In this Terry Tao's blog post, he claims that if one has a solution to the Schrödinger equation $$i\,\partial_t u +\Delta u=Vu $$ with a "reasonably smooth and localised $V$", $u$ has bounded energy ($L^2$ norm) and "$u$ is spatially localised in the sense that there exists a compact set $K$ and $\varepsilon>0$ such that $$\int_{\mathbb{R}^d-K}|u|^2(t,x)\,dx\le \varepsilon $$ for all times $t$, then one can conclude that $u$ is a combination of eigenfunctions.

Does anyone have a proof or reference for this (sort of) result?

mass, while the energy is reserved for say the $H^1$ norm. $\endgroup$