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Universality in the Thomas-Fermi-von Weizsäcker model of atoms and molecules. (English) Zbl 0708.35071
The Thomas-Fermi-von Weizsäcker (TFW) theory is a nonlinear model for the ground state of an atom or a molecule given (in the atomic case) by the energy functional \[ {\mathcal E}(\psi)=A\int | \nabla \psi |^ 2dx+\frac{3}{5}\gamma \int | \psi |^{10/3}dx-z\int \frac{| \psi |^ 2}{| x|}dx+\frac{1}{2}\int \frac{| \psi (x)|^ 2| \psi (y)|^ 2}{| x-y|}dx dy \] on functions \(\psi\) in \({\mathbb{R}}^ 3\). The minimization problem for this model has been studied by R. Benguria, H. Brezis and E. H. Lieb [Commun. Math. Phys. 79, 167-180 (1981; Zbl 0478.49035)]. In this paper we prove that as \(z\to \infty\) all physical quantities converge. The idea is to renormalize the infinite nuclear charge.
In the atomic case above one of the results is that the unique absolute minimizer \(\psi_ z\) converges, i.e., \(\psi_ z(x)\to \psi_{\infty}(x)\), \(| x| \neq 0\). \(\psi^ 2_{\infty}\) is what we call the density for the universal infinite atom. The main step in the renormalization is to characterize the singularity of \(\psi_{\infty}\) at \(x=0\). This is done by showing that both \(\psi_ z\) and \(\psi_{\infty}\) solve a system of partial differential equations. Solutions to this TFW-system have either weak \((z<\infty)\) or strong \((z=\infty)\) singularities at \(x=0\), and the strong singularities have a unique Laurent expansion. Recently we have shown by a completely different method that the highest order term in this expansion also holds for the much more complicated Hartree-Fock model (without the exchange term).
Reviewer: J.P.Solovej

MSC:
35Q40 PDEs in connection with quantum mechanics
81V55 Molecular physics
35A15 Variational methods applied to PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
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[1] Benguria, R., Brezis, H., Lieb, E.H.: The Thomas-Fermi-von Weizsäcker theory of atoms and molecules. Commun. Math. Phys.79, 167–180 (1981) · Zbl 0478.49035
[2] Benguria, R., Lieb, E.H.: The most negative ion in the Thomas-Fermi-von Weizsäcker theory of atoms and molecules. J. Phys. B18, 1045–1059 (1985)
[3] Brezis, H., Lieb, E.H.: Long range potentials in Thomas-Fermi theory. Commun. Math. Phys.65, 231–246 (1979) · Zbl 0416.35066
[4] Dreizler, R.M.: Private communication
[5] Gagliardo, E.: Ulteriori propieta di alcune classi di funzioni in piu variabili. Ric. Mat.8, 24–51 (1959) · Zbl 0199.44701
[6] Hoffmann-Ostenhof, T.: A comparison theorem for differential inequalities with applications in quantum mechanics. J. Phys. A13, 417–424 (1980) · Zbl 0432.35080
[7] Hughes, W.: An atomic energy lower bound that gives Scott’s correction. PhD thesis, Princeton, Department of Mathematics, 1986
[8] Liberman, D.A., Lieb, E.H.: Numerical calculation of the Thomas-Fermi-von Weizsäcker function for an infinite atom without electron repulsion. Los Alamos National Laboratory Report, LA 9186-MS, 1981
[9] Lieb, E.H.: Bound on the maximum negative ionization of atoms and molecules. Phys. Rev. A29, 3018–3028 (1984)
[10] Lieb, E.H.: Thomas-Fermi and related theories of atoms and molecules. Rev. Mod. Phys.53, 603–640 (1981) · Zbl 1114.81336
[11] Lieb, E.H., Simon, B.: The Thomas-Fermi theory of atoms, molecules, and solids. Adv. Math.23, 22–116 (1977) · Zbl 0938.81568
[12] Nirenberg, L.: On elliptic partial differential equations. Ann. Scu. Norm. Sup. Pisa13, 115–162 (1959) · Zbl 0088.07601
[13] Rother, W.: Zur Thomas-Fermi-von Weizsäcker Theorie für Atome und Moleküle. Bayreuther Mathematische Schriften18, 39–145 (1985) · Zbl 0572.35034
[14] Seco, L.A., Sigal, I.M., Solovej, J.P.: Bound on the ionization energy of large atoms. Commun. Math. Phys. (to appear) · Zbl 0714.35059
[15] Siedentop, H., Weikard, R.: On the leading correction of the Thomas-Fermi model: Lower bound – with an appendix by A.M.K. Müller. Invent. Math.97, 159–193 (1989) · Zbl 0689.34011
[16] Siedentop, H., Weikard, R.: On the leading energy correction for the statistical model of the atom: interacting case. Commun. Math. Phys.112, 471–490 (1987) · Zbl 0920.35120
[17] Solovej, J.P.: Universality in the Thomas-Fermi-von Weizsäcker Theory of Atoms and Molecules. PhD thesis, Princeton, Department of Mathematics, June 1989 · Zbl 0708.35071
[18] Sommerfeld, A.: Asymptotische Integration der Differentialgleichung des Thomas-Fermischen Atoms. Z. Phys.78, 283–308 (1932) · JFM 58.1353.03
[19] Stich, W., Gross, E.K.U., Malzacher, P., Dreizler, R.M.: Accurate solution of the Thomas-Fermi-Dirac-Weizsäcker variational equations for the case of neutral atoms and positive ions. Z. Phys. A309, 5–11 (1982)
[20] Veron, L.: Singular solutions of some nonlinear elliptic equations. Nonlinear Analysis5, 225–242 (1981) · Zbl 0457.35031
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