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

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 $V\subset \mathrm{\Omega }$ of positive measure, define $$C_2(V;\mathrm{\Omega }):=\underset{\lambda \in \sigma (-{\mathrm{\Delta }}_{\mathrm{\Omega }})}{\inf }\underset{0\mathrm{\ne }u\in E_{\lambda }(\mathrm{\Omega })}{\inf }\frac{\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2(V)}}{\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })}}\mathrm{.}$$ We prove that $C_2(V;\mathrm{\Omega })>0$ 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 $\alpha =\mathrm{\mid }V\mathrm{\mid }\mathrm{/}\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }$, we derive an explicit sufficient threshold ${\lambda }_{\ast }(\mathrm{\Omega },V)$ such that every eigenfunction with $\lambda \ge {\lambda }_{\ast }(\mathrm{\Omega },V)$ satisfies $$\frac{\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2(V)}}{\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })}}\ge {\left[\frac{\alpha }{2}(1-\frac{\sin (\pi \alpha )}{\pi \alpha })\right]}^{1\mathrm{/}2}\mathrm{.}$$ 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.