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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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