Growth of Sobolev norms for $2d$ NLS with harmonic potential

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