arxiv.1304.5698,
title = {Liouvillian Propagators, Riccati Equation and Differential Galois Theory},
author = {Primitivo B. Acosta-Humánez and Erwin Suazo},
journal= {arXiv preprint arXiv:1304.5698},
year = {2015}
}
Comments
21 pages