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Boundary Delocalization and Spectral Packets for Dirichlet Eigenfunctions

Anton Alexa

TL;DR

On a bounded $C^ abla$ strictly convex domain $\Omega$, high-frequency Dirichlet eigenfunctions cannot accumulate most of their boundary energy into a single mode when averaged over a sublinear spectral window. The authors develop a mode-to-packet comparison using the Rellich identity on the boundary and, for the multi-mode case, the boundary local Weyl law to obtain zero-mean cancellation. They prove that for any $N_k\to\infty$ with $N_k=o(k)$, $E_k/\sum_{m=k}^{k+N_k-1}E_m\to 0$, and that no positive fraction of modes in a sublinear packet can maintain a fixed zero-mean boundary bias, independent of eigenvalue monotonicity and stable under crossings. This boundary-delocalization principle provides a deterministic tool for boundary spectral statistics, highlighting that averaging over short spectral packets controls single-mode boundary concentration without invoking microlocal or dynamical assumptions.

Abstract

We establish a boundary delocalization principle for high-frequency Dirichlet eigenfunctions on smooth strictly convex domains. The main result excludes persistent boundary concentration at the level of individual eigenmodes when compared to short spectral packets of sublinear length. Quantitatively, we compare boundary energies of single eigenfunctions to packet sums over frequency windows of size N_k = o(k), without asserting any asymptotic gain in magnitude. The main mode-to-packet estimate relies only on the Rellich identity. For the multi-mode bias exclusion we additionally use the boundary local Weyl law to obtain a packet zero-mean cancellation estimate. This mode-to-packet comparison is independent of eigenvalue monotonicity and is stable under eigenvalue crossings.

Boundary Delocalization and Spectral Packets for Dirichlet Eigenfunctions

TL;DR

On a bounded strictly convex domain , high-frequency Dirichlet eigenfunctions cannot accumulate most of their boundary energy into a single mode when averaged over a sublinear spectral window. The authors develop a mode-to-packet comparison using the Rellich identity on the boundary and, for the multi-mode case, the boundary local Weyl law to obtain zero-mean cancellation. They prove that for any with , , and that no positive fraction of modes in a sublinear packet can maintain a fixed zero-mean boundary bias, independent of eigenvalue monotonicity and stable under crossings. This boundary-delocalization principle provides a deterministic tool for boundary spectral statistics, highlighting that averaging over short spectral packets controls single-mode boundary concentration without invoking microlocal or dynamical assumptions.

Abstract

We establish a boundary delocalization principle for high-frequency Dirichlet eigenfunctions on smooth strictly convex domains. The main result excludes persistent boundary concentration at the level of individual eigenmodes when compared to short spectral packets of sublinear length. Quantitatively, we compare boundary energies of single eigenfunctions to packet sums over frequency windows of size N_k = o(k), without asserting any asymptotic gain in magnitude. The main mode-to-packet estimate relies only on the Rellich identity. For the multi-mode bias exclusion we additionally use the boundary local Weyl law to obtain a packet zero-mean cancellation estimate. This mode-to-packet comparison is independent of eigenvalue monotonicity and is stable under eigenvalue crossings.
Paper Structure (5 sections, 8 theorems, 35 equations)

This paper contains 5 sections, 8 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. Rellich identity and single--mode bounds
  4. Packet scales and a contextual Weyl remark
  5. Boundary delocalization and spectral packet comparison

Key Result

theorem thmcountertheorem

Let $\Omega \subset \mathbb{R}^d$ be a bounded $C^\infty$ strictly convex domain, $d \ge 2$, and let $\{(\lambda_k,u_k)\}_{k\ge1}$ be the Dirichlet eigenpairs of $-\Delta_\Omega$. Let $N_k$ be any sublinear spectral window with $N_k\to\infty$ and $N_k=o(k)$. Then In particular, no individual high--frequency eigenfunction can asymptotically carry a fixed positive proportion of the boundary energy

Theorems & Definitions (24)

  • theorem thmcountertheorem: Main result - Introductory Form
  • remark thmcounterremark: Sharpness of the packet length condition
  • definition thmcounterdefinition: Spectral packets
  • definition thmcounterdefinition: Boundary correlation coefficient
  • remark thmcounterremark: Eigenvalue crossings
  • lemma thmcounterlemma: Single--mode boundary energy bound
  • proof
  • definition thmcounterdefinition: Integrated boundary energy
  • remark thmcounterremark: Integrated boundary Weyl expansion (context)
  • lemma thmcounterlemma: Packet zero--mean cancellation
  • ...and 14 more