D(\lambda) &\equiv& C  W_\lambda , \qquad  
D^{\rm P}(\lambda) \equiv C^{\rm P} [-\psi'_\lambda(0) / \psi_\lambda(0)], \\
W_\lambda &\defi& \mbox{Wronskian} \{ \psi_\lambda(-q),\psi_\lambda(q) \}
\equiv -2 \psi_\lambda(0) \psi'_\lambda(0) \end{aligned}$$ for some constants $C,\ C^{\rm P}$; to identify these, we test a characteristic property of the spectral determinants: the canonical large-$\lambda$ asymptotics (eq.(\[das\])) of their logarithms down to constants included; as regards the right-hand sides in eq.(\[IDP\]), the WKB formulae (\[WKB\],\[Sib’\]) (good for large $\lambda$) supply these asymptotic forms, $$\label{CL}
W_\lambda / 2 \sim \e^{ 2 \int_0^{+\infty} \Pi_\lambda(q) \d q }
\equiv D_{\rm cl} (\lambda), \quad 
 -\psi'_\lambda(0) / \psi_\lambda(0) \sim \Pi_\lambda(0) \quad
(\lambda \to +\infty),$$ both of which have canonical ($\lambda \to +\infty$) logarithms ($D_{\rm cl} (\lambda)$ by analogy with $D(\lambda)$, and $\Pi_\lambda(0)$ by inspection); hence necessarily, $$\label{ID0}
D(\lambda) \equiv W_\lambda/2 \equiv -\psi_\lambda(0) \psi'_\lambda(0), \qquad
D^{\rm P}(\lambda) \equiv -\psi'_\lambda(0) / \psi_\lambda(0).$$ Incidentally, the second eq.(\[CL\]) then naturally specifies a “classical skew determinant", $D_{\rm cl}^{\rm P}(\lambda) \defi \Pi_\lambda(0)$, from which classical analogs of $D^\pm(\lambda)$ also follow as $$\label{DPMC}
D_{\rm cl}^\pm(\lambda ) \defi 
\Pi_\lambda(0)^{\pm 1/2} \e^{\int_0^{+\infty} \Pi_\lambda(q) \d q} .$$

The main conclusion however concerns the quantum spectral determinants $D^\pm$ themselves: upon a straightforward simplification of eq.(\[ID0\]), they get expressed in terms of $\psi_\lambda (q)$ by 