Efficiency and localisation for the first Dirichlet eigenfunction

Bounds are obtained for the efficiency or mean to max ratio E(\Omega) for the first Dirichlet eigenfunction (positive) for open, connected sets \Omega with finite measure in Euclidean space \mathbb{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(n^{-2/3}\log n) . This disproves some claims in the literature. A key technical tool is the Feynman–Kac formula.