$\lambda=\lambda_1(\Omega,V)$, and Lemma 2 and its proof in [@vdBC] hold for the semigroup associated with $L$. Finally Lemma 3 and Theorem 1 and their proofs in [@vdBC] hold with $\lambda=\lambda_1(\Omega,V)$. This proves the upper bound in . It remains to prove the lower bound. Let $\Omega_R=\Omega\cap B(0;R)$. Then $0<|\Omega_R|\le \omega_mR^m$. Let $L_R$ be the restriction of $-\Delta+V$ acting in $L^2(\Omega_R)$ with Dirichlet boundary conditions on $\partial \Omega_R$. If we denote $V_R={\bf1}_{B(p;R)}V,$ then $L_R$ is also the operator $-\Delta+V_R$ acting in $L^2(\Omega_R)$ with Dirichlet boundary conditions on $\partial \Omega_R$. Then $L_R$ is self-adjoint, and its spectrum is discrete. Since the first Dirichlet eigenfunction $\varphi_{1,\Omega_R,V_R}$ is non-negative, and $\| \varphi_{1,\Omega_R,V_R} \|_2 =1 $, one has, by the Cauchy-Schwarz inequality, $$\label{e31}
0<\int_{\Omega_R}\varphi_{1,\Omega_R,V_R}\le |\Omega_R|^{1/2}.$$ By self-adjointness $$\begin{aligned}
\label{e32}
\int_{\Omega_R}\varphi_{1,\Omega_R,V_R}
&=\int_{\Omega_R}\varphi_{1,\Omega_R,V_R}L_Rv_{\Omega_R,V_R}=
\int_{\Omega_R}\big(L_R\varphi_{1,\Omega_R,V_R}\big)v_{\Omega_R,V_R}\nonumber \\
&=\lambda_1(\Omega_R,V_R)\int_{\Omega_R}\varphi_{1,\Omega_R,V_R}v_{\Omega_R,V_R}\nonumber \\
& \le \lambda_1(\Omega_{R},V_R)\|v_{\Omega_R,V_R}\|_{\infty}\int_{\Omega_R}\varphi_{1,\Omega_R,V_R}.\end{aligned}$$ By – we conclude that $$\lambda_1(\Omega_R,V_R) \, \|v_{\Omega_R,V_R}\|_{\infty} \ge 1.$$ Since $\|v_{\Omega_R,V_R}\|_{\infty}\le \|v_{\Omega,V}\|_{\infty}$ we have that $$\lambda_1(\Omega_R,V_R) \|v_{\Omega,V}\|_{\infty}\ge 1.$$ The assertion follows since $R\mapsto \lambda_1(\Omega_R,V_R)$ is decreasing to $\lambda_1(\Omega,V)$ as $R\rightarrow\infty$. $\square$

\[lem2\] Let $\Omega\subset {\mathbb{R}}^m$ be open with $0 <  |\Omega | < \infty$. If $V : \Omega \rightarrow {\mathbb{R}}^+$ is measurable with $0 \le c_1 \le V(x) \le c_2 < \infty$, $x \in \Omega$, then 