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

We prove polynomial upper bounds on the growth of solutions to the 2d cubic nonlinear Schrödinger equation where the Laplacian is confined by the harmonic potential. Due to better bilinear effects, our bounds improve on those available for the 2d cubic nonlinear Schrödinger equation in the periodic setting: our growth rate for a Sobolev norm of order s is t2(s−1)/3+ε, for s=2k and k≥1 integer. In the appendix we provide a direct proof, based on integration by parts, of bilinear estimates associated with the harmonic oscillator.