We present a theory for carrying out homogenization limits for quadratic functions (called "energy densities") of solutions of initial value problems (IVPs) with anti-self-adjoint (spatial) pseudo-differential operators (PDOs). The approach is based on the introduction of phase space Wigner (matrix) measures that are calculated by solving kinetic equations involving the spectral properties of the PDO. The weak limits of the energy densities are then obtained by taking moments of the Wigner measure.The very general theory is illustrated by typical examples like (semi)classical limits of Schrödinger equations (with or without a periodic potential), the homogenization limit of the acoustic equation in a periodic medium, and the classical limit of the Dirac equation. IntroductionWe consider the following type of initial value problems:where ε is a small parameter, u ε (t, x) is a vector-valued L 2 -function on R m x , and P ε is an anti-self-adjoint, matrix-valued (pseudo)-differential operator with a Weyl symbol given by P 0 (x, x/ε, εξ) + O(ε). Here P 0 = P 0 (x, y, ξ) is a smooth function that is periodic with respect to y. By ξ we denote the conjugate variable to the position x; that is, ξ = −i∇ x .The main assumptions are that the data u ε I are bounded in L 2 as ε goes to 0 and that u ε I oscillates at most at frequency 1/ε; for instance,A more general formulation of the assumptions on u ε I is given in definitions (1.26) and (1.27) below.

Abstract. This work is devoted to the derivation of a nonlinear 1-particle equation from a linear AT-particle Schrodinger equation in the time dependent case. It emphazises the role of a so-called "finite Schrodinger hierarchy" and of a limiting (infinite) "Schrodinger hierarchy". Convergence of solutions of the first to solutions of the second is established by using "physically relevant" estimates (L 2 and energy conservation) under very general assumptions on the interaction potential, including in particular the Coulomb potential. In the case of bounded potentials, a stability theorem for the infinite Schrodinger hierarchy is proved, based on Spohn's idea of using the trace norm and elementary techniques pertaining to the abstract Cauchy-Kowalewskaya theorem. The core of this program is to prove that if the limiting AT-particle distribution function is factorized at time t = 0, it remains factorized for all later times.We offer this contribution to Cathleen Morawetz as an expression of our admiration and friendship and in recognition of the influence that her work on the interaction of mathematics and physics has exerted on us.

The time-dependent Hartree-Fock equations are derived from the N -body linear Schrödinger equation with mean-field scaling in the limit N → ∞ and for initial data that are like Slater determinants. Only the case of bounded, symmetric interaction potentials is treated in this work. We prove that, as N → ∞, the first partial trace of the Nbody density operator approaches the solution of the time-dependent Hartree-Fock equations (in operator form) in the trace norm.

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Under natural assumptions on the initial density matrix of a mixed quantum state (Hermitian, non-negative definite, uniformly bounded trace, Hilbert-Schmidt norm and kinetic energy) we prove that accumulation points (as the scaled Planck constant tends to zero) of solutions of a corresponding slightly regularized Wigner-Poisson system are distributional solutions of the classical Vlasov-Poisson system. The result holds for the gravitational and repulsive cases. Also, for every phase-space density in [Formula: see text] (with bounded kinetic energy) we prepare a sequence of density matrices satisfying the above assumptions, such that the given density is the limit of the Wigner transforms of these density matrices.

A rigorous derivation of the semiclassical Liouville equation for electrons which move in a crystal lattice (without the influence of an external field) is presented herein. The approach is based on carrying out the semiclassical limit in the band-structure Wigner equation. The semiclassical macroscopic densities are also obtained as limits of the corresponding quantum quantities.

We present a steepest descent energy minimization scheme for micromagnetics. The method searches on a curve that lies on the sphere which keeps the magnitude of the magnetization vector constant. The step size is selected according to a modified Barzilai-Borwein method. Standard linear tetrahedral finite elements are used for space discretization. For the computation of static hysteresis loops the steepest descent minimizer is faster than a Landau-Lifshitz micromagnetic solver by more than a factor of two. The speed up on a graphic processor is 4.8 as compared to the fastest single-core CPU implementation.

We propose to quantify the correlation inherent in a many-electron (or many-fermion) wave function psi by comparing it to the unique uncorrelated state that has the same 1-particle density operator as does /|psi>