Finite Element Approximations to the Discrete Spectrum of the Schrödinger Operator with the Coulomb Potential

In the present paper, the authors consider the Schrödinger operator H with the Coulomb potential defined in R3m, where m is a positive integer. Both bounded domain approximations to multielectron systems and finite element approximations to the helium system are analyzed. The spectrum of H becomes completely discrete when confined to bounded domains. The error estimate of the bounded domain approximation to the discrete spectrum of H is obtained. Since numerical solution is difficult for a higher-dimensional problem of dimension more than three, the finite element analyses in this paper are restricted to the S-state of the helium atom. The authors transform the six-dimensional Schrödinger equation of the helium S-state into a three-dimensional form. Optimal error estimates for the finite element approximation to the three-dimensional equation, for all eigenvalues and eigenfunctions of the three-dimensional equation, are obtained by means of local regularization. Numerical results are shown in the last section.