Sharp regularity results for many-electron wave functions

TL;DR

This paper establishes a universal factorization for many-electron wave functions in atoms and molecules, revealing their regularity properties and deriving cusp conditions, with implications for quantum chemistry and mathematical physics.

Contribution

It introduces a novel representation of wave functions as a product of a universal factor and a smoother function, based on new elliptic regularity results.

Findings

Wave functions can be expressed as Psi=F*phi with F universal and phi smoother.

The representation is optimal, demonstrated with hydrogenic wave functions.

New elliptic regularity results underpin the analysis.

Abstract

We show that electronic wave functions Psi of atoms and molecules have a representation Psi=F*phi, where F is an explicit universal factor, locally Lipschitz, and independent of the eigenvalue and the solution Psi itself, and phi has locally bounded second derivatives. This representation turns out to be optimal as can already be demonstrated with the help of hydrogenic wave functions. The proofs of these results are, in an essential way, based on a new elliptic regularity result which is of independent interest. Some identities that can be interpreted as cusp conditions for second order derivatives of Psi are derived.