---
abstract: 'We consider the torsion function for the Dirichlet Laplacian $-\Delta$, and for the Schrödinger operator $- \Delta + V$ on an open set $\Omega\subset {\mathbb{R}}^m$, with Lebesgue measure $0<|\Omega|<\infty$, with a real-valued, non-negative, measurable potential $V.$ We investigate the phenomena of vanishing efficiency and localisation, and large efficiency for the torsion function and the first Dirichlet eigenfunction.'
author:
- |
    [M. van den Berg]{}\
    School of Mathematics, University of Bristol\
    Fry Building, Woodland Road\
    Bristol BS8 1UG, United Kingdom\
    `mamvdb@bristol.ac.uk`\
    \
    [D. Bucur]{}\
    Laboratoire de Mathématiques, Université Savoie Mont Blanc\
    UMR CNRS 5127\
    Campus Scientifique, 73376 Le-Bourget-Du-Lac, France\
    `dorin.bucur@univ-savoie.fr`\
    \
    [T. Kappeler]{}\
    Institut für Mathematik, Universität Zürich\
    Winterthurerstrasse 190, CH-8057 Zürich, Switzerland\
    `thomas.kappeler@math.uzh.ch`
date: 13 May 2020
title: On efficiency and localisation for the torsion function
---

3truecm

     [ Mathematics Subject Classification (2000)]{}: 35J05, 35J25, 35K05, 35P99.

**Keywords**: Torsion function, first Dirichlet eigenfunction, Schrödinger operator, Dirichlet boundary condition, localisation, efficiency.

Introduction and main results \[sec1\]
======================================

Let $\Omega$ be an open set in ${\mathbb{R}}^m$, with finite Lebesgue measure, $0 < |\Omega| < \infty$, and boundary $\partial\Omega$, and let $V:\Omega\rightarrow {\mathbb{R}}^+,\, {\mathbb{R}}^+= [0,\infty) $ be measurable. Let $$L=-\Delta+V,$$ be the Schrödinger operator acting in $L^2(\Omega)$. Throughout this paper we 