c}}{\nu_l}$$ for all sufficiently large $\sigma$. Thus, this holds for $l$ satisfying $a\sigma/6>\nu_l> a \sigma /M$ if $\sigma $ is sufficiently large, providing a lower bound on $\| \vhalf_0 T \vhalf_0 P_l\|$, and thus on $\|B_{1,\pm V_0}(i\sigma)P_l\|$.

\[l:spcaselb\] Let $V_0=\epsilon \chi_a$, where $\epsilon, \; a>0$ and $\chi_a$ is the characteristic function of the ball of radius $a$ and center $0$. Then for $m_0\not =0$, $m_0\in \Integers$, there is a $c>0$ so that for $\sigma>0$ sufficiently large $$F_{m_0,\pm V_0}(i\sigma)\geq c\exp(c \sigma^d).$$ The constant $c$ depends on $a,\;\epsilon$, and $m_0$.

Recall that $$|F_{m_0,\pm V_0}(i\sigma)|= |\det( I\pm i m_0 B_{1, \pm, V_0}(i\sigma))|$$ and that for sufficiently large $\sigma>0$ $B_1(i\sigma)$ is a self-adjoint operator. Thus for sufficiently large $\sigma$ $$\label{eq:Fmeigenvalues}
|F_{m_0,\pm V_0}(i\sigma)|= \prod_{j=1}^{\infty} \sqrt{1+m_0^2 \lambda_j^2}$$ where $\lambda_j$ are the nonzero eigenvalues of $B_{1,\pm, V_0}(i\sigma)$. The $\lambda_j$ of course depend on $\sigma$, but we omit this in our notation.

A decomposition of $B_{1,\pm, V_0}$ using spherical harmonics shows that $B_{1,\pm, V_0}$ has eigenvalue $\| B_{1,\pm, V_0} P_l\|$ with multiplicity (at least) $\mu(l)$. Thus using (\[eq:Fmeigenvalues\]) and the fact that $\lambda_j^2> 0$, we get $$|F_{m_0}(i\sigma)|^2 
\geq \prod_{l=1}^{\infty} (1+m_0^2\| B_{1,\pm, V_0} P_l\|^2)^{\mu(l)}$$ for sufficiently large $\sigma$. From Lemma \[l:B1Pl\], we see $$\begin{aligned}
|F_{m_0}(i\sigma)|^2 & \geq \prod_{a\sigma/6>\nu_l> a 