Regularity for Eigenfunctions of Schrödinger Operators

We prove a regularity result in weighted Sobolev (or Babuška-Kondratiev) spaces for the eigenfunctions of certain Schrödinger-type operators. Our results apply, in particular, to a non-relativistic Schrödinger operator of an N-electron atom in the fixed nucleus approximation. More precisely, let K ^m _a(ℝ ^3N, r _S) be the weighted Sobolev space obtained by blowing up the set of singular points of the potential, satisfies (-Δ+V)u=λu in distribution sense, then u ∈K ^m _a for all m∈ℤ ₊ and all a ≤ 0. Our result extends to the case when b _j and c _ij are suitable bounded functions on the blown-up space. In the single-electron, multi-nuclei case, we obtain the same result for all a < 3/2.