Regularity for eigenfunctions of Schr\"odinger operators

We prove a regularity result in weighted Sobolev spaces (or Babuska–Kondratiev spaces) for the eigenfunctions of a Schr"odinger operator. More precisely, let K_{a}^{m}($\mathbb{R}$^{3N}) be the weighted Sobolev space obtained by blowing up the set of singular points of the Coulomb type potential V(x) = \sum_{1 \le j \le N} \frac{b_j}{|x_j|} + \sum_{1 \le i < j \le N} \frac{c_{ij}}{|x_i-x_j|}, x in $\mathbb{R}$^{3N}, b_j, c_{ij} in $\mathbb{R}$. If u in L^2($\mathbb{R}$^{3N}) satisfies (-\Delta + V) u = \lambda u in distribution sense, then u belongs to K_{a}^{m} for all m \in $\mathbb{Z}$+ and all a \le 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.