Scale-free unique continuation for arbitrary compact energy intervals

Let $d\geq 1$, let $E_0,G>0$, let $\delta\in(0,G/2)$, and let $(G,\delta)$-equidistributed sequences define the observation set $S_\delta$. Let $V:\mathds{R}^d\to\mathds{R}$ be measurable and bounded, and consider the Schrödinger operator $-\Delta+V$ on $\mathds{R}^d$. For a suitable constant $C_{\mathrm{sfUC}}^{G}$ of the same form as in the preceding scale-free unique-continuation estimate, the proposed extension. There is a constant $N=N(d)$ such that, for all $E_0,G>0$, all $\delta\in(0,G/2)$, all $(G,\delta)$-equidistributed sequences, all measurable and bounded $V:\mathds{R}^d\to\mathds{R}$, all $E_0>0$, and all $\phi\in\operatorname{Ran}\bigl(\chi_{(-\infty,E_0]}(-\Delta+V)\bigr)$, one has

$$\lVert\phi\rVert_{S_\delta}^2\geq C_{\mathrm{sfUC}}^{G}\lVert\phi\rVert_{\mathds{R}^d}^2,$$

with $C_{\mathrm{sfUC}}^{G}$ as above. The preceding theorem establishes this type of estimate only for short energy intervals; extending it to arbitrary compact intervals is not immediate from the existing proof, but a generalized eigenfunction expansion is expected to yield such an extension.