R_0 \vhalf_1)^{-1} \frac{\vhalf_1}
{\vhalf_2}(I \pm \vhalf_2 R_0 \vhalf_2)^{1/2}.$$ This proves the lemma, since the norm of the right hand side is the square of the norm of the operator in question.

\[l:eigenvaluebd\] Let ${\mathcal H}$ be an infinite dimensional complex separable Hilbert space, $A,\; B\in {\mathcal L }({\mathcal H})$, with $B=B^*$, and $\|A\| \leq 1$. Let $|\lambda_1|\geq |\lambda_2|\geq...$ be the norms of the eigenvalues of $A^*BA$, and $|\mu_1|\geq |\mu_2|\geq...$ be the norms of the eigenvalues of $B$. In both cases we repeat according to multiplicity. Then $|\mu_j|\geq |\lambda_j|$ for all $j$.

One way to prove this it that by noting that since $B$ and $A^*BA$ are self-adjoint, the norms of the the eigenvalues are the characteristic values. Then this lemma is an immediate application of the bound for the characteristic values of a product found, for example, in [@simonti Theorem 1.6].

The next lemma shows that $F_{m,\pm V}(i\sigma)$ depends monotonically on $V$ in some sense.

\[l:mono\] Let $V_1,\; V_2\in L^{\infty}(\Real^d)$ and suppose the support of $V_j$ is contained in $\overline{B}(R)$ for $j=1,\;2$. Suppose $V_2(x)\geq V_1(x)\geq 0$ for all $x\in \Real^d$. Then $|F_{m,V_1}(i\sigma)|\leq |F_{m,V_2}(i\sigma)|$ for all $\sigma \in \Real_+$. Moreover, if $\sigma \geq 2(\|V_2\|_{\infty}+1)$, then $|F_{m,-V_1}(i\sigma)|\leq |F_{m,-V_2}(i\sigma)|$.

For any compactly supported 