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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?

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  • $\begingroup$ Just a comment to say that usually the $L^2$ norm is referred to as the mass, while the energy is reserved for say the $H^1$ norm. $\endgroup$
    – Dispersion
    Commented Jun 25 at 20:33
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    $\begingroup$ Additionally, you have your quantifiers wrong I believe. The correct phrasing is that once you fix $\varepsilon$, there exists a compact set $K$ such that your stated inequality holds for all time. $\endgroup$
    – Dispersion
    Commented Jun 25 at 20:57

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The statement Terry alluded to is a consequence of the RAGE Theorem.

For a statement and proof of the (abstract) RAGE Theorem, see Section 5.2 of the 2nd edition of Teschl's Mathematical Methods in Quantum Mechanics. (A copy of this book is freely available on the author's website.)

To get the precise result you are looking for, you need to realize that the projection of a wave function to a compact spatial subset is a compact operator. This is Corollary 10.3 in the same book.

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