  In this paper we study a model of the two-dimensional quantum harmonic
oscillator in a 3-dimensional anti-de Sitter background. We use a generalized
Schr\"odinger picture in which the analogs of the Schr\"odinger operators of
the particle are independent of both the time and the space coordinates in
different representations. The spacetime independent operators of the particle
induce the Lie algebra of Killing vector fields of the $AdS_3$ spacetime. In
this picture, we have a metamorphosis of the Heisenberg's uncertainty
relations.
