Threshold Hierarchy for Packet-Scale Boundary Cancellation of Dirichlet Eigenfunctions
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866912914832621568 |
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| author | Alexa, Anton |
| author_facet | Alexa, Anton |
| contents | 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 $\int_{\partialΩ} w,dσ=0$, while a strictly weaker threshold applies when additionally $\int_{\partialΩ} H,w,dσ=0$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_11605 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Threshold Hierarchy for Packet-Scale Boundary Cancellation of Dirichlet Eigenfunctions Alexa, Anton Spectral Theory Analysis of PDEs 35P20, 58J50, 35P15 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 $\int_{\partialΩ} w,dσ=0$, while a strictly weaker threshold applies when additionally $\int_{\partialΩ} H,w,dσ=0$, 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. |
| title | Threshold Hierarchy for Packet-Scale Boundary Cancellation of Dirichlet Eigenfunctions |
| topic | Spectral Theory Analysis of PDEs 35P20, 58J50, 35P15 |
| url | https://arxiv.org/abs/2601.11605 |