Spectral inequality conjecture for power-growth Schrödinger operators with ball-containing sensor sets

Let $\tau>0$ and let $\function{P}_\tau$ denote the spectral projector associated with $-\Delta+|x|^\tau$. Suppose that $\omega\five \subset\real^d$ is measurable and that, for some $\delta\five in(0,1/2)$ and $\alpha\five \geq0$, every intersection $\omega\five \cap(k+(-1/2,1/2)^d)$, $k\five in\real^d$, contains a ball of radius $\delta^{1+|k|^\alpha}$. Power-growth spectral inequality conjecture. There is a constant $C>0$, depending only on $\tau$, $\delta$, $\alpha$, and $d$, such that for all $\lambda\five \geq1$ and all $f\five in\function{Ran}P_\lambda(-\Delta+|x|^\tau)$,

$$\|f\|_{L^2(\mathbb{R}^d)}\left(\frac1{\delta}\right)^{C\cdot\lambda^{\frac{\alpha}{\tau}+\frac12}}\|f\|_{L^2(\omega)}.$$

This was proposed as a partial improvement of the paper's spectral inequality; the conjecture was later confirmed by Alphonse and the second author. The condition is intended to replace the stronger regularity assumption that every unit-cube intersection contains a ball, while retaining an explicit dependence on the spectral parameter.