American Institute of Mathematical Sciences

Advanced
March  2010, 28(1): 41-65. doi: 10.3934/dcds.2010.28.41

A convexified energy functional for the Fermi-Amaldi correction

Gisèle Ruiz Goldstein 1, , Jerome A. Goldstein 1, and  Naima Naheed 2,

1. 

Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, United States, United States
2. 

Department of Mathematics and Computer Science, Benedict College, Columbia, SC 29204, United States

Received  October 2009 Revised  January 2010 Published  April 2010

Consider the Thomas-Fermi energy functional $E$ for a spin polarized atom or molecule with $N_{1} $ [resp. $N_{2}$] spin up [resp. spin down] electrons and total positive molecular charge Z. Incorporating the Fermi-Amaldi correction as Benilan, Goldstein and Goldstein did, $E$ is not convex. By replacing $E$ by a well-motivated convex minorant $ \mathcal{E}$ ,we prove that $ \mathcal{E} $ has a unique minimizing density $( \rho _{1},\rho _{2}) \ $ when $N_{1}+N_{2}\leq Z+1\ $and $N_{2}\ $is close to $N_{1}.$
Keywords: $L^{1} $constrained minimization, ground state electron density, Fermi-Amaldi correction, convex minorant, spin polarized system, degree theory, Thomas-Fermi theory.
Mathematics Subject Classification: Primary: 35J47, 35J91, 49S05; Secondary: 81Q99, 81V55, 92E1.
Citation: Gisèle Ruiz Goldstein, Jerome A. Goldstein, Naima Naheed. A convexified energy functional for the Fermi-Amaldi correction. Discrete & Continuous Dynamical Systems, 2010, 28 (1) : 41-65. doi: 10.3934/dcds.2010.28.41
[1]

Leif Arkeryd. A kinetic equation for spin polarized Fermi systems. Kinetic & Related Models, 2014, 7 (1) : 1-8. doi: 10.3934/krm.2014.7.1
[2]

John P. Perdew, Adrienn Ruzsinszky. Understanding Thomas-Fermi-Like approximations: Averaging over oscillating occupied orbitals. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 5319-5325. doi: 10.3934/dcds.2013.33.5319
[3]

Simone Paleari, Tiziano Penati. Equipartition times in a Fermi-Pasta-Ulam system. Conference Publications, 2005, 2005 (Special) : 710-719. doi: 10.3934/proc.2005.2005.710
[4]

Yingying Li, Stanley Osher, Richard Tsai. Heat source identification based on $l_1$ constrained minimization. Inverse Problems & Imaging, 2014, 8 (1) : 199-221. doi: 10.3934/ipi.2014.8.199
[5]

Antonio Giorgilli, Simone Paleari, Tiziano Penati. Local chaotic behaviour in the Fermi-Pasta-Ulam system. Discrete & Continuous Dynamical Systems - B, 2005, 5 (4) : 991-1004. doi: 10.3934/dcdsb.2005.5.991
[6]

Mathias Wilke. $L_p$-theory for a Cahn-Hilliard-Gurtin system. Evolution Equations & Control Theory, 2012, 1 (2) : 393-429. doi: 10.3934/eect.2012.1.393
[7]

Riccardo Adami, Diego Noja, Nicola Visciglia. Constrained energy minimization and ground states for NLS with point defects. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1155-1188. doi: 10.3934/dcdsb.2013.18.1155
[8]

Carlos J. Garcia-Cervera, Xiao-Ping Wang. Spin-polarized transport: Existence of weak solutions. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 87-100. doi: 10.3934/dcdsb.2007.7.87
[9]

Fabio Cipriani, Gabriele Grillo. On the $l^p$ -agmon's theory. Conference Publications, 1998, 1998 (Special) : 167-176. doi: 10.3934/proc.1998.1998.167
[10]

Xinliang An, Avy Soffer. Fermi's golden rule and $ H^1 $ scattering for nonlinear Klein-Gordon equations with metastable states. Discrete & Continuous Dynamical Systems, 2020, 40 (1) : 331-373. doi: 10.3934/dcds.2020013
[11]

Jacopo De Simoi. Stability and instability results in a model of Fermi acceleration. Discrete & Continuous Dynamical Systems, 2009, 25 (3) : 719-750. doi: 10.3934/dcds.2009.25.719
[12]

Luchuan Ceng, Qamrul Hasan Ansari, Jen-Chih Yao. Extragradient-projection method for solving constrained convex minimization problems. Numerical Algebra, Control & Optimization, 2011, 1 (3) : 341-359. doi: 10.3934/naco.2011.1.341
[13]

Liren Lin, I-Liang Chern. A kinetic energy reduction technique and characterizations of the ground states of spin-1 Bose-Einstein condensates. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1119-1128. doi: 10.3934/dcdsb.2014.19.1119
[14]

Yanfei You, Suhong Jiang. Improved Lagrangian-PPA based prediction correction method for linearly constrained convex optimization. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021174
[15]

Stefan Possanner, Claudia Negulescu. Diffusion limit of a generalized matrix Boltzmann equation for spin-polarized transport. Kinetic & Related Models, 2011, 4 (4) : 1159-1191. doi: 10.3934/krm.2011.4.1159
[16]

Carlos J. García-Cervera, Xiao-Ping Wang. A note on 'Spin-polarized transport: Existence of weak solutions'. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2761-2763. doi: 10.3934/dcdsb.2015.20.2761
[17]

Lei Wu, Zhe Sun. A new spectral method for $l_1$-regularized minimization. Inverse Problems & Imaging, 2015, 9 (1) : 257-272. doi: 10.3934/ipi.2015.9.257
[18]

Alexander Pankov, Vassilis M. Rothos. Traveling waves in Fermi-Pasta-Ulam lattices with saturable nonlinearities. Discrete & Continuous Dynamical Systems, 2011, 30 (3) : 835-849. doi: 10.3934/dcds.2011.30.835
[19]

Alexander Pankov. Traveling waves in Fermi-Pasta-Ulam chains with nonlocal interaction. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2097-2113. doi: 10.3934/dcdss.2019135
[20]

Piernicola Bettiol. State constrained $L^\infty$ optimal control problems interpreted as differential games. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 3989-4017. doi: 10.3934/dcds.2015.35.3989

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (66)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy