\lambda)^\pm$.

The final Sec.4 (Exercise III) gives a complete treatment of the similar determinants for a different class of binomial potentials, $V(q)=q^N+v q^{N/2-1}$ (for $N$ even). Here the formalism yields fully closed forms for the zero-energy spectral determinants: eq.(\[DNMF\]) for $\det[ -\d^2/\d q^2 + V(|q|)]^\pm$ in terms of Gamma functions (plus non-trivial exponential prefactors) and, when $N$ is multiple of 4, the still simpler eq.(\[DNME\]) for the zero-energy determinant of the potential $V(q)$ itself on the whole real line; exact quantization formulae follow for the corresponding generalized spectra (in the $v$ variable: eqs.(\[EQNM\]) and (\[EQNE\]) respectively). A broad generalization of the familiar harmonic-oscillator exact results is thus obtained; for $N \equiv 2 \ [{\rm mod} \ 4]$, this seems to describe a zero-energy cross-section of the formalism for [*supersymmetric*]{}$^2$ (including some [*quasi-exactly solvable*]{}) models.

Although all three problems rely on the same background formalism, they can be approached fairly independently from one another. Accordingly, there are no global conclusions but each Section carries its own concluding remarks.

Introduction
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General results on spectral functions [@V5; @VZ]
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Here, an [*admissible*]{} spectrum is a purely discrete countable set $\{E_k\}_{k=0,1,2,\cdots}$ with $E_k >0$, $E_k \uparrow +\infty$, such that its partition function $$\label{theta}
