{c_\rho^\pm \over \G(1-\rho)} \quad(\rho \ne 0),
\qquad {b_0^\pm \over 2} \pm {1 \over 4} \equiv c_0^\pm \equiv Z^\pm(0)$$ (the latter specifies the $s=0$ trace identities); here, moreover, $$\label{tr0}
b_{-\rho}^+ \equiv b_{-\rho}^- \mbox{ for all } \rho \le 0 \qquad
\Longrightarrow \qquad Z(0) = b_0^\pm, \quad Z^{\rm P}(0) = 1/2 ;$$ then the same trace identity follows for all $Z(s,\lambda)$ and $Z^{\rm P}(s,\lambda)$ by eq.(\[Tr0\]), except for $N=2$ (eq.(\[Tr20\]) below).

By the same token, $D^\pm(\lambda)$ become directly specifiable as [*functionals of the spectrum*]{}: in the simplest case $\mu<1$, i.e. $N>2$, $$\begin{aligned}
\label{EMc}
\log D^\pm(\lambda)  &=& \lim _{K \to +\infty} \Bigl\{ \sum_ {k<K}
\log (E_{k} + \lambda) + {1\over 2} \log (E_{K} + \lambda) \\
&&\qquad -{1 \over 2} \sum_{ \{ \rho <0 \} } 
b_{-\rho}^\pm E_{K}^{-\rho}
\Bigl( \log E_{K}+{1 \over \rho}\Bigr) \Bigr\} \quad
{\rm for\ }k,K\ \textstyle{\rm even \atop  odd}
\nonumber\end{aligned}$$ (the expansion coefficients $b_{-\rho}^\pm$ being themselves functions of the spectrum $\{E_k\}$).

$\bullet$ $D^\pm(\lambda)$ are also related to an exact solution $\psi_\lambda(q)$ of eq.(\[Schr\]) defined by a particular WKB normalization, $$\label{WKB}
\psi_\lambda (q) \sim \Pi_\lambda(q)^{-1/2} 
\e^{ \int_q^{+\infty} \Pi_\lambda(q') \d q'} \qquad 
\mbox{for }\Pi_\lambda(q)  \to +\infty ;$$ here $\Pi_\lambda(q)$ is the classical momentum function (\[PI\]), and the divergent integral $\int_q^{+\infty} \Pi_\lambda(q') \d 