# Regularity for Eigenfunctions of Schrödinger Operators

Ontology type: schema:ScholarlyArticle      Open Access: True

### Article Info

DATE

2012-03-18

AUTHORS ABSTRACT

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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{K}_{a}^{m}(\mathbb{R}^{3N},r_S)}$$\end{document} be the weighted Sobolev space obtained by blowing up the set of singular points of the potential \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${V(x) = \sum_{1 \le j \le N} \frac{b_j}{|x_j|} + \sum_{1 \le i < j \le N} \frac{c_{ij}}{|x_i-x_j|}}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${x \in \mathbb{R}^{3N}}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${b_j, c_{ij} \in \mathbb{R}}$$\end{document} . If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${u \in L^2(\mathbb{R}^{3N})}$$\end{document} satisfies \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(-\Delta + V) u = \lambda u}$$\end{document} in distribution sense, then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${u \in \mathcal{K}_{a}^{m}}$$\end{document} for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${m \in \mathbb{Z}_+}$$\end{document} and all a ≤ 0. Our result extends to the case when bj and cij 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. More... »

PAGES

49-84

### References to SciGraph publications

• 2010-06-06. A New Proof of the Analyticity of the Electronic Density of Molecules in LETTERS IN MATHEMATICAL PHYSICS
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• 2007-01-17. The hyperbolic cross space approximation of electronic wavefunctions in NUMERISCHE MATHEMATIK
• 2010. Regularity and Approximability of Electronic Wave Functions in NONE
• 2008-06-02. Semiclassical Resolvent Estimates for Schrödinger Operators with Coulomb Singularities in ANNALES HENRI POINCARÉ
• 2005-01-11. Sharp Regularity Results for Coulombic Many-Electron Wave Functions in COMMUNICATIONS IN MATHEMATICAL PHYSICS
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• 1966. Perturbation theory for linear operators in NONE
• 1995. Perturbation Theory for Linear Operators in NONE
• 1996. Partial Differential Equations II, Qualitative Studies of Linear Equations in NONE
• 2008-12-10. Analytic Structure of Many-Body Coulombic Wave Functions in COMMUNICATIONS IN MATHEMATICAL PHYSICS
• 1981-03. Pointwise bounds on eigenfunctions and wave packets inN-body quantum systems in COMMUNICATIONS IN MATHEMATICAL PHYSICS
• 2003-06. Bounded imaginary powers of differential operators on manifolds with conical singularities in MATHEMATISCHE ZEITSCHRIFT
• 2011-02-05. Explicit Green operators for quantum mechanical Hamiltonians. I. The hydrogen atom in MANUSCRIPTA MATHEMATICA
• 1998-04. On Eigenfunction Decay for Two Dimensional¶Magnetic Schrödinger Operators in COMMUNICATIONS IN MATHEMATICAL PHYSICS
• 2001. Basics of the b-Calculus in APPROACHES TO SINGULAR ANALYSIS
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• 1966. Multiple Integrals in the Calculus of Variations in NONE
• ### Journal

TITLE

Letters in Mathematical Physics

ISSUE

1

VOLUME

101

### Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/s11005-012-0551-z

DOI

http://dx.doi.org/10.1007/s11005-012-0551-z

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1052081297

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