  We use exact WKB analysis to derive some concrete formulae in singular
quantum perturbation theory, for Schr\"odinger eigenvalue problems on the real
line with polynomial potentials of the form $(q^M + g q^N)$, where $N>M>0$
even, and $g>0$. Mainly, we establish the $g \to 0$ limiting forms of global
spectral functions such as the zeta-regularized determinants and some spectral
zeta functions.
