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USA) 76: 6062 (1979); M. Levy, Universal functionals of the density and first-order density matrix, Bull. Amer. Phys. So~. 24:626 (1979). M. Levy, Electron densities in search of hamiltonians, Phys. Rev. A 26:1200 (1982). 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. E. H. Lieb, Density functionals for coulomb systems, in "Physics as Natural Philosophy: Essays in Honor of Laszlo Tisza on his 75th Birthday", H. Feshbach and A. T. Press, Cambridge (1982); E. H. Lieb, Density functionals for coulomb systems, Int.
For example, a minimizing ~ will not exist if v is an attractive square well and if N is too large; the extra, unbound electrons will simply "leak away" to infinity. In such a case, E(v) would still have physical signifiance. It would be the ground state energy for fewer than N particles. 9) for all Vl, V2, and v O~a~l = aVl + (1-a)v2. (ii) E(v) is monotone decreasing: that is, if all x, then E(Vl)~E(V2)' Vl(X)~V2(X) for (iii) E(v) is continuous in the L3/2+ Loo norm and is, moreover, locally Lipschitz.
Vp makes sense. We restrict ourselves here to L3 / 2+Loo for simplicity of exposition. The class includes Coulomb potentials because Ixl- l = e(x) Ixl- l + [t-e(x)] Ixl- l with e(x)=1 iflxl~1,e(x)= a,lxl>1. The two terms on the right are in L3 / 2 and in Loo,respectively. 7) Technical remark. ~ is a subset of the Banach space X = L 3 ('\ Ll. Y = L3/2 + L00 • However, the dual of Y 1S • X* , the dual of X, 1S not X oo oo beca~se while L is the dual of Ll, Ll is not the dual of L • However, X4:Y .