Geodesic
Lie algebraic treatment of the~quadratic invariants for a~quantum system
Teoretičeskaâ i matematičeskaâ fizika, Tome 159 (2009) no. 1, pp. 142-161

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We consider the problem of the time-dependent degenerate parametric amplifier. We obtain the quadratic invariant and use it to derive the wave function via its $su(1,1)$ algebraic basis and a unitary transformation to the time-dependent Schrödinger equation for the parametric amplifier. We obtain the real and the complex invariants, which we use to solve the time-dependent Cauchy problem. Using different integrability conditions, we find the most general solution, which we analyze extensively, providing details of the calculations.
Keywords: wave function, Lie algebra. 
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     author = {M. Sebawe Abdalla and P. G. L. Leach},
     title = {Lie algebraic treatment of the~quadratic invariants for a~quantum system},
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M. Sebawe Abdalla; P. G. L. Leach. Lie algebraic treatment of the~quadratic invariants for a~quantum system. Teoretičeskaâ i matematičeskaâ fizika, Tome 159 (2009) no. 1, pp. 142-161. http://geodesic.mathdoc.fr/item/TMF_2009_159_1_a8/