Any (extended) abstract of this compact paper should remind the
readers about the efficiency and about the key relevance of the
concept of spectral functions (and, in particular, of the Selberg's
zeta-function $Z$ of eq. (5) and of the zeta-regularized spectral
determinant $D$ of eq. (6)). Implying the ``exact WKB solvability"
of {\em any} one-dimensional Schr\"{o}dinger equation with
polynomial potential as nicely reviewed by A. Voros, in ``Recent
Trends in Exponential Asymptotics", RIMS Kokyuroku series, vol. 1424
(2005) pp. 214 - 231 (edited by Y. Takei). The earlier, 2004
application ref.~[1] of this language to a special 1D
Schr\"{o}dinger singular perturbation problem (1) (i.e., to
interactions $V=q^N+vq^M$ where $N < 2M+2$ and $v \gg 1$) is now
complemented by a few important amendments. In particular, it is
demonstrated that the apparent breakdown of the exact quantization
in the ``weak perturbation" limit $v \to \infty$ (i.e., the
discontinuity puzzle as formulated in 2000 in [11]) is inessential.
It is shown that the role of the underlying Wronskian identity (50)
itself (i.e., of a basic and beautiful bilinear relation between
spectral determinants which, by the way, involves the conjugate --
complex-rescaled -- Schr\"{o}dinger eigenvalue problems) remains
unaffected and compatible with the continuous evolution of the
spectral determinants in the limit $v \to \infty$.

MR2758917 Voros, André From exact-WKB toward singular quantum
perturbation theory II. Algebraic analysis of differential equations
from microlocal analysis to exponential asymptotics, 321--334,
Springer, Tokyo, 2008. 81Q20 (34M60 81Q15)