$v \to +\infty$ properties of Airy type. The third study addresses the potentials $V(q)=q^N+v q^{N/2-1}$ of even degree: their zero-energy spectral determinants prove computable in closed form, and the generalized eigenvalue problems with $v$ as spectral variable admit exact quantization formulae which are perfect extensions of the harmonic oscillator case (corresponding to $N=2$); these results probably reflect the presence of supersymmetric$^2$ potentials in the family above.

Exact quantization, or exact WKB analysis, supplies new tools for the analytical study of the 1D Schrödinger equation, now including arbitrary polynomial potentials. Here we initiate applications of an exact method to miscellaneous concrete problems and models of analytical interest, emphasizing exact and asymptotic relations for the spectral determinants and related spectral zeta functions.

We have chosen three rather different quantum potentials to illustrate a variety of situations. These have some basic common features (besides their required one-dimensional and polynomial nature): they are rather simple, with a minimal number of parameters, to remain concretely manageable; one crucial parameter (discrete or continuous) governs the transition to a singular limit, creating an interesting dynamical and analytical situation; some uniform principles for tackling those problems can be issued at the most general level.

In contrast to earlier 