In this article, we study for p∈(1,∞) the Lp‐realization of the vector‐valued Schrödinger operator Lu:=prefix div (Q∇u)+Vu. Using a noncommutative version of the Dore–Venni theorem due to Monniaux and Prüss, we prove that the Lp‐realization of scriptL, defined on the intersection of the natural domains of the differential and multiplication operators which form scriptL, generates a strongly continuous contraction semigroup on Lp(double-struckRd;double-struckCm). We also study additional properties of the semigroup such as extension to L1, positivity, ultracontractivity and prove that the generator has compact resolvent.

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In this paper we consider vector-valued Schrödinger operators of the form div(Q∇u) − V u, where V = (v ij ) is a nonnegative locally bounded matrix-valued function and Q is a symmetric, strictly elliptic matrix whose entries are bounded and continuously differentiable with bounded derivatives. Concerning the potential V , we assume an that it is pointwise accretive and that its entries are in L ∞ loc (R d ). Under these assumptions, we prove that a realization of the vector-valued Schrödinger operator generates a C 0 -semigroup of contractions in L p (R d ; C m ). Further properties are also investigated.2010 Mathematics Subject Classification. Primary: 35K40, 47D08; Secondary: 47D06.

We study Density Functional Theory models for systems which are transitionally invariant in some directions, such as a homogeneous 2-d slab in the 3-d space. We show how the different terms of the energy are modified and we derive reduced equations in the remaining directions. In the Thomas-Fermi model, we prove that there is perfect screening, and provide precise decay estimates for the electronic density away from the slab. In Kohn-Sham models, we prove that the Pauli principle is replaced by a penalization term in the energy. In the reduced Hartree-Fock model in particular, we prove that the resulting model is well-posed, and provide some properties for the minimizer.

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