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We consider the $q$-deformed Schr\"odinger equation of the harmonicoscillator on the $N$-dimensional quantum Euclidian space. The creation andannihilation operator are found, which systematically produce all energy levelsand eigenfunctions of the Schr\"odinger equation. In order to get the$q$-series representation of the eigenfunction, we also give an alternative wayto solve the Schr\"odinger equation which is based on the $q$-analysis. Werepresent the Schr\"odinger equation by the $q$-difference equation and solveit by using $q$-polynomials and $q$-exponential functions.Comment: 19pages, Latex, TU-44

THE q-DEFORMED SCHRÖDINGER EQUATION OF THE HARMONIC OSCILLATOR ON THE QUANTUM EUCLIDEAN SPACE