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 $V\subset\Omega$ of positive measure, define \[ C_2(V;\Omega) := \inf_{\lambda\in\sigma(-\Delta_\Omega)} \inf_{0\neq u\in E_\lambda(\Omega)} \frac{\|u\|_{L^2(V)}}{\|u\|_{L^2(\Omega)}}. \] We prove that $C_2(V;\Omega)>0$ 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 $\alpha=|V|/|\Omega|$, we derive an explicit sufficient threshold $\lambda_*(\Omega,V)$ such that every eigenfunction with $\lambda\geq\lambda_*(\Omega,V)$ satisfies \[ \frac{\|u\|_{L^2(V)}}{\|u\|_{L^2(\Omega)}} \geq \left[ \frac{\alpha}{2} \left( 1-\frac{\sin(\pi\alpha)}{\pi\alpha} \right) \right]^{1/2}. \] 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.