Supersymmetric Pair of q-Deformed Nonlocal Operators

A simple version of the q-deformed calculus is used to generate a pair ofq-nonlocal, second-order difference operators by means of deformed counterpartsof Darboux intertwining operators for the Schrödinger—Hermite oscillators atzero factorization energy. These deformed nonlocal operators may be consideredas supersymmetric partners and their structure contains contributions originatingin both the Hermite operator and the quantum harmonic oscillator operator. Thereare also extra ±x contributions. The undeformed limit, in which allq-nonlocalities wash out, corresponds to the usual supersymmetric pair of quantum mechanicalharmonic oscillator Hamiltonians. The more general case of negative factorizationenergy is briefly discussed as well.