q'$ is specifically “renormalized" through eq.(\[INTP\]); however, eq.(\[REs\]) also makes $\psi_\lambda(q)$ directly specified as [*the*]{} (unique) solution of the Schrödinger equation (\[Schr\]) that decays for $q \to +\infty$ with the precise behaviour $$\label{Sib}
\psi_\lambda (q) \sim \e^{\mathcal C} q^{-N/4 \, -\beta_{-1}(0)} 
\e^{-{\mathcal S}_\lambda(q)}
(\equiv \e^{\mathcal C} q^{NZ^-(0,\lambda)} \e^{-{\mathcal S}_\lambda(q)}),
\qquad q \to +\infty$$ (the latter form comes from eqs.(\[Tres\],\[tr0\])). Remark: in parallel to eqs.(\[WKB\],\[Sib\]), $$\begin{aligned}
\label{Sib'}
-\psi_\lambda '(q) &\sim& \Pi_\lambda(q)^{+1/2} 
\e^{ \int_q^{+\infty} \Pi_\lambda(q') \d q' } 
\qquad \mbox{for }\Pi_\lambda(q) \to +\infty \\
\bigl( &\sim& \e^{\mathcal C} q^{+N/4 \, -\beta_{-1}(0)} 
\e^{-{\mathcal S}_\lambda(q)}
= \e^{\mathcal C} q^{NZ^+(0,\lambda)} \e^{-{\mathcal S}_\lambda(q)} 
\quad \mbox{for }q \to +\infty \bigr) . \nonumber\end{aligned}$$

We refer to $\psi_\lambda (q)$ as the “canonical recessive" solution. (It is proportional to the “subdominant" solution of [@S], ch.2, but only equal to it if $\beta_{-1}(s) \equiv 0$. The current normalization also differs from [@V1], and from [@V0; @V6] — where it suffers localized inconsistencies. The discrepancies, which for $N>2$ involve $\lambda$-independent factors only, ultimately cancel out in all results involving only spectral determinants.)

The extension of $\psi_\lambda (q)$ to the whole real line, as a solution of eq.(\[Schr\]) with the potential $V(|q|)$, is shown by integrations to identically satisfy ([@V1] Apps.A,D) $$\begin{aligned}
\label{IDP}
