Quantum Harmonic Oscillators with Nonlinear Effective Masses in the weak density approximation

We study the eigen-energy and eigen-function of a quantum particle acquiring the probability density-dependent effective mass (DDEM) in harmonic oscillators. Instead of discrete eigen-energies, continuous energy spectra are revealed due to the introduction of a nonlinear effective mass. Analytically, we map this problem into an infinite discrete dynamical system and obtain the stationary solutions in the weak density approximation, along with the proof on the monotonicity in the perturbed eigen-energies. Numerical results not only give agreement to the asymptotic solutions stemmed from the expansion of Hermite-Gaussian functions, but also unveil a family of peakon-like solutions without linear counterparts. As nonlinear Schr ¨odinger wave equation has served as an important model equation in various sub-fields in physics, our proposed generalized quantum harmonic oscillator opens an unexplored area for quantum particles with nonlinear effective masses.

Структура электронных состояний соединений Gd-=SUB=-5-=/SUB=-Sb-=SUB=-3-=/SUB=- и Gd-=SUB=-5-=/SUB=-Ge-=SUB=-2-=/SUB=-Sb по данным зонных расчетов и оптической спектроскопии

Results of investigations of the electronic structure and optical properties of Gd5Sb3 and Gd5Ge2Sb compounds are presented. Calculations of band spectra were carried out in frame of local density approximation with a correction for strong correlation effects in 4f shell of rare-earth ion (DFT+U+SO method). Optical constants of these materials were measured by ellipsometric technique in wide wavelength interval. Energy dependencies of a number of spectral parameters were determined. The nature of quantum light absorption is discussed on the base of comparative analysis of the experimental and calculated spectra of optical conductivity.

Density Functional Theory of Material Design: Fundamentals and Applications - I

This article is part-I of a review of density-functional theory (DFT) that is the most widely used method for calculating electronic structure of materials. The accuracy and ease of numerical implementation of DFT methods has resulted in its extensive use for materials design and discovery and has thus ushered in the new field of computational material science. In this article we start with an introduction to Schrödinger equation and methods of its solutions. After presenting exact results for some well-known systems, difficulties encountered in solving the equation for interacting electrons are described. How these difficulties are handled using the variational principle for the energy to obtain approximate solutions of the Schrödinger equation is discussed. The resulting Hartree and Hartree-Fock theories are presented along with results they give for atomic and solid-state systems. We then describe Thomas-Fermi theory and its extensions which were the initial attempts to formulate many-electron problem in terms of electronic density of a system. Having described these theories, we introduce modern density functional theory by discussing Hohenberg-Kohn theorems that form its foundations. We then go on to discuss Kohn-Sham formulation of density-functional theory in its exact form. Next, local density approximation is introduced and solutions of Kohn-Sham equation for some representative systems, obtained using the local density approximation, are presented. We end part-I of the review describing the contents of part-II.

A Riemannian Newton trust-region method for fitting Gaussian mixture models

Gaussian Mixture Models are a powerful tool in Data Science and Statistics that are mainly used for clustering and density approximation. The task of estimating the model parameters is in practice often solved by the expectation maximization (EM) algorithm which has its benefits in its simplicity and low per-iteration costs. However, the EM converges slowly if there is a large share of hidden information or overlapping clusters. Recent advances in Manifold Optimization for Gaussian Mixture Models have gained increasing interest. We introduce an explicit formula for the Riemannian Hessian for Gaussian Mixture Models. On top, we propose a new Riemannian Newton Trust-Region method which outperforms current approaches both in terms of runtime and number of iterations. We apply our method on clustering problems and density approximation tasks. Our method is very powerful for data with a large share of hidden information compared to existing methods.