  We study the trapping of Abelian anyons (quasiholes and quasiparticles) by a
local potential (e.g., induced by an AFM tip) in a microscopic model of
fractional quantum Hall liquids with long-range Coulomb interaction and edge
confining potential. We find, in particular, at Laughlin filling fraction $\nu
= 1/3$, both quasihole and quasiparticle states can emerge as the ground state
of the system in the presence of the trapping potential. As expected, we find
the presence of an Abelian quasihole has no effect on the edge spectrum of the
quantum liquid, unlike in the non-Abelian case [Phys. Rev. Lett. {\bf 97},
256804 (2006)]. Although quasiholes and quasiparticles can emerge generically
in the system, their stability depends on the strength of the confining
potential, the strength and the range of the trapping potential. We discuss the
relevance of the calculation to the high-accuracy generation and control of
individual anyons in potential experiments, in particular, in the context of
topological quantum computing.
