# Third derivative of the one-electron density at the nucleus

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- Department of Mathematical Sciences, Aalborg University
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AALBORG UNIVERSITY ' & $ % Third derivative of the one-electron density at the nucleus by S. Fournais, M. Hoffmann-Ostenhof and T. Østergaard Sørensem R-2006-26 July 2006 Department of Mathematical Sciences Aalborg University Fredrik Bajers Vej 7G DK - 9220 Aalborg Øst Denmark Phone: +45 96 35 80 80 Telefax: +45 98 15 81 29 URL: http://www.math.aau.dk e ISSN 1399–2503 On-line version ISSN 1601–7811 THIRD DERIVATIVE OF THE ONE-ELECTRON DENSITY AT THE NUCLEUS S. FOURNAIS, M. HOFFMANN-OSTENHOF, AND T. ØSTERGAARD SØRENSEN Abstract. We study electron densities of eigenfunctions of ato- mic Schro¨dinger operators. We prove the existence of ρ˜ ′′′(0), the third derivative of the spherically averaged atomic density ρ˜ at the nucleus. For eigenfunctions with corresponding eigen- value below the essential spectrum we obtain the bound ρ˜ ′′′(0) ≤ −(7/12)Z3ρ˜(0), where Z denotes the nuclear charge. This bound is optimal. 1. Introduction and results In a recent paper [5] the present authors (together with T. Hoff- mann-Ostenhof (THO)) proved that electron densities of atomic and molecular eigenfunctions are real analytic away from the positions of the nuclei. Concerning questions of regularity of ρ it therefore re- mains to study the behaviour of ρ in the vicinity of the nuclei. A general (optimal) structure-result was obtained recently [2]. For more detailed information, two possible approaches are to study limits when approaching a nucleus under a fixed angle ω ∈ S2, as was done in [2], and to study the spherical average of ρ (here denoted ρ˜ ), which is mostly interesting for atoms. The existence of ρ˜ ′(0), the first derivative of ρ˜ at the nucleus, and the identity ρ˜ ′(0) = −Zρ˜(0) (see (1.12) be- low) follow immediately from Kato’s classical result [12] on the ‘Cusp Condition’ for the associated eigenfunction (see also [15], [10]). Two of the present authors proved (with THO) the existence of ρ˜ ′′(0), and, for densities corresponding to eigenvalues below the essentia

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