(s,\lambda/\alpha)
\quad \Longrightarrow \quad
D(\lambda \vert \alpha) \equiv \alpha^{Z(0,\lambda/\alpha)} D(\lambda/\alpha)$$ (a behaviour hence mainly governed by the leading trace identity, eq.(\[Tr0\]));

– finally, all those results extend to analogous complex spectra [@QHS].

1D Schrödinger operators with polynomial potentials
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We subsequently specialize to 1D Schrödinger equations involving a polynomial potential $V(q)$ (adjusted to $V(0)=0$), [@S; @O] $$\label{Schr}
(-\d^2/\d q^2\,+[V(q)+\lambda]) \psi = 0, \quad 
V(q) = +q^N+ \mbox{[lower order terms]} .$$ We call $\hat H^+$ (resp. $\hat H^-$) the Schrödinger operator on the half-line $\{q>0\}$ with the Neumann (resp. Dirichlet) boundary condition at $q=0$, and $\hat H$ the Schrödinger operator on the whole line with the potential $V(|q|)$, whose spectrum we denote by $\{E_k\}$. Then $\{E_k\}_{k\ {\rm even}}$ (resp. $\{E_k\}_{k\ {\rm odd}}$) is the spectrum of $\hat H^+$ (resp. $\hat H^-$), each one is an admissible spectrum in the previous sense, with $$\begin{aligned}
\label{Ord}
\mbox{growth order } \mu &=& \textstyle{1 \over 2} + \textstyle{1 \over N} \qquad 
(\mu<1 \mbox{ generically, i.e. for } N>2) , \\
\mbox{exponents }\{\rho\} &=& \{ -\mu +j/N \}_{j=0,1,2,\cdots} ;\end{aligned}$$ their spectral functions $Z^+,\ D^+$ (resp. $Z^-,\ D^-$) are our basic concern, and have properties as above (with exceptions in the singular case $N=2$). However, a few 