We consider a nonlinear Schrödinger equation
$\displaystyle iu_{t} -h_{0}u + \beta ( \vert u\vert^{2} )u=0 , (t,x)\in \mathbb{R}\times \mathbb{R}, $
with $ h_{0}= -\frac{d^{2}}{dx^{2}} +P(x)$ a Schrödinger operator with finitely many spectral bands. We assume the existence of an orbitally stable family of ground states. Exploiting dispersive estimates in Cuccagna (2008), Cuccagna and Visciglia (2009), and following the argument in Cuccagna (to appear) we prove that under appropriate hypotheses the ground states are asymptotically stable.
