large-$q$ asymptotic behaviour, as $$\begin{aligned}
\label{REs}
\int_q^{+\infty} \Pi_\lambda(q') \d q' &=&
-{\mathcal S}_\lambda(q) - \beta_{-1}(0) \log q + {\mathcal C} + o(1) \qquad 
(q \to +\infty) \\
\mbox{where} \quad {\mathcal S}_\lambda(q) &\defi&
\sum_{\{\sigma>-1\}} \beta_\sigma(0) {q^{\sigma+1} \over \sigma+1}, \qquad
{\mathcal C} \defi {1 \over N} 
\Bigl( - 2 \log 2 \, \beta_{-1}(0) 
+ \partial_s \Bigl[ {\beta_{-1}(s) \over 1-2s} \Bigr]_{s=0} \Bigr). \nonumber\end{aligned}$$

Classical analogs will also arise for $D^+$ and $D^-$ separately (eq.(\[DPMC\]) below).

### Special features of 1D Schrödinger determinants [@V0; @V6]

The quantum determinants $D^\pm$ for eq.(\[Schr\]) can be specified in two additional ways.

$\bullet$ Either spectrum of $\hat H^+$ or $\hat H^-$ obeys a semiclassical (high-energy Bohr–Sommerfeld) quantization condition of the form $$\label{BS}
\sum_\rho b_{-\rho}^\pm E_k^{-\rho} \sim k+1/2 \quad 
\mbox{ for } \textstyle{\rm even \atop odd} \ k \to +\infty,$$ which implies Euler–Maclaurin continuation formulae for the zeta functions; e.g., down to $\Re(s) \ge 0$, $$\label{EML}
Z^\pm (s) = \lim _{K \to +\infty} 
\Bigl\{ \sum_ {k<K} E_k^{-s} +{1\over 2} E_K^{-s}
-{1\over 2} \sum_{ \{ \rho <0 \} } {\rho \,b_{-\rho}^\pm \over s+\rho}
E_K^{-s-\rho}
\Bigr\} \quad
\mbox{ for } \textstyle{\rm even \atop odd} \ k,K \ .$$ The resulting residues are compatible with eq.(\[res\]) provided $$\label{COMP}
{b_{-\rho}^\pm \over 2} \equiv 