Regularity for eigenfunctions of Schrödinger operators
This provides a regularity result for eigenfunctions of Schrödinger operators, which is important for numerical analysis and theoretical understanding in quantum chemistry and physics.
The paper proves that eigenfunctions of Schrödinger operators with Coulomb-type potentials belong to weighted Sobolev spaces (Babuska-Kondratiev spaces) for all positive integers m and a ≤ 0, extending to bounded function coefficients and improving the exponent a in the single-electron case.
We prove a regularity result in weighted Sobolev spaces (or Babuska--Kondratiev spaces) for the eigenfunctions of a Schrödinger 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 (-Δ+ V) u = λu 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.