and let $(V_n)$ be a sequence of measurable potentials, $V_n:\Omega_n\rightarrow{\mathbb{R}}^+, n\in {\mathbb{N}}$, be such that $\lambda_1(\Omega_n,V_n)$ has multiplicity $1$. If $(v_{\Omega_n,V_n})$ has vanishing efficiency, $\lim_{n \to \infty}\Phi(\Omega_n,V_n) = 0$, then $(\varphi_{1, \Omega_n,V_n})$ localises. If $(v_{\Omega_n,V_n})$ localises, then $(\varphi_{1,\Omega_n,V_n})$ localises.

This paper is organised as follows. The proofs of Theorems \[the1\], \[the2\], \[the3\], \[the5\], and \[the8\] will be given in Section \[sec2\]. Examples \[exa1\] and \[exa2\] will be analysed in Sections \[sec3\] and \[sec4\] respectively. The proofs of Theorems \[the7\] and \[the9\] will be given in Sections \[sec5\] and \[sec6\] respectively.

Proofs of Theorems \[the1\], \[the2\], \[the3\], \[the5\], and \[the8\] \[sec2\]
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To prove Theorem \[the1\] and Theorem \[the2\], we first need to establish some auxiliary results. It is well known that the torsion function can be expressed in terms of the heat kernel of $L$. Let $\Omega \subset {\mathbb{R}}^m$ be open with $0 < |\Omega| < \infty$, and let $V:\Omega\rightarrow {\mathbb{R}}^+$ be measurable. Denote by $p_{\Omega,V}(x,y;t)$, $x\in \Omega, y\in \Omega,t > 0$ the heat kernel of $$\frac{\partial u}{\partial t} = Lu,  \qquad u\in H_0^1(\Omega).$$ The torsion function of $L$ then satisfies $$\label{e24}
v_{\Omega, V}(x) = \int_{\Omega} dy \int_{{\mathbb{R}}^+} dt\, p_{\Omega,V}(x,y;t),\qquad \forall x \in \Omega\ .$$ In case $V=0$ we write 