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Regularity for Eigenfunctions of Schrödinger Operators

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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 \({\mathcal{K}_{a}^{m}(\mathbb{R}^{3N},r_S)}\) be the weighted Sobolev space obtained by blowing up the set of singular points of the 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 \({u \in \mathcal{K}_{a}^{m}}\) for all \({m \in \mathbb{Z}_+}\) 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.

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Authors and Affiliations

  1. Fakultät für Mathematik, Universität Regensburg, 93040, Regensburg, Germany

    Bernd Ammann

  2. Mathematics Department, Instituto Superior Técnico, UTL, Av. Rovisco Pais, 1049-001, Lisbon, Portugal

    Catarina Carvalho

  3. Mathematics Department, Pennsylvania State University, University Park, PA, 16802, USA

    Victor Nistor

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  1. Bernd Ammann
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  2. Catarina Carvalho
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  3. Victor Nistor
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Correspondence to Bernd Ammann.

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Ammann’s manuscripts are available from http://www.berndammann.de/publications. Carvalho’s manuscripts are available from http://www.math.ist.utl.pt/~ccarv. Nistor’s Manuscripts available from http://www.math.psu.edu/nistor/. Nistor was partially supported by the NSF Grants DMS-0713743, OCI-0749202, and DMS-1016556.

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Ammann, B., Carvalho, C. & Nistor, V. Regularity for Eigenfunctions of Schrödinger Operators. Lett Math Phys 101, 49–84 (2012). https://doi.org/10.1007/s11005-012-0551-z

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