\|v_{\Omega}\|_{\infty}\le
    \frac{(m+2)^2}{4m^2} \bigg(\frac{|\Omega|}{\omega_m}\bigg)^{2/m}(1-\Phi(\Omega)),$$

    $$\label{p22b}
    \lambda_1(\Omega)\ge
    \frac{4m^2}{(m+2)^2}  \bigg(\frac{\omega_m}{|\Omega |}\bigg)^{2/m}(1-\Phi(\Omega))^{-1}.$$

-   If $\Omega$ is connected, then $$\label{p22c}
    \lambda_1(\Omega)\ge
    \frac{4m^2}{(m+2)^2}  \bigg(\frac{\omega_m}{|\Omega |}\bigg)^{2/m}(1-E(\Omega))^{-1}.$$

We now introduce the notion of $(\kappa, \delta)$-localisation of the torsion function of a Schrödinger operator on a non-empty open set $\Omega$ with finite Lebesgue measure. We refer to [@grebenkov] for a related notion of localisation for Dirichlet eigenfunctions.

\[def2\] Let $\Omega$ be an open set in ${\mathbb{R}}^m$ with $0 <|\Omega|<\infty$, and let $V:\Omega \to {\mathbb{R}}^+$ be measurable. Let $0 < \kappa<1, 0<\delta < 1.$ We say that $v_{\Omega, V}$ $(\kappa, \delta)$-localises on a measurable subset $A \subset \Omega$ if $$\frac{|A|}{|\Omega|} \le \delta,\qquad
\frac{\int_A v_{\Omega, V}}{\|v_{\Omega, V}\|_1}
\ge \kappa.$$ We say that $v_{\Omega, V}$ $(\kappa, \delta)$-localises if there exists a measurable subset $A \subset \Omega$ so that $v_{\Omega, V}$ $(\kappa, \delta)$-localises on $A$.

In Example \[exa1\] in Section \[sec3\] we analyse a one parameter family of Schrödinger operators in one dimension, where $(\kappa,\delta)$-localisation occurs for certain parameter choices.

The following definition defines localisation of the torsion function for sequences $(\Omega_n,V_n)$, and complements Definition \[def2\] above.

\[def3\] Let $(\Omega_n)$ be a sequence of open sets in ${\mathbb{R}}^m$ with $0 < |\Omega_n|<\infty$, and let $V_n:\Omega_n\to {\mathbb{R}}^+,$ be a sequence of measurable functions. Furthermore, 