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

This chapter is complementary to previous work of the authors, see (Acosta-Humanez 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 phi = 0 and phi = 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 Schrodinger equation in R-d: i partial derivative psi/partial derivative t = -1/2 Delta psi + Sigma(d)(j=1) b(j) (t)/2 x(j)(2)psi - f(j)(t)x (j) psi + ig (j) (t) partial derivative psi/partial derivative x(j) - i c(j) (t)/2 ( 2x(j) partial derivative psi/partial derivative x(j) + psi) using differential Galois theory.