and determinants. Still, we identify several possibly unnoticed properties and formulae in the $N=\infty$ limit which arise as regular limits of nontrivial finite-$N$ properties.

Sec.3 (Exercise II) deals with the quartic anharmonic oscillator family $V(q)=q^4 + v q^2$ which is the most common model for singular perturbation theory (the free harmonic oscillator emerges in the $v \to +\infty$ limit). We single out a pair of one-parameter spectral functions for their remarkably numerous and simple explicit properties: the zero-energy determinants $\Qi^\pm (v) \defi \det( -\d^2/\d q^2 + q^4 + v q^2)^\pm$ ($+$ corresponds to the even-state sector, $-$ to the odd-state sector). We present a simple WKB technique allowing to express asymptotic relations between $v$-dependent determinants such as $\det( -\d^2/\d q^2 + q^4 + v q^2)^\pm$ and $\det( -\d^2/\d q^2 + v q^2)^\pm$ when $v \to +\infty$. Then, practically all the analytical results available for the homogeneous quartic case ($V(q)=q^4$) have counterparts for the functions $\Qi^\pm$ (and their associated spectral zeta functions), while their asymptotic properties remind of the Airy functions (see fig.1). Several special values are computable and listed in Table 1 against the analogous results for the Airy functions and the quartic determinants $\det( -\d^2/\d q^2 + q^4 + 