Wave Packets and Eigenvalue Estimates for Limiting Operators on the Disk

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 $R&gt;1$, $S\subset\mathbb{R}<sup>2$ 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 $L<sup>2(D(R))$ with frame bounds uniform in $R$ using Gevrey-$s$ cutoffs ($s&gt;1$) 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 $s&gt;1$ 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 $R<sup>{-ν}$ for a fixed $ν&gt; 0$. By an affine transformation, the same result holds if $D(R)$ is a scaled ellipse.

• The paper demonstrates a quantitative bound on transition eigenvalues $M_\varepsilon(T_R)$ for spatially bounded elliptical domains and well-shaped frequency domains, with a bound of $R(\log(R/\varepsilon))^{1+2s}$.
• The study introduces a wave packet frame adapted to the disk, with a linear phase in the bulk and polar-linear phase boundary layer, achieving Gevrey regularity and $R$-independent frame constants.
• Advanced governing heuristics: The framework used acknowledges prior limitations in eigenvalue behavior at boundaries in different dimensions.

Context and problem

The paper studies the eigenvalue distribution of two-dimensional spatio–spectral limiting operators (SSLOs)

$$T_R := P_{F(R)} B_S P_{F(R)} : L^2(\mathbb{R}^2) \to L^2(\mathbb{R}^2),$$

where $P_{F(R)}$ is multiplication by the indicator of the dilated spatial domain $F(R)$ and $B_S$ is the Fourier multiplier projecting onto a frequency domain $S$. For $F$ an ellipse centered at the origin and $S$ a "well-shaped" domain (finite boundary length plus a Minkowski-type bound on the volume of $t$-neighborhoods of $\partial S$, covering convex domains, Lipschitz domains, and domains with maximally Ahlfors regular boundaries), the authors prove a quantitative bound on the size of the plunge region — the set of eigenvalues lying in $(\varepsilon, 1-\varepsilon)$.

Landau's Weyl-type law gives the asymptotic count of eigenvalues above any fixed threshold as $P_{F(R)}$0, but without rates. The present work contributes to the program of non-asymptotic bounds on $P_{F(R)}$1 initiated in one dimension by Landau–Widom and continued in higher dimensions in recent work. The main theorem states that for every Gevrey parameter $P_{F(R)}$2 there is a constant $P_{F(R)}$3 such that

$P_{F(R)}$4

The result holds verbatim for scaled ellipses by affine invariance of both the Fourier transform and the well-shaped class. In the regime where $P_{F(R)}$5 scales polynomially with $P_{F(R)}$6 (i.e., $P_{F(R)}$7), this improves prior estimates: it saves nearly one power of $P_{F(R)}$8 over the bound $P_{F(R)}$9 of Hughes–Israel–Mayeli, which itself saved one power over Marceca–Romero–Schmidt's exponent $F(R)$0. For fixed $F(R)$1, however, the earlier hypercube estimate of Israel–Mayeli, with log-power $F(R)$2, remains sharper than $F(R)$3 when $F(R)$4; the paper acknowledges this regime dependence explicitly rather than claiming uniform superiority.

Wave packet frame on the disk

The central analytic contribution is a disk-adapted wave packet system answering Open Problem 2 from the authors' earlier SampTA work in dimension $F(R)$5. The construction begins with a Whitney-type radial–angular sectorization: the disk $F(R)$6 (with $F(R)$7 dyadic, $F(R)$8) is partitioned into sectors $F(R)$9 at radial scale $B_S$0 carrying $B_S$1 angular arcs each. Sectors with $B_S$2 lie in the interior annulus away from $B_S$3; sectors with $B_S$4 form a boundary layer of fixed thickness inside $B_S$5.

Two families of packets are defined:

• Interior packets ($B_S$6): $B_S$7, with plane-wave phase.
• Boundary packets ($B_S$8): $B_S$9, whose phase is linear in polar coordinates but nonlinear in Cartesian coordinates — the key device for handling tangential localization near the curved boundary.

The cutoffs $S$0 and $S$1 are constructed explicitly in Gevrey class $S$2, $S$3, satisfying partition-of-unity, support, and derivative-growth conditions (R1)–(R3) and (A1)–(A3). Gevrey regularity yields near-exponential decay of Fourier transforms at rate $S$4, which is what drives all subsequent energy estimates.

The frame property (Proposition 2) is proved by reducing to orthonormal exponential bases on bounding boxes: interior sectors fit in squares of side $S$5, while boundary boxes are handled in $S$6-coordinates via norm equivalence between $S$7 and $S$8, using a standard frame-perturbation lemma. The resulting family is a unit-norm frame for $S$9 with absolute frame constants independent of both $F$0 and $F$1 — notably, the proof uses only the exact partition of unity and Plancherel, so no Gevrey decay enters here. Unlike the hypercube construction of Israel–Mayeli, the system is a frame rather than an orthonormal basis; the authors note this suffices for their counting argument.

Energy concentration estimates

The bridge from frames to eigenvalue counts is an energy concentration statement (Proposition 1): for each $F$2 the index set splits as $F$3 with

$F$4

For interior packets, the analysis rests on two lemmas. First, the envelope $F$5 is shown to be Gevrey-$F$6 at scale $F$7, via an induction establishing that Cartesian derivatives expand into trigonometric polynomials times powers of $F$8 acting on radial and angular derivatives; the factor $F$9 absorbs the curvature singularity at the origin. Second, Lemma 4.2 gives pointwise Fourier decay $S$0. Packets whose center frequency $S$1 lies within $S$2 of $S$3 are declared residual; the residual cardinality is controlled by lattice-point counting near dilated boundaries, using the well-shaped Minkowski condition. Summing the tail integrals gives total leakage $S$4 for out-of-band energy and $S$5 for in-band energy.

For boundary packets, the Fourier transform is expanded via the Jacobi–Anger decomposition into radial factors $S$6 (Hankel-type transforms involving $S$7) and angular factors $S$8. Three ingredients combine: elementary Bessel bounds $S$9 for $t$0 and $t$1; Gevrey decay of the angular Fourier coefficients at rate $t$2; and Gevrey decay of the radial factor in $t$3. A binning argument over the Bessel index $t$4 yields the joint decay

$t$5

from which the boundary residual set — those packets with $t$6 and $t$7 — has cardinality $t$8, matching the interior contribution up to constants.

Choosing $t$9 balances the two error sources and produces the stated residual bound. The final step applies the frame-based eigenvalue counting lemma (extending a lemma of Israel–Ward): if a unit-norm frame with lower constant $\partial S$0 satisfies the energy concentration inequality at level $\partial S$1, then $\partial S$2. Since $\partial S$3 acts on the supported frame elements exactly as $\partial S$4, the SSLO structure enters only through this reduction.

Limitations and open questions

Several restrictions are intrinsic to the argument. The plunge-region exponent $\partial S$5 degrades as the Gevrey index grows, and for fixed $\partial S$6 the bound is weaker than the hypercube estimate's $\partial S$7 whenever $\partial S$8; whether the disk case admits a uniformly better log-power remains open. The proof requires $\partial S$9 to be a power of two; extension to general $(\varepsilon, 1-\varepsilon)$0 presumably follows by monotonicity but is not carried out. The well-shaped hypothesis excludes domains whose boundaries have infinite length or fail the Minkowski condition, and the underlying heuristic — that $(\varepsilon, 1-\varepsilon)$1 should be governed by the codimension-one Hausdorff measure of $(\varepsilon, 1-\varepsilon)$2 and $(\varepsilon, 1-\varepsilon)$3 up to polylogarithmic losses — is asserted as consistent with the result rather than proved. Most significantly, the companion open problem concerning ball-adapted frames with residual cardinality $(\varepsilon, 1-\varepsilon)$4 remains unresolved in dimensions $(\varepsilon, 1-\varepsilon)$5; the polar-coordinate phase construction used here does not obviously generalize, since the boundary-layer geometry becomes genuinely higher-codimensional.

Conclusion

The paper establishes that for planar SSLOs with elliptical spatial domain and well-shaped frequency domain, the number of transition eigenvalues obeys $(\varepsilon, 1-\varepsilon)$6, improving prior general-domain bounds in the regime $(\varepsilon, 1-\varepsilon)$7. The methodological advance is a Gevrey-regularized, two-phase wave packet frame adapted to curved geometry — linear phases in the bulk, polar-linear phases in the boundary layer — with $(\varepsilon, 1-\varepsilon)$8-independent frame constants, together with sharp control of frequency leakage through lattice counting against the well-shaped boundary condition. The result confirms the boundary-measure heuristic for the plunge region in dimension two and reduces the higher-dimensional analogue to the construction of analogous frames on balls in $(\varepsilon, 1-\varepsilon)$9.