bounds of Proposition \[p:lb\], we shall need a special case of that proposition, in which the potential is of the form $V(x)=\epsilon\chi_B(x)$, and $\chi_B(x)$ is the characteristic function of a ball centered at the origin. To study such a special case, we will introduce spherical coordinates in $\Real^d$ (polar coordinates in the case $d=2$).

In spherical coordinates, $$-\Delta = -\frac{\partial^2}{\partial r^2}-\frac{d-1}{r}\frac{\partial }{\partial r} +\frac{1}{r^2}\Delta_{\Sphere^{d-1}}.$$ The eigenvalues of of the Laplacian on $\Sphere^{d-1}$, $\Delta_{\Sphere^{d-1}}$, are $l(l+d-2)$, $l\in \Natural_0$ with multiplicity $$\mu(l)=\frac{2l +d-2}{d-2}\left( \begin{array}{c} l+d-3\\ 
d-3 \end{array}\right) = \frac{2l^{d-2}}{(d-2)!}(1+O(l^{-1})).$$ Denote by $Y_l^\mu$, $1\leq \mu \leq \mu(l)$, $l=0,1,2,...$ a complete orthonormal set of spherical harmonics on $\Sphere^{d-1}$ with eigenvalue $l(l+d-2)$.

We denote by $P_l$ projection onto the span of $$\{ h(|x|)Y^\mu_l(x/|x|): \; 1\leq \mu \leq \mu(l),\; h(|x|)\in L^2(\Real^d; r^{d-1}dr)\}.$$ Thus writing $x=r\theta$, with $r>0$ and $\theta \in \Sphere^{d-1}$ $$\label{eq:Pl}
(P_l g)(r\theta)=\sum_{\mu =1}^{\mu(l)} \int_{\Sphere^{d-1}} g(r\omega) 
Y^{\mu}_l(\theta)\overline{Y}^\mu_l(\omega ) d S_{\omega}.$$

\[l:B\_1approx\] Let $V\geq 0$, $V\in L^{\infty}_{\comp}(\Real^d)$ be a radial function, so that $V(x)=f(|x|)$ for some function $f\in L^{\infty}_{\comp}([0,\infty))$. Then for $\sigma>0$ sufficiently large, with $B_1=B_{1,\pm, V}$ the operator defined in (\[eq:B1\]), $$\left\| \large(\vhalf T(i\sigma)\vhalf -B_{1,\pm ,V}(i\sigma)\large)
P_l \right\| \leq \frac{C}{\sigma^2}\| \vhalf T(i\sigma)
\vhalf P_l\|$$ where $C$ depends on $V$ but not $\sigma$ or $l$.

To 