$i\sigma$ to denote the point in the physical region with norm $\sigma$ and argument $\pi/2$. Taking the positive sign, $I+V^{1/2}R_0(i\sigma)|V|^{1/2}=I+V^{1/2}R_0(i\sigma)V^{1/2} $ is a positive operator for $\sigma>0$. When we choose the negative sign, we will additionally assume that $\sigma$ is chosen large enough that $I- \vhalf R_0(i\sigma) \vhalf $ is a positive invertible operator; this is possible by, for example, insisting $\sigma>2(\| V \|_{\infty}+1)$ since $\|R_0(i\sigma)\| \leq 1/\sigma^2$. With these assumptions on $\sigma$, using the properties of the determinant and the fact that $V\geq 0$ we may rewrite the function $F_{m,\pm V}(i\sigma)$ from (\[eq:FmV\]) as $$\begin{aligned}
\label{eq:Fmrewrite}
F_{m}(i\sigma)& =F_{ m,\pm  V}(i\sigma)\\ &= 
\det \left( I \pm im (I\pm \vhalf R_0(i\sigma) \vhalf )^{-1/2} \vhalf T(i\sigma)
 \vhalf 
(I\pm \vhalf R_0(i\sigma) \vhalf )^{-1/2} \right). \nonumber\end{aligned}$$ We shall obtain a lower bound on $F_m(i\sigma)$ as $\sigma \rightarrow \infty$.

The following proposition is central to the proof of Theorem \[thm:lbd\] and is the main result of this section. Related results were obtained in odd dimensions in [@scv Section 5].

\[p:lb\] Let $V\in L^{\infty}_{\comp}(\Real^d)$, $V\geq 0$, and let $V$ be bounded below by $\epsilon \chi_{B}$ where $\epsilon>0$ and $\chi_B$ is the characteristic function of a nontrivial open ball. Let $m\in \Integers$, $m\not =0$. Then there is 