Liouvillian Propagators, Riccati Equation and Differential Galois Theory

TL;DR

This paper introduces a Galoisian method to construct propagators via Riccati equations, linking Galois integrability of Schrödinger equations to the solvability of their differential Galois groups, with applications in quantum optics.

Contribution

It establishes a novel connection between differential Galois theory and quantum propagator construction, applying it to solve complex differential equations like Ince's equation.

Findings

Successfully solved Ince's differential equation using Hamiltonian Algebrization and Kovacic Algorithm.

Linked Galois integrability to the solvability of the differential Galois group of Schrödinger equations.

Presented toy models of propagators based on integrable Riccati equations.

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.