SOq(N) covariant differential calculus on quantum space and quantum deformation of schrödinger equation

We construct a differential calculus on theN-dimensional non-commutative Euclidean space, i.e., the space on which the quantum groupSO_q(N) is acting. The differential calculus is required to be manifestly covariant underSO_q(N) transformations. Using this calculus, we consider the Schrödinger equation corresponding to the harmonic oscillator in the limit ofq?1. The solution of it is given byq-deformed functions.