  We prove polynomial upper bounds on the growth of solutions to 2d cubic NLS
where the Laplacian is confined by the harmonic potential. Due to better
bilinear effects our bounds improve on those available for the $2d$ cubic NLS
in the periodic setting: our growth rate for a Sobolev norm of order s=2k,
$k\in \mathbb{N}$, is $t^{2(s-1)/3+\varepsilon}$. In the appendix we provide an
direct proof, based on integration by parts, of bilinear estimates associated
with the harmonic oscillator.
