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

TL;DR

This paper develops a q-analog of the free Schrödinger equation on homogeneous spaces of the Euclidean quantum group, analyzing eigenfunctions and eigenstates using q-exponentials and Hahn-Exton functions.

Contribution

It introduces a natural q-analog of the free Schrödinger equation on quantum homogeneous spaces and explores its eigenfunctions and eigenstates using harmonic analysis techniques.

Findings

Eigenfunctions factorized as products of q-exponentials

Eigenstates expressed via Hahn-Exton functions

Hahn-Exton and Jackson q-Bessel functions derived as matrix elements

Abstract

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