\[the2\], and if either $(v_{\Omega_n})$ or $(v_{\Omega_n,V_n})$ have non-vanishing efficiencies then both $(v_{\Omega_n})$ and $(v_{\Omega_n,V_n})$ are not localising.

[*Proof of Theorem* ]{} Using and we obtain $$v_{\Omega,V}(x)=\sum_{j =1}^{\infty} \lambda_j(\Omega,V)^{-1} \bigg(\int_{\Omega}\varphi_{j, \Omega,V}\bigg)\varphi_{j, \Omega,V}(x).$$ Integrating with respect to $x$ over $\Omega$ gives $$\begin{aligned}
\int_{\Omega}v_{\Omega,V}&=\sum_{j =1}^{\infty} \lambda_j(\Omega,V)^{-1} \bigg(\int_{\Omega}\varphi_{j, \Omega,V}\bigg)^2\nonumber \\ &
\ge \lambda_1(\Omega,V)^{-1} \bigg(\int_{\Omega}\varphi_{1, \Omega,V}\bigg)^2.\end{aligned}$$ Multiplying both sides with $\lambda_1(\Omega,V)$, and using the definition of $\mathfrak{d}_m$ in gives $$\mathfrak{d}_m\frac{\|v_{\Omega,V}\|_1}{\|v_{\Omega,V}\|_{\infty}}\ge \bigg(\int_{\Omega}\varphi_{1, \Omega,V}\bigg)^2.$$ This implies that $$\mathfrak{d}_m\Phi(\Omega,V)\ge \frac{1}{|\Omega|}\bigg(\int_{\Omega}\varphi_{1, \Omega,V}\bigg)^2.$$ If $\lim_{n\rightarrow\infty}\Phi(\Omega_n,V_n)=0$, then it follows that $\lim_{n\rightarrow\infty}\frac{1}{|\Omega_n|}\big(\int_{\Omega_n}\varphi_{1, \Omega_n,V_n}\big)^2=0$. This implies localisation by Lemma 3 in [@vdBBDPG]. Since by Theorem \[the5\] $(v_{\Omega_n,V_n})$ has vanishing efficiency if it localises, the last assertion in Theorem \[the8\] follows. $\square$

 Example \[exa1\] {#sec3}
=================

In this section we analyse a one parameter family of Schrödinger operators in one dimension, where $(\kappa,\delta)$-localisation for certain parameter choices occurs.

\[exa1\] If $\Omega=(-1, 1)$, and $$V_\nu(x) = \nu^4 1_{(-1, -\frac{1}{\nu})}(x) +
\nu^4 1_{(\frac{1}{\nu}, 1)}(x)
\, , \quad x \in (-1, 1)\, ,$$ where $\nu>1$ is a real parameter, then the torsion function $v_\nu := v_{\Omega, V_\nu}$ satisfies $$\label{fundamental estimate}
\frac{\int_{1/\nu}^1 v_\nu}{\int_{0}^{1/\nu} v_\nu}
\le \frac{6}{\nu}\, .$$ As a consequence, for any $0 < \kappa < 1,$ $0 < \delta < 1$ there exists $\nu_{\kappa, \delta} >1$ 