Wave Packets and Eigenvalue Estimates for Limiting Operators on the Disk

We study two-dimensional spatio-spectral limiting operators $$T_R:=P_{D(R)}{B}_S{P}_{D(R)}:L^2({\mathbb{R}}^2)\to L^2({\mathbb{R}}^2),$$ where $D(R)$ is a disk of radius $R>1$, $S\subset {\mathbb{R}}^2$ is a domain with well-shaped boundary, $P_{D(R)}$ is the orthogonal projection on the subspace of functions supported on $D(R)$, and $B_S$ is the orthogonal projection on the subspace of functions whose Fourier transform is supported on $S$. We construct a disk-adapted wave-packet frame for $L^2(D(R))$ with frame bounds uniform in $R$ using Gevrey-$s$ cutoffs ($s>1$) to obtain near-exponential Fourier localization. Exploiting these localization estimates, we bound the size of the eigenvalue plunge-region for $T_R$ and prove that for each $s>1$ and each $\epsilon \in (0,1\mathrm{/}2)$, $$\mathrm{\#}\{k:{\lambda }_k(T_R)\in (\epsilon ,1-\epsilon )\}=O\text{\hspace{0.17em}⁣}(R(\log (R\mathrm{/}\epsilon ){)}^{1+2s}),$$ with constants depending on $s$ and the geometric parameters of $S$. This bound improves existing plunge-region estimates in the classical setting where both domains are disks, when $\epsilon$ scales like $R^{-\nu }$ for a fixed $\nu >0$. By an affine transformation, the same result holds if $D(R)$ is a scaled ellipse.