Wave Packets and Eigenvalue Estimates for Limiting Operators on the Disk
Kevin Hughes, Arie Israel, Azita Mayeli
TL;DR
The paper addresses the problem of quantifying the plunge region in the spectrum of two-dimensional spatio-spectral limiting operators by constructing a disk-adapted wave-packet frame with Gevrey cutoffs to achieve near-Fourier localization. The authors develop a Whitney-type radial-angular sectorization of the disk, splitting into interior and boundary packets with linear and nonlinear phases, respectively, and prove a unit-norm frame for $L^2(D(R))$ with frame constants independent of $R$ and $s$. Using precise Fourier localization estimates, they show that the number of eigenvalues in the plunge interval $(\varepsilon,1-\varepsilon)$ satisfies $\#\{k: \lambda_k(T_R)\in(\varepsilon,1-\varepsilon)\} \le C_{s,S} \, R \, (\log(R/\varepsilon))^{1+2s}$, improving prior bounds in the disk-disk setting for certain scaling regimes of $\varepsilon$. The results extend to scaled ellipses by affine maps and advance quantitative spectral understanding of SSLOs, with potential implications for time-frequency analysis in bounded domains and for multi-scale spatio-spectral concentration. The methods combine Gevrey-cutoff based localization with a frame-based eigenvalue counting approach, yielding robust, scale-invariant bounds tied to boundary geometry.
Abstract
We study two-dimensional spatio-spectral limiting operators \[ T_R := P_{D(R)} B_S P_{D(R)} : L^2(\mathbb{R}^2) \rightarrow L^2(\mathbb{R}^2), \] 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 $\varepsilon\in(0,1/2)$, \[ \#\{k : λ_k(T_R)\in(\varepsilon,1-\varepsilon)\} = O\!\left(R (\log(R/\varepsilon))^{1+2s}\right), \] 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 $\varepsilon$ scales like $R^{-ν}$ for a fixed $ν> 0$. By an affine transformation, the same result holds if $D(R)$ is a scaled ellipse.