results take a neater or a more regular form upon recombined functions instead, $$\begin{aligned}
\label{SC}
Z \defi Z^+ + Z^-, \quad D \defi D^+ D^- \quad
\mbox{ (spectral functions of $\hat H$)}, \\
\label{SK}
Z^{\rm P} \defi Z^+ - Z^-, \quad D^{\rm P} \defi D^+ / D^- 
\mbox{ (``skew" spectral functions)}.\end{aligned}$$

### Classical “spectral" functions [@V0]

The quantum spectral functions of the problem (\[Schr\]) admit natural classical counterparts with parallel properties. The Weyl–Wigner correspondence, for instance, associates the following classical partition function to the quantum one of eq.(\[theta\]), $$\label{TC}
\theta_{\rm cl}(t) = \int_{\Bbb R ^2}
{\d p\,\d q \over 2 \pi} \e^{-[(p^2+V(|q|)]t} \equiv
{1 \over \sqrt{\pi t}} \int_0^{+\infty} \e^{-V(q)t} \d q ,$$ with an expansion coinciding with eq.(\[tas\]) as long as $\rho \le 0$. The same Mellin transforms as from the quantum $\theta(t)$ to $Z(s,\lambda)$ then yield $$\begin{aligned}
\label{ZC} Z_{\rm cl}(s,\lambda) &\defi&
{1 \over \G(s)} \int_0^{+\infty} \theta_{\rm cl}(t) \e^{-\lambda t} t^{s-1} \d t
= {\G(s-1/2) \over \G(s) \sqrt\pi} I_0(s,\lambda) \\
\label{IQ}
\mbox{where} \quad I_q(s,\lambda) &\defi& 
\int_q^{+\infty} (V(q')+\lambda) ^{-s+1/2} \d q' \qquad (\Re(s) >\mu); \\
\label{DC}
\mbox{and} \quad
D_{\rm cl}(\lambda) &\defi& \exp [- \partial_s Z_{\rm cl}(s,\lambda) ] _{s=0},\end{aligned}$$ with properties induced by eq.(\[tas\]) similar to the quantum case (but $D_{\rm cl}(\lambda)$ 