Regularity for eigenfunctions of Schrödinger operators

Abstract

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.

to appear in Lett. Math. Phys

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• Sharp regularity results for many-electron wave functions
• Magnetic Pseudodifferential Operators
• Analytic structure of many-body Coulombic wave functions
• Localizations at infinity and essential spectrum of quantum Hamiltonians: I. General theory
• Weighted Sobolev spaces and regularity for polyhedral domains
• Semiclassical resolvent estimates for Schroedinger operators with Coulomb singularities
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Cited by in corpus (4)

• Analysis on singular spaces: Lie manifolds and operator algebras
• The diagonal behaviour of the one-particle Coulombic density matrix
• A regularity result for the bound states of $N$-body Schrödinger operators: Blow-ups and Lie manifolds
• A comparison of the Georgescu and Vasy spaces associated to the N-body problems and applications