Strong instability of standing waves for nonlinear Schr\"odinger equations with harmonic potential

We study strong instability of standing waves $e^{i\omega t} \phi_{\omega}(x)$ for nonlinear Schr\"odinger equations with $L^2$-supercritical nonlinearity and a harmonic potential, where $\phi_{\omega}$ is a ground state of the corresponding stationary problem. We prove that $e^{i\omega t} \phi_{\omega}(x)$ is strongly unstable if $\partial_{\lambda}^2 E(\phi_{\omega}^{\lambda}) |_{\lambda=1}\le 0$, where $E$ is the energy and $v^{\lambda}(x)=\lambda^{N/2} v(\lambda x)$ is the $L^2$-invariant scaling.