  The explicit semiclassical treatment of the logarithmic perturbation theory
for the bound-state problem for the spherical anharmonic oscillator and the
screened Coulomb potential is developed. Based upon the $\hbar$-expansions and
suitable quantization conditions a new procedure for deriving perturbation
expansions is offered. Avoiding disadvantages of the standard approach, new
handy recursion formulae with the same simple form both for ground and excited
states have been obtained. As examples, the perturbation expansions for the
energy eigenvalues of the quartic anharmonic oscillator and the Debye potential
are considered.
