can further rewrite (\[rhoe2\]) $$\rho_2(E') - \rho_2^0(E') = -\frac{N}{\pi} \frac{\partial}
{\partial z}
F(z) |_{z=E'+i\epsilon},
\label{rhof}$$ with $$F(z)=\frac{1}{N} \sum_{\vec{K}} \mathop{\rm arg}\nolimits
{\cal R}'(z,\vec{K}).
\label{fdef}$$

From (\[rhoe2\]) and (\[r2\]) one sees that only $E'$ values occur in (\[b2e2\]) which lead to complex values of ${\cal R}'(z,\vec{K})$. These occur when $z=E'+i\epsilon$ is a pole in the $\vec{k}$-integration in (\[r2\]). Thus the combined bandwidth of $\varepsilon(
\vec{K}/2+\vec{k}) +\varepsilon(\vec{K}/2 -\vec{k}) $ determines the range of $E'$ values. This is twice the bandwidth of $\varepsilon(\vec{k})$ just as in the ideal Bose contribution (\[bl0\]).

The one-dimensional case {#oned}
========================

We interrupt the general discussion for the treatment of the one-dimensional case of (\[b2e2\])-(\[fdef\]), as this case is interesting, completely analyzable, and elucidating for the structure of the functions occurring in (\[b2e2\])-(\[fdef\]).

The $d=1$ band structure is given by (for a lattice constant $a=1$) $$\varepsilon(k)=2t(1-\cos k),
\label{1deps}$$ which gives for $b_1(\beta)$ the expression $$b_1(\beta)=\frac{1}{2\pi} \int_{-\pi}^{\pi} dk \,
e^{-2\beta t (1-\cos k)}=
e^{-2\beta t} I_0(2 \beta t),
\label{1db1}$$ where $I_0(x)$ is a modified Bessel function of the first kind.

For the evaluation of ${\cal R}(z,K)$ we rearrange the Brillouin zone in such a way that its boundaries in $K,k$ space are convenient, and also such that no reciprocal lattice 