Neumann Data Mass on Perturbed Triangles

TL;DR

This paper extends the analysis of Neumann data for eigenfunctions on triangles to perturbed triangles and those with small potentials, showing the Neumann data norm remains close to a geometric ratio with small deviations.

Contribution

It introduces the study of Neumann data mass on perturbed triangles and with small potentials, expanding previous work on standard triangles.

Findings

Neumann data norm approximates side length over triangle area

Deviation from the approximation is bounded by the perturbation size $psilon$

Results hold for both geometric perturbations and small potentials

Abstract

Based on a previous paper [Chr17] on Neumann data for Dirichlet eigenfunctions on triangles, we extend the study in two ways. First, we investigate the (semi-classical) Neumann data mass on perturbed triangles. Specifically, we replace one side of a triangle by adding a smooth perturbation, and assume that the disparity between the perturbation and the original side is bounded by a small value $\epsilon$. Second, we add a small $\epsilon$ sized potential to the (semi-classical) Laplacian and see how the results change on triangles. In both cases, we find that the $L^2$ norm of Neumann data on each side is close to the length of the side divided by the area of the triangle, and the difference is dominated by $\epsilon$.