  It is shown that for the one-dimensional anharmonic oscillator with potential
$V(x)= a x^2 + b g x^3 +\ldots=\frac{1}{g^2}\,\hat{V}(gx)$, as well as for the
radial oscillator $V(r)=\frac{1}{g^2}\,\hat{V}(gr)$ and for the perturbed
Coulomb problem $V(r)=\frac{\alpha}{r}+ \beta g r + \ldots = g\,\tilde{V}(gr)$,
the Perturbation Theory in powers of the coupling constant $g$ (weak coupling
regime) and the semiclassical expansion in powers of $\hbar^{1/2}$ for the
energies coincide. This is related to the fact that the dynamics developed in
two spaces: $x\ (r)$-space and $gx\ (gr)$-space, lead to the same energy
spectra. The equations which govern dynamics in these two spaces, the
Riccati-Bloch equation and the Generalized Bloch equation, respectively, are
presented. It is shown that the perturbation theory for the logarithmic
derivative of the wavefunction in $gx\ (gr)$- space leads to (true)
semiclassical expansion in powers of $\hbar^{1/2}$; for the one-dimensional
case this corresponds to the flucton calculus for the density matrix in the
path integral formalism in Euclidean (imaginary) time proposed by one of the
authors, Shuryak(1988). Matching the perturbation theory in powers of $g$ and
the semiclassical expansion in powers of $\hbar^{1/2}$ for the wavefunction
leads to a highly accurate local approximation in the entire coordinate space,
its expectation value for the Hamiltonian provides a prescription for the
summation of the perturbative (trans)-series.
