arXiv subjects
Publications and source records attributed to Abdallah Maichine.
In this work, we characterize self-adjoint operators that commute with magnetic translations. We use this characterization to derive effective kinetic energy functionals for homogeneous electron gases and three-dimensional electronic systems with two-dimensional symmetries in the presence of a magnetic field.
This paper studies DFT models for homogeneous 1D materials in the 3D space. It follows our previous work about DFT models for homogeneous 2D materials in 3D. We show how to reduce the problem from a 3D energy functional to a 2D energy functional. The kinetic energy is treated as in the 2D case by diagonilizing admissible states, and writing the kinetic energy as the infimum of a modified kinetic energy functional on reduced states. Besides, we treat here the Hartree interaction term in 2D, and show how to properly define the mean-field potential, through Riesz potential. We then show the well posdness of the reduced model and present some numerical illustrations.
This paper studies DFT models for homogeneous 2D materials in 3D space, under a constant perpendicular magnetic field. We show how to reduce the three--dimensional energy functional to a one--dimensional one, similarly as in our previous work. This is done by minimizing over states invariant under magnetic translations and that commute with the Landau operator. In the reduced model, the Pauli principle no longer appears. It is replaced by a penalization term in the energy.
We study Density Functional Theory models for systems which are translationally 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 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.
In this paper we establish generation of analytic strongly continuous semigroup in $L^p$--spaces for the symmetric matrix Schrödinger operator $div(Q\nabla u)-Vu$, where, for every $x\in\mathbb{R}^d$, $V(x)=(v_{ij}(x))$ is a semi-definite positive and symmetric matrix. The diffusion matrix $Q(\cdot)$ is supposed to be strongly elliptic and bounded and the potential $V$ satisfies the weak condition $v_{ij}\in L^1_{loc}(\mathbb{R}^d)$, for all $i,j\in\{1,\dots,m\}$. We also characterize positivity of the semigroup and we investigate on its compactness.
In the paper \cite{KLMR} the $L^p$-realization $L_p$ of the matrix Schrödinger operator $\mathcal{L}u=div(Q\nabla u)+Vu$ was studied. The generation of a semigroup in $L^p(\R^d,\C^m)$ and characterization of the domain $D(L_p)$ has been established. In this paper we perturb the operator $L_p$ of by a scalar potential belonging to a class including all polynomials and show that still we have a strongly continuous semigroup on $L^p(\R^d,\C^m)$ with domain embedded in $W^{2,p}(\R^d,\C^m)$. We also study the analyticity, compactness, positivity and ultracontractivity of the semigroup and prove Gaussian kernel estimates. Further kernel estimates and asymptotic behaviour of eigenvalues of the matrix Schrödinger operator are investigated.
In this article we study for $p\in (1,\infty)$ the $L^p$-realization of the vector-valued Schrödinger operator $\mathcal{L}u := \mathrm{div} (Q\nabla u) + V u$. Using a noncommutative version of the Dore-Venni theorem due to Monniaux and Prüss, we prove that the $L^p$-realization of $\mathcal{L}$, defined on the intersection of the natural domains of the differential and multiplication operators which form $\mathcal{L}$, generates a strongly continuous contraction semigroup on $L^p(\mathbb{R}^d; \mathbb{R}^m)$. We also study additional properties of the semigroup such as extension to $L^1$, positivity, ultracontractivity and prove that the generator has compact resolvent.