a constant $c_0>0$ so that $|F_{m,\pm V}(i\sigma)|\geq 
c_0\exp(c_0 \sigma ^d)$ for all sufficiently large $\sigma>0$.

The proof is similar to the proofs of some results of [@l-p; @vasy] in that it uses both a property of monotonicity in $V$ and the fact that for potentials which are positive multiples of the characteristic function of a ball much can be said by using a decomposition into spherical harmonics and special functions. However, the implementation of these underlying ideas is rather different here.

The proof of Proposition \[p:lb\] uses the following lemma, a monotonicity result reminiscent of results of [@l-p; @vasy]. In fact, the proof of this lemma uses a result from [@vasy].

\[l:normbd\] Let $V_1,\; V_2\in L^{\infty}(\Real^d)$ and suppose the support of $V_j$ is contained in $\overline{B}(R)=
\{ x\in \Real^d: |x| \leq R\}$ for $j=1,\;2$. Suppose $V_2(x)\geq V_1(x) \geq 0$ for all $x\in \Real^d$. We use the convention that $\vhalf_1/\vhalf_2$ is $0$ outside the support of $V_1$. Then $$\left\| (I + \vhalf_1 R_0(i\sigma) \vhalf_1)^{-1/2}\frac{ \vhalf_1}{\vhalf_2}
(I + \vhalf_2 R_0(i\sigma) \vhalf_2)^{1/2} \right\| \leq 1.$$ If $\sigma \geq 2(\|V_2\|_{\infty}+1)$, then $$\left\| (I - \vhalf_1 R_0(i\sigma) \vhalf_1)^{-1/2} \frac{\vhalf_1}{\vhalf_2}
(I - \vhalf_2 R_0(i\sigma) \vhalf_2)^{1/2} \right\| \leq 1.$$

When $\sigma>0$ is sufficiently large that $I\pm \vhalf_j 