continuation to $\Lambda$, and $G_{\nu_l}(r,r';\lambda)$ does as well.

Now we use [@ab-st 9.1.35, 9.1.36] to obtain $$J_\nu(e^{i\pi}z)= e^{i\pi \nu}J_{\nu}(z).$$ Specializing [@ab-st 9.1.36] to the case of $\nu$ an integer we have $$Y_{\nu_l}(e^{i\pi}z)= e^{-\nu_l \pi i}(Y_{\nu_l}(z)+2i J_{\nu_l}(z))$$ giving $$H_{\nu_l}^{(1)}(e^{i\pi }z)= e^{i\nu_l \pi}(-J_{\nu_l} (z)+i Y_{\nu_l}(z)).$$ Thus $$\label{eq:gnudiff}
\tilde{G}_{\nu_l}(r,r';\lambda){\stackrel{\rm{def}}{=}}G_{\nu_l}(r,r';e^{i\pi} \lambda)-G_{\nu_l}(r,r';\lambda)
= i\pi (rr')^{-(d-2)/2}J_{\nu_l}(\lambda r)J_{\nu_l}(\lambda r').$$ Together, (\[eq:R0exp\]) and (\[eq:gnudiff\]) give us an expression for the Schwartz kernel of $R_0(e^{i\pi}\lambda)-R_0(\lambda)$ in spherical coordinates: with $r, \; r'>0$, $\theta \in \Sphere^{d-1}$, $$\begin{gathered}
\label{eq:shexp12}
\left((R_0(e^{i\pi \lambda})-R_0(\lambda))g\right)(r\theta)  \\=
\sum_{l=0}^{\infty} \sum_{\mu =1}^{\mu(l)}
\int_0^\infty  \int_{\Sphere^{d-1}} \tilde{G}_{\nu_l}(r,r';\lambda) Y^{\mu}_l( \theta)
\overline{Y}^\mu_l(\omega) g(r'\omega) (r')^{d-1}dS_\omega dr'.\end{gathered}$$

We continue to denote by $P_l$ the operator given in (\[eq:Pl\]).

\[l:B1Pl\] Let $B_1$ be the operator defined in (\[eq:B1\]). 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 $0$. Fix a constant $M>3$. Then there is a constant $c>0$ independent of $\sigma$ so that $$\| B_{1,\pm ,V_0}(i\sigma)P_l\| \geq c \frac{e^{c\nu_l}}{\nu_l}$$ for all $l\in \Natural$ which satisfy $a\sigma/6>\nu_l> a \sigma /M$ for all sufficiently large $\sigma>0$.

Before beginning the proof, we note that the constant $c$ does depend on $\epsilon$ and on $a$.

From Lemma \[l:B\_1approx\] it suffices to prove an analogous lower bound for $\| 