Liouvillian Propagators and Degenerate Parametric Amplification with Time-Dependent Pump Amplitude and Phase

Abstract

This chapter is complementary to previous work of the authors, see (Acosta-Humánez et al., http://arxiv.org/abs/1311.2479, 2013; J. Phys. A Math. Theor. 46(45):455203–455219, 2013). We present in detail missed computations using differential Galois theory dealing with the construction of one-dimensional propagators for the degenerate parametric amplification with time-dependent pump amplitude and phase $\varphi=0$ and $\varphi=\pi/2$. Also presented is a generalization of Liouvillian propagators for the n-dimensional case, which concerns to the study of explicit solutions for the Cauchy problem of the Schrödinger equation in $\mathbb{R}^{d}:$

$$\begin{aligned} \text{i}\frac{\partial \psi}{\partial t}=-\frac{1}{2}\Delta \psi +\sum_{j=1}^{d}\frac{b_{j}\left( t\right) }{2}x_{j}^{2}\psi -f_{j}(t)x_{j}\psi +\text{i}g_{j}(t)\frac{\partial \psi}{\partial x_{j}}-\text{i}\frac{c_{j}\left( t\right) }{2}\left( 2x_{j}\frac{\partial \psi}{\partial x_{j}}+\psi \right)\end{aligned}$$

using differential Galois theory.