Regularity for eigenfunctions of Schr\"odinger operators

TL;DR

This paper establishes a regularity result for eigenfunctions of Schr"odinger operators with Coulomb-type potentials, showing they belong to weighted Sobolev spaces, which enhances understanding of their smoothness near singularities.

Contribution

The authors prove that eigenfunctions of Schr"odinger operators with Coulomb potentials are in weighted Sobolev spaces, extending regularity results to cases with bounded potential coefficients.

Findings

Eigenfunctions belong to weighted Sobolev spaces for all positive orders and non-positive weights.

Regularity extends to cases with bounded potential functions on the blown-up space.

Results apply to multi-electron systems with Coulomb interactions, including single-electron multi-nuclei cases.

Abstract

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.