\sqrt{-\lambda}, \quad 
\Delta_\infty^-(\lambda) = {\sin \sqrt{-\lambda} \over \sqrt{-\lambda}}
\quad \Longrightarrow \quad 
\Delta_\infty(\lambda) = {\sin 2\sqrt{-\lambda} \over 2\sqrt{-\lambda}}$$ and the plain spectral zeta functions, $$\label{Zinf}
Z_\infty^+(s) = {2^{2s}-1 \over \pi^{2s}} \zeta(2s), \quad
Z_\infty^-(s) = {1 \over \pi^{2s}} \zeta(2s)
\quad \Longrightarrow \quad
Z_\infty(s) = \Bigl( {2 \over \pi} \Bigr) ^{2s} \zeta(2s).$$ The latter formulae imply the explicit computability of $Z_\infty^\pm(s)$ at all integers $s \in \Bbb Z$, e.g., $$\label{Zi0}
Z_\infty^+(0) = 0, \quad Z_\infty^-(0) = -1/2
\quad \Longrightarrow \quad Z_\infty(0) =  -1/2,$$ and ultimately that of the spectral determinants themselves (even though these will not serve here) through $D_\infty^\pm (\lambda) = 
\exp [-(Z_\infty^\pm)'(0)] \Delta_\infty^\pm (\lambda)$, using $$\label{Di0}
\exp [-(Z_\infty^\pm)'(0)] \equiv D_\infty^\pm (0) =2
\quad \Longrightarrow \quad  D_\infty(0)=4.$$

The transitional behaviour problem
----------------------------------

Regarding the behaviour of the spectral functions, a first task is to seek conditions ensuring the regular behaviour of a quantity (meaning that it has a finite limit, [*and*]{} this is the correct value for the limiting problem). For the other (singular or pathological) quantities then comes the additional task of describing their precise behaviours.

A fundamental quantity in these problems is the growth order $\mu$. Here, the limiting (square-well) value $\mu_\infty = {1 \over 2}$ agrees with the limit 