On the connection between perturbation theory and new semiclassical expansion in quantum mechanics

It is shown that for the one-dimensional anharmonic oscillator with potential V(x)=ax2+bgx3+…=1g2Vˆ(gx), as well as for the radial oscillator V(r)=1g2Vˆ(gr) and for the perturbed Coulomb problem V(r)=αr+βgr+…=gV˜(gr), the Perturbation Theory in powers of the coupling constant g (weak coupling regime) and the semiclassical expansion in powers of ħ1/2 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 ħ1/2; 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 [5]. Matching the perturbation theory in powers of g and the semiclassical expansion in powers of ħ1/2 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.