One of paradoxes characterizing the interface between Quantum
Mechanics and the analytic theory of ODEs (in the language, say,
of the Sibuya's book [6]) is that the insight obtained in the
latter context (like the results on the limiting forms of the
various spectral functions as presented in the paper under
consideration) may be perceived strongly academic in the former
perspective. Figure~2 shows the bridge: In it, the results of the
purely numerical, brute-force evaluation of the energies of the
popular quartic and sextic quantum anharmonic oscillators are
shown to comply, very well, with the vanishing-coupling limit
behaviour of the so called zeta spectral functions as predicted by
the Voros' analysis. The remarkable difference between the
regular and singular modes is emphasized, with the prediction of 
the latter pattern [viz., eq. (5.13)] being a climax of the 
text. Its derivation (relying on the explicit evaluation of the
action integrals) is interesting by itself. The announced
continuation of this study ``to complex asymptotics" looks
promising to all who know how important was the role of certain
particular global spectral functions (called spectral determinants
of the Schroedinger differential operators) in the proof of the
reality of the spectrum of the imaginary cubic oscillator [cf. P.
Dorey, C. Dunning and R. Tateo, A reality proof in PT-symmetric
quantum mechanics, Czech. J. Phys. 54 (2004) 35-41].


MR2074707
Voros, André . 
From exact-WKB towards singular quantum perturbation theory. 
Publ. Res. Inst. Math. Sci. 40 (2004), no. 3, 973--990