Quantitative and Uniform $L^2$ Non-Localization on Integrable Polygons

Abstract

We study uniform $L^2$ non-localization of Dirichlet Laplacian eigenfunctions on planar integrable polygons, with particular emphasis on spectral degeneracy and quantitative dependence on the observation set. For a measurable set $V\subsetΩ$ of positive measure, define \[ C_2(V;Ω) := \inf_{λ\inσ(-Δ_Ω)} \inf_{0\neq u\in E_λ(Ω)} \frac{\|u\|_{L^2(V)}}{\|u\|_{L^2(Ω)}}. \] We prove that $C_2(V;Ω)>0$ for rectangles, isosceles right triangles, equilateral triangles, and hemi-equilateral triangles, uniformly over the complete eigenspaces and hence independently of spectral multiplicity. For rectangles, we obtain a quantitative refinement. If the reflected extension of $V$ has finite perimeter and $α=|V|/|Ω|$, we derive an explicit sufficient threshold $λ_*(Ω,V)$ such that every eigenfunction with $λ\geqλ_*(Ω,V)$ satisfies \[ \frac{\|u\|_{L^2(V)}}{\|u\|_{L^2(Ω)}} \geq \left[ \fracα{2} \left( 1-\frac{\sin(πα)}{πα} \right) \right]^{1/2}. \] We further establish stability under bounded real-valued potentials on the rectangular branch and under controlled spectral defects, yielding corresponding non-localization results for sufficiently accurate quasimodes and narrow spectral clusters.