Uniform Non-Localization for Laplace Eigenfunctions on the Equilateral Triangle: Dirichlet, Neumann, and Robin Boundary Conditions

Abstract

We study uniform non-localization for complete Laplace eigenspaces on the equilateral triangle under Dirichlet, Neumann, and Robin boundary conditions, with particular emphasis on stability under Robin perturbations of the Neumann problem. As a stationary reference estimate, we give a self-contained proof that every measurable observation set $V\subset T$ of positive measure captures a frequency-independent positive proportion of the $L^2$ mass of every vector in every Dirichlet or Neumann eigenspace, even in the presence of arbitrarily large arithmetic multiplicity. The proof unfolds the triangle to a flat torus and combines a two-dimensional Fourier-cluster argument with a Jarník-type lattice lemma. Our principal perturbative result shows that the Neumann estimate persists uniformly for $0\leqσ\leqσ_0$ under Robin boundary conditions. McCartin's exact Robin parametrization yields modal-space estimates uniform in the spectral indices, while the small-parameter simplicity theorem of Rudnick and Wigman prevents distinct desymmetrized spectral classes from merging. Finally, for an arbitrary fixed $σ>0$, we show that bounded coincidence complexity of Robin modal classes is sufficient for uniform eigenspace-level observation via a multivariate Turán--Nazarov inequality. Thus the unconditional small-Robin theorem is separated from the remaining global spectral-complexity question at a general fixed Robin parameter.