Systematics of perturbative semiclassical quantum defect expansions probed by RKR-QDTand a Fisher-information-based criterion

Abstract

The systematics of perturbative semiclassical quantum defect expansions corresponding to a hydrogenic potential plus a perturbing term of the form -A/2r^κ, $\kappa \geqslant 2$, are studied as a function of expansion order N. Towards this task the expansions μ _Nare first used as input for constructing associated N-dependent atomic RKR-QDT potential curves. Subsequently the coordinate Fisher information for the energy levels supported by those curves as well as its rate ε with respect to N is semiclassically computed. Then, the plot of relative quantum defect error between successive orders, δμ _N+1,N, with respect to ε serves as convergence indicator for both approximate potentials and quantum defects. For a given κ and when the quantum defect expansion proves to be of limited accuracy the plot reveals an A- and N-dependent scatter of points and “saturation” (the relative error remains almost constant with respect to ε). More importantly, when ε is equal to or lower than the value of ε (N=1) for which πμ $_{1}\leqslant 1/2$ the relative error exhibits a κ-, A- and N-independent power-law dependence, δμ _N+1,N∝ ε ^m, clearly distinguishing the N=1 order (m=1/2) from all other N>1 orders (m=1). These power-laws may be employed for setting-up confidence level bounds on perturbative expansions.