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Defocusing cubic nonlinear Schrödinger equation in two dimensions

Harmonic-oscillator energy space

The embedding $\Sigma\hookrightarrow L^2(\mathbb R^d)$ is compact. Rellich compactness controls a fixed ball, while
$$
\int_{|x|>R}|v|^2\leq R^{-2}\||x|v\|_2^2
$$
controls the tail uniformly.


Compact embedding of the harmonic-oscillator energy space

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 154 / 1 / 4 / Solution

Compact embedding of the harmonic-oscillator energy space

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Compact embedding of the harmonic-oscillator energy space

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