  The long-time asymptotics is analyzed for finite energy solutions of the 1D
Schr\"odinger equation coupled to a nonlinear oscillator; mathematically the
system under study is a nonlinear Schr\"odinger equation, whose nonlinear term
includes a Dirac delta. The coupled system is invariant with respect to the
phase rotation group U(1). This article, which extends the results of a
previous one, provides a proof of asymptotic stability of solitary wave
solutions in the case that the linearization contains a single discrete
oscillatory mode satisfying a non-degeneracy assumption of the type known as
the Fermi Golden Rule.
