Efficiency and localisation for the first Dirichlet eigenfunction

TL;DR

This paper investigates the efficiency and localization properties of the first Dirichlet eigenfunction in Euclidean spaces, establishing bounds, implications, and examples that challenge previous claims using the Feynman-Kac formula.

Contribution

It provides new bounds and characterizations of eigenfunction localization, including counterexamples to existing literature, using probabilistic techniques.

Findings

Localization implies vanishing efficiency.

Vanishing efficiency bounds imply localization.

Eigenfunctions localize for elongated convex sets and certain quadrilaterals.

Abstract

Bounds are obtained for the efficiency or mean to peak ratio $E(\Omega)$ for the first Dirichlet eigenfunction (positive) for open, connected sets $\Omega$ with finite measure in Euclidean space $\R^m$. It is shown that (i) localisation implies vanishing efficiency, (ii) a vanishing upper bound for the efficiency implies localisation, (iii) localisation occurs for the first Dirichlet eigenfunctions for a wide class of elongating bounded, open, convex and planar sets, (iv) if $\Omega_n$ is any quadrilateral with perpendicular diagonals of lengths $1$ and $n$ respectively, then the sequence of first Dirichlet eigenfunctions localises, and $E(\Omega_n)=O\big(n^{-2/3}\log n\big)$. This disproves some claims in the literature. A key technical tool is the Feynman-Kac formula.