The [*efficiency*]{} of $v_{\Omega,V}$ is $$\Phi(\Omega,V)=\frac{\|v_{\Omega,V}\|_1}{|\Omega|
\|v_{\Omega,V}\|_\infty}.$$ Let $(\Omega_n)$ be a sequence of open sets in ${\mathbb{R}}^m$ with $0<|\Omega_n|<\infty$, and let $V_n:\Omega_n\rightarrow {\mathbb{R}}^+$ be a sequence of measurable functions. We say that $(v_{\Omega_n,V_n})$ has vanishing efficiency if $\lim_{n\rightarrow\infty}\Phi(\Omega_n,V_n)=0$.

If $V=0$, then $\Phi(\Omega,0)\equiv \Phi(\Omega)$ coincides with the definition in [@DPGGLB].

Our first two results concern the comparison of $\Phi(\Omega,V)$ and $\Phi(\Omega)$.

\[the1\] If $\Omega$ is an open set in ${\mathbb{R}}^m$ with $0<|\Omega|<\infty$, and if $V: \Omega \to {\mathbb{R}}^+$ is measurable with $0\le V \le c$, then $$\label{e7}
\Phi(\Omega,V)\le 2^{2(3m+4)c/\lambda_1(\Omega)}\frac{8c+\lambda_1(\Omega)}{\lambda_1(\Omega)}\bigg(\frac{8c+\lambda_1(\Omega)}{8c}\bigg)^{8c/\lambda_1(\Omega)}\Phi(\Omega),$$ and $$\label{e8}
\Phi(\Omega,V)\ge 2^{-2(3m+4)c/\lambda_1(\Omega)}\frac{\lambda_1(\Omega)}{8c+\lambda_1(\Omega)}\bigg(\frac{8c}{8c+\lambda_1(\Omega)}\bigg)^{8c/\lambda_1(\Omega)}\Phi(\Omega).$$ Furthermore for fixed $\Omega$ the right-hand sides of and converge to $\Phi(\Omega)$ as $c\downarrow 0$.

\[the2\] 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. If there exists $\eta<\infty$ such that $\sup_{n\in{\mathbb{N}}}\frac{\| V_n \|_{\infty}}{\lambda_1(\Omega_n)} \le \eta,$ then $\lim_{n\rightarrow\infty}\Phi(\Omega_n,V_n)=0$ if and only if $\lim_{n\rightarrow\infty}\Phi(\Omega_n)=0$.

In [@HLP] it was shown that if $m\ge 2$, then $$\inf\{\Phi(\Omega):\Omega\subset {\mathbb{R}}^m,\,\Omega\,\,\,\textup{open},\, 0 < |\Omega|<\infty\}=0,$$ and $$\label{e17a}
\sup\{\Phi(\Omega):\Omega\subset {\mathbb{R}}^m,\,\Omega\,\,\,\textup{open},\, 0 < |\Omega|<\infty\}=1.$$

The analogous result for Schrödinger operators reads as follows.

\[the3\] If $\Omega$ is a fixed open set in ${\mathbb{R}}^m$ with $0 < | \Omega | < \infty$, then 