On connection between perturbation theory and semiclassical expansion in quantum mechanics

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.