Threshold Hierarchy for Packet-Scale Boundary Cancellation of Dirichlet Eigenfunctions

TL;DR

This paper establishes geometry-dependent thresholds for boundary correlation cancellation of high-frequency Dirichlet eigenfunctions on convex domains, revealing how boundary weights influence eigenfunction behavior at different scales.

Contribution

It introduces a threshold hierarchy based on boundary weights and curvature moments, providing new conditions for boundary correlation decay independent of eigenvalue ordering.

Findings

Boundary correlation coefficients vanish beyond certain packet scales.

Threshold conditions depend on boundary weight integrals and curvature moments.

Results are stable under eigenvalue crossings and do not rely on eigenvalue monotonicity.

Abstract

We identify geometry--dependent minimal packet scales required for cancellation of boundary correlations of high--frequency Dirichlet eigenfunctions on smooth strictly convex domains. The main result is a threshold hierarchy: for zero--mean boundary weights, the energy--weighted packet average of boundary correlation coefficients vanishes once the packet length exceeds a scale determined by the vanishing order of curvature moments of the weight. In particular, the threshold $N_k/k^{1-2/d}\to\infty$ suffices when $\int_{\partial\Omega} w,d\sigma=0$, while a strictly weaker threshold applies when additionally $\int_{\partial\Omega} H,w,d\sigma=0$, reducing in dimension $d=3$ to the minimal condition $N_k\to\infty$. The thresholds follow from the boundary local Weyl law. As a structural consequence of the Rellich identity alone, the single--mode share of boundary energy within any sublinear spectral packet is of order $1/N_k$. All estimates are independent of eigenvalue monotonicity and remain stable under eigenvalue crossings.