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In this paper we study a model of the two-dimensional quantum harmonicoscillator in a 3-dimensional anti-de Sitter background. We use a generalizedSchr\"odinger picture in which the analogs of the Schr\"odinger operators ofthe particle are independent of both the time and the space coordinates indifferent representations. The spacetime independent operators of the particleinduce the Lie algebra of Killing vector fields of the $AdS_3$ spacetime. Inthis picture, we have a metamorphosis of the Heisenberg's uncertaintyrelations.

A model of the two-dimensional quantum harmonic oscillator in an $AdS_3$ background