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

  • Primitivo B. Acosta-HumánezEmail author
  • Erwin Suazo
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 121)


This chapter is complementary to previous work of the authors, see (Acosta-Humánez et al.,, 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.


Cauchy initial value problem Degenerate harmonic oscillator Differential Galois theory Mehler’s formula Linear Schrödinger equation Liouvillian propagator 



The first author was partially supported by the MICIIN/FEDER grant number MTM2009–06973, the Generalitat de Catalunya grant number 2009SGR859, and DIDI - Universidad del Norte (Raimundo Abello). The second author was partially supported by a grant from the Simons Foundation (#316295 to Erwin Suazo), Arizona State University, and University of Puerto Rico, Mayaguez. The authors thank to Greisy Morillo by their hospitality during the final process of this chapter.


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© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of MathematicsUniversidad del Atlántico and Intelectual. CoBarranquillaColombia
  2. 2.Department of Mathematical SciencesUniversity of Puerto RicoPuerto RicoUSA
  3. 3.School of Mathematics and StatisticsArizona State UniversityAZUSA

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