Free q-Schrodinger Equation from Homogeneous Spaces of the 2-dim Euclidean Quantum Group

摘要

After a preliminary review of the definition and the general properties ofthe homogeneous spaces of quantum groups, the quantum hyperboloid qH and thequantum plane qP are determined as homogeneous spaces of Fq(E(2)). Thecanonical action of Eq(2) is used to define a natural q-analog of the freeSchro"dinger equation, that is studied in the momentum and angular momentumbases. In the first case the eigenfunctions are factorized in terms of productsof two q-exponentials. In the second case we determine the eigenstates of theunitary representation, which, in the qP case, are given in terms of Hahn-Extonfunctions. Introducing the universal T-matrix for Eq(2) we prove that theHahn-Exton as well as Jackson q-Bessel functions are also obtained as matrixelements of T, thus giving the correct extension to quantum groups of wellknown methods in harmonic analysis.