On Green's functions, propagators, and sturmians for the nonrelativistic coulomb problem

Abstract

Recent progress in the mathematical physics and quantum chemistry of Coulomb Green’s functions is summarized. Analogy with the defining relation for the Green’s function has led to a finite model for the Fermi contact interaction which avoids spurious divergences in second-order perturbation calculations. The Hamilton-Jacobi mechanics of the Coulomb problem is reviewed. A compact parametrization for Hamilton’s principal and characteristic functions provides a key element in further developments. These include a semiclassical representation for the Coulomb propagator in Feynman’s formalism and a new propagator in the domain of Coulomb Sturmian eigenstates. In projected applications, approximate many-electron Green’s functions constructed from combinations of oneparticle Coulomb propagators provide a basis for computation of atomic and molecular eigenvalue spectra. Coulomb Green’s Functions The hydrogen atom has played a key historical role in the development of the quantum theory of matter, from the Bohr model to Schrodinger’s wave mechanics to Dirac’s relativistic theory to modem quantum electrodynamics to current models exploiting higher dynamical symmetry. The literature on Coulomb Green’s functions has been correspondingly extensive, dating back to Meixner’s partial solution for G(r,,r2,E) in the limits r2 =O or [l]. A number of integral representations and partial-wave expansions for the time-independent Coulomb Green’s function were subsequently developed [2]. Schwinger [3] gave an elegant representation for the Green’s function in momentum space. Hostler [4] finally discovered a closed form for the Coulomb Green’s function G(rl,r2,E) and also derived approximate relativistic Green’s functions for both the Klein-Gordon and Dirac equations. The Coulomb Green’s function is the solution under specified boundary conditions of the inhomogeneous differential equation