Threshold Hierarchy for Packet-Scale Boundary Cancellation of Dirichlet Eigenfunctions

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.