Analytic treatment of the Coulomb potential in the path integral formalism by exact summation of a perturbation expansion

A straightforward analytical calculation of the s-like energy spectrum of the hydrogen atom is performed entirely within Feynman's path integral formalism. For this purpose the integral transform W = ≮d³r _βK(r_β,0), where K(r_β,0) is the density matrix of the hydrogen atom written as a path integral, is calculated by means of the exact summation of a "modified" perturbation expansion (W is expanded as a power series in √β with β = 1/kΤ). Performing this summation is equivalent to solving a problem of moments with infinite moments. For a wide class of potentials the perturbation expansion for W converges faster than the power-series expansion for the exponential function (for the Coulomb potential the convergence rate of both expansions is the same). It is shown how the complete energy spectrum can be obtained by this method. It is also illustrated how the wavefunctions might be obtained by transforming W.