We show that electronic wave functions ψ of atoms and molecules have a representation ψ = F φ, where F is an explicit universal factor, locally Lipschitz, and independent of the eigenvalue and the solution ψ itself, and φ 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 ψ are derived.

We prove the first correction to the leading Thomas-Fermi energy for the ground state energy of atoms and molecules in a model where the kinetic energy of the electrons is treated relativistically. The leading Thomas-Fermi energy, established in [25], as well as the correction given here, are of semiclassical nature. Our result on atoms and molecules is proved from a general semiclassical estimate for relativistic operators with potentials with Coulomb-like singularities. This semiclassical estimate is obtained using the coherent state calculus introduced in [36]. The paper contains a unified treatment of the relativistic as well as the nonrelativistic case.

We prove that the electronic densities of atomic and molecular eigenfunctions are real analytic in R 3 away from the nuclei.Date: November 22, 2002.

Abstract. We prove that the electron densities of electronic eigenfunctions of atoms and molecules are smooth away from the nuclei.

We investigate the analytic structure of solutions of non-relativistic Schrödinger equations describing Coulombic manyparticle systems. We prove the following: Let ψ(x) with x = (x 1 , . . . , x N ) ∈ R 3N denote an N -electron wavefunction of such a system with one nucleus fixed at the origin. Then in a neighbourhood of a coalescence point, for which x 1 = 0 and the other electron coordinates do not coincide, and differ from 0, ψ can be represented locally as ψ(x) = ψ (1) (x) + |x 1 |ψ (2) (x) with ψ (1) , ψ (2) real analytic. A similar representation holds near two-electron coalescence points. The Kustaanheimo-Stiefel transform and analytic hypoellipticity play an essential role in the proof.