Exact analytic relation between quantum defects and scattering phases with applications to Green’s functions in quantum defect theory

Abstract

The relation between the quantum defects, µ _λ , and scattering phases, δ _λ , in the single-channel Quantum Defect Theory (QDT) is discussed with an emphasis on their analyticity properties for both integer and noninteger values of the orbital angular momentum parameter λ. To derive an accurate relation between µ _λ and δ _λ for asymptotically-Coulomb potentials, the QDT is formally developed for the Whittaker equation in its general form “perturbed” by an additional short-range potential. The derived relations demonstrate that µ _λ is a complex function for above-threshold energies, which is analogous to the fact that δ _λ is complex for below-threshold energies. The QDT Green’s function, G _λ , of the “perturbed” Whittaker equation is parameterized by the functions δ _λ and µ _λ for the continuous and discrete spectrum domains respectively, and a number of representations for G _λ are presented for the general case of noninteger λ. Our derivations and analyses provide a more general justification of known results for nonrelativistic and relativistic cases involving Coulomb potentials and for a Coulomb plus point dipole potential.