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We construct the Perelomov number coherent states for any three $su(1,1)$ Liealgebra generators and study some of their properties. We introduce threeoperators which act on Perelomov number coherent states and close the $su(1,1)$Lie algebra. We show that the most general $SU(1,1)$ coherence-preservingHamiltonian has the Perelomov number coherent states as eigenfunctions, and weobtain their time evolution. We apply our results to obtain the non-degenerateparametric amplifier eigenfunctions, which are shown to be the Perelomov numbercoherent states of the two-dimensional harmonic oscillator.

The SU(1,1) Perelomov number coherent states and the non-degenerate parametric amplifier