The Quantum Group as a Symmetry - The Schrödinger equation of the $N$-dimensional $q$-deformed Harmonic Oscillator -

Abstract

With the aim to construct a dynamical model with quantum group symmetry, the $q$-deformed Schrödinger equation of the harmonic oscillator on the $N$-dimensional quantum Euclidian space is investigated. After reviewing the differential calculus on the $q$-Euclidian space, the $q$-analog of the creation-annihilation operator is constructed. It is shown that it produces systematically all eigenfunctions of the Schrödinger equation and eigenvalues. We also present an alternative way to solve the Schrödinger equation which is based on the $q$-analysis. We represent the Schrödinger equation by the $q$-difference equation and solve it by using $q$-polynomials and $q$-exponential functions. The problem of the involution corresponding to the reality condition is discussed.