Quasiclassical approach to the shifting parameter of the 1/Nmethod

Proof is given that 1/N expansions for energy levels can be interpreted in terms of the quasiclassical minimizations of corresponding Hamiltonian forms like scrH(r,p=ħ$d_0$/r) and vice versa. This mutual connection enables us to show that the fixing condition for the expansion parameter (k=N+2l-a) proposed previously is implied to first order only. In general, the a parameter should be chosen, order by order, so that the sum of corrections to the zeroth-order result vanishes. This requirement leads to algebraic equations which produce self-consistent $d_0$=k/2 evaluations. ($d_0$ denotes the underlying phase-space quantum.) Particular fixing conditions can also be proposed. First-order $d_0$ estimates are discussed in more detail by using as an example the linear-plus-Coulomb potential. We also find the covariance behavior of $d_0$ under quasiclassical symmetry transformations.