Analytic structure of solutions to multiconfiguration equations

TL;DR

This paper analyzes the regularity of solutions to multiconfiguration equations in Coulomb systems, showing they can be locally decomposed into analytic functions plus a term involving the distance to nuclei.

Contribution

It proves a local regularity decomposition for solutions to multiconfiguration equations near nuclei, extending previous methods to Coulomb systems.

Findings

Solutions are locally expressible as sums of analytic functions and a term involving |x - R_k|.

The decomposition applies to wavefunctions and electron densities.

Uses Kustaanheimo--Stiefel transformation for regularity analysis.

Abstract

We study the regularity at the positions of the (fixed) nuclei of solutions to (non-relativistic) multiconfiguration equations (including Hartree--Fock) of Coulomb systems. We prove the following: Let {phi_1,...,phi_M} be any solution to the rank--M multiconfiguration equations for a molecule with L fixed nuclei at R_1,...,R_L in R^3. Then, for any j in {1,...,M} and k in {1,...,L}, there exists a neighbourhood U_{j,k} in R^3 of R_k, and functions phi^{(1)}_{j,k}, phi^{(2)}_{j,k}, real analytic in U_{j,k}, such that phi_j(x) = phi^{(1)}_{j,k}(x) + |x - R_k| phi^{(2)}_{j,k}(x), x in U_{j,k} A similar result holds for the corresponding electron density. The proof uses the Kustaanheimo--Stiefel transformation, as applied earlier by the authors to the study of the eigenfunctions of the Schr"odinger operator of atoms and molecules near two-particle coalescence points.