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The explicit semiclassical treatment of the logarithmic perturbation theoryfor the bound-state problem for the spherical anharmonic oscillator and thescreened Coulomb potential is developed. Based upon the $\hbar$-expansions andsuitable quantization conditions a new procedure for deriving perturbationexpansions is offered. Avoiding disadvantages of the standard approach, newhandy recursion formulae with the same simple form both for ground and excitedstates have been obtained. As examples, the perturbation expansions for theenergy eigenvalues of the quartic anharmonic oscillator and the Debye potentialare considered.

The logarithmic perturbation theory for bound states in spherical-symmetric potentials via the $\hbar$-expansions