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 Schrödinger 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.

21 pages

References in corpus (4)

• Propagator of a Charged Particle with a Spin in Uniform Magnetic and Perpendicular Electric Fields
• The time-dependent quantum harmonic oscillator revisited: Applications to Quantum Field Theory
• The Minimum-Uncertainty Squeezed States for for Atoms and Photons in a Cavity
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Cited by in corpus (3)

• Generalized Lennard-Jones Potentials, SUSYQM and Differential Galois Theory
• A Note on a Differential Galois Approach to Path Integrals
• Quantum-based solution of time-dependent complex Riccati equations