Harmonic-oscillator energy space ($\Sigma$)

The harmonic-oscillator energy space isThe spatial moment prevents mass from escaping to infinity.

The embedding $\Sigma \hookrightarrow L^2({\mathbb{R}}^d)$ is compact. Rellich compactness controls a fixed ball, whilecontrols the tail uniformly.

Critical points of the trapped defocusing cubic energysatisfy $-\Delta v+|x{|}^{2}v-\omega v+v^3=0$.