q deformation by intertwining with application to the singular oscillator

Abstract

We present a version of q-deformed calculus based on deformed counterparts of Darboux intertwining operators. The case in which the deformed transformation function is of the vacuum type is detailed, but the extension to counterparts of excited states used as Darboux transformation functions is also formally discussed. The method leads to second-order Fokker-Planck-like deformed operators which may be considered as supersymmetric partners, though for a sort of q-deformed open systems, i.e., those possessing q nonlocal drift terms, potential part, as well as q-spreaded vacuum fluctuations. The undeformed limit corresponds to the conservative case, since all q nonlocalities wash out. The procedure is applied to the x^{-2} singular oscillator, for which we also present a formal q generalization of the Bagrov-Samsonov coherent states