2 00 2 Reply to Comment “ On large - N expansion ” Omar

Fernandez Comment [1] on our pseudo-perturbative shifted-l expansion technique [2,3] is either unfounded or ambiguous. In his comment [1] on our pseudo-perturbation shifted l expansion technique (PSLET) [2,3], Fernandez strived to prove that (I) PSLET is just a version of SLNT, (II) it is not true that PSLET enables one to obtain more perturbation corrections than SLNT, and (III) it seems that SLNT ( and, consequently, also PSLET) is divergent. We explain below why we believe criticisms (I) and (II) to be unfounded and criticism (III) to be ambiguous. • Our statement “ the difficulty of calculating higher-order corrections in SLNT through Rayleigh-Schrõdinger perturbation theory (RSPT) results in a loss of accuracy” is clear and need not be misleading. We refer to the comprehensive, historical, account in the work of Imbo et al [4], indicating the actual novelty of SLNT ( which could handle, via RSPT, only the first four terms of the energy series). Fernandez and co-workers ( in [6-8] of [1]) have used the hypervirial perturbation method (HPM) to calculate higher-order corrections in SLNT. Therefore one would call their method HPM-SLNT, or, at least, Modified SLNT (as they themselves named it) and not SLNT. • We did not claim that PSLET is completely different from SLNT [4] ( c.f. our comment following equation (31) in [3]). At the top of page 3063 in [5] we commented on the higher accuracy of the Fernandez HPM-SLNT method (although we had reservations about the order-dependent shift approach to the Klein-Gordon and Dirac equations). • Fernandez derived relations between a and β, k̄ and l̄, · · ·etc. However, that work simply illustrates part of the message which we tried to deliver to readers, i.e., SLNT is not an expansion in large-N but, in effect, an expansion in large-l ( c.f. C M Bender et al [6]); hence we preferred the abbreviation PSLET . • It is true, of course, that our conclusions in [3] about numerical accuracy referred to calculations for state wavefunctions with at most one node. However, the comment by