denote by $\Vert\cdot\Vert_p$ the standard $L^p$ norm, $1\le p\le \infty$. It is well-known ([@vdBC],[@vdB]) that the first eigenvalue of the Dirichlet Laplacian defined by $$\lambda_1(\Omega)=\inf_{\varphi\in
H_0^1(\Omega)\setminus\{0\}}\frac{\Vert \nabla\varphi\Vert_2^2}{
\Vert \varphi\Vert_2^2},$$ is bounded away from $0$. Furthermore $$\label{e3}
-\Delta v=1,\, \qquad v\in H_0^{ 1 }(\Omega),$$ has a unique solution denoted by $v_{\Omega}$. The function $v_{\Omega}$ is non-negative, pointwise increasing in $\Omega$, and satisfies, $$\label{e4}
\lambda_1(\Omega)^{-1}< \|v_{\Omega}\|_{\infty}\le
(4+3m\log 2)\lambda_1(\Omega)^{-1}.$$ The $m$-dependent constant in the right-hand side of has subsequently been improved ([@GS],[@HV]). We denote the sharp constant by $$\label{e5}
\mathfrak{c}_m=\sup\{\lambda_1(\Omega)\|v_{\Omega}\|_{\infty}:\Omega\,\textup{ open in }\,{\mathbb{R}}^m,\,|\Omega|<\infty\}.$$

Similarly, $Lv = 1$ has a unique solution $v_{\Omega, V}\in H^1_0(\Omega)$, referred to as the torsion function for $L$.

In this paper we consider the [*efficiency*]{} of the torsion function, and its localisation. The notion of efficiency, or mean to max ratio, goes back to [@ps] and [@sperb], where it was introduced for the first Dirichlet eigenfunction. Here we state the analogous definition for the torsion function of a Schrödinger operator on an open set $\Omega \subset {\mathbb{R}}^m $ with $0 < | \Omega| < \infty$.

\[def1\] Let $\Omega$ be an open set in ${\mathbb{R}}^m$ with $0 < |\Omega | < \infty,$ and let $V: \Omega \to {\mathbb{R}}^+$ be measurable. 