American Institute of Mathematical Sciences

Advanced
January  2006, 6(1): 1-16. doi: 10.3934/dcdsb.2006.6.1

Homogenization of the Schrödinger equation with a time oscillating potential

Grégoire Allaire 1, and  M. Vanninathan 2,

1. 

Centre de Mathématiques Appliquées, Ecole Polytechnique, 91128 PALAISEAU Cedex, France
2. 

TIFR Center, P.O. Box 1234, Bangalore - 560012, India

Received  March 2005 Revised  August 2005 Published  October 2005

We study the homogenization of a Schrödinger equation in a periodic medium with a time dependent potential. This is a model for semiconductors excited by an external electromagnetic wave. We prove that, for a suitable choice of oscillating (both in time and space) potential, one can partially transfer electrons from one Bloch band to another. This justifies the famous "Fermi golden rule" for the transition probability between two such states which is at the basis of various optical properties of semiconductors. Our method is based on a combination of classical homogenization techniques (two-scale convergence and suitable oscillating test functions) and of Bloch waves theory.
Keywords: Bloch waves., Schrödinger equation, Homogenization.
Mathematics Subject Classification: Primary: 35B27; Secondary: 35Q4.
Citation: Grégoire Allaire, M. Vanninathan. Homogenization of the Schrödinger equation with a time oscillating potential. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 1-16. doi: 10.3934/dcdsb.2006.6.1
[1]

Grégoire Allaire, Tuhin Ghosh, Muthusamy Vanninathan. Homogenization of stokes system using bloch waves. Networks & Heterogeneous Media, 2017, 12 (4) : 525-550. doi: 10.3934/nhm.2017022
[2]

Yue Liu. Existence of unstable standing waves for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2008, 7 (1) : 193-209. doi: 10.3934/cpaa.2008.7.193
[3]

Reika Fukuizumi. Stability and instability of standing waves for the nonlinear Schrödinger equation with harmonic potential. Discrete & Continuous Dynamical Systems, 2001, 7 (3) : 525-544. doi: 10.3934/dcds.2001.7.525
[4]

François Genoud. Existence and stability of high frequency standing waves for a nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2009, 25 (4) : 1229-1247. doi: 10.3934/dcds.2009.25.1229
[5]

Sevdzhan Hakkaev. Orbital stability of solitary waves of the Schrödinger-Boussinesq equation. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1043-1050. doi: 10.3934/cpaa.2007.6.1043
[6]

Grégoire Allaire, Carlos Conca, Luis Friz, Jaime H. Ortega. On Bloch waves for the Stokes equations. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 1-28. doi: 10.3934/dcdsb.2007.7.1
[7]

Alex H. Ardila. Stability of standing waves for a nonlinear SchrÖdinger equation under an external magnetic field. Communications on Pure & Applied Analysis, 2018, 17 (1) : 163-175. doi: 10.3934/cpaa.2018010
[8]

Reika Fukuizumi, Louis Jeanjean. Stability of standing waves for a nonlinear Schrödinger equation wdelta potentialith a repulsive Dirac. Discrete & Continuous Dynamical Systems, 2008, 21 (1) : 121-136. doi: 10.3934/dcds.2008.21.121
[9]

Jun-ichi Segata. Initial value problem for the fourth order nonlinear Schrödinger type equation on torus and orbital stability of standing waves. Communications on Pure & Applied Analysis, 2015, 14 (3) : 843-859. doi: 10.3934/cpaa.2015.14.843
[10]

Nan Lu. Non-localized standing waves of the hyperbolic cubic nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3533-3567. doi: 10.3934/dcds.2015.35.3533
[11]

Minoru Murai, Kunimochi Sakamoto, Shoji Yotsutani. Representation formula for traveling waves to a derivative nonlinear Schrödinger equation with the periodic boundary condition. Conference Publications, 2015, 2015 (special) : 878-900. doi: 10.3934/proc.2015.0878
[12]

Jun-ichi Segata. Well-posedness and existence of standing waves for the fourth order nonlinear Schrödinger type equation. Discrete & Continuous Dynamical Systems, 2010, 27 (3) : 1093-1105. doi: 10.3934/dcds.2010.27.1093
[13]

Jerry L. Bona, Thierry Colin, Colette Guillopé. Propagation of long-crested water waves. Ⅱ. Bore propagation. Discrete & Continuous Dynamical Systems, 2019, 39 (10) : 5543-5569. doi: 10.3934/dcds.2019244
[14]

Camille Laurent. Internal control of the Schrödinger equation. Mathematical Control & Related Fields, 2014, 4 (2) : 161-186. doi: 10.3934/mcrf.2014.4.161
[15]

Claude Bardos, François Golse, Peter Markowich, Thierry Paul. On the classical limit of the Schrödinger equation. Discrete & Continuous Dynamical Systems, 2015, 35 (12) : 5689-5709. doi: 10.3934/dcds.2015.35.5689
[16]

D.G. deFigueiredo, Yanheng Ding. Solutions of a nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2002, 8 (3) : 563-584. doi: 10.3934/dcds.2002.8.563
[17]

Frank Wusterhausen. Schrödinger equation with noise on the boundary. Conference Publications, 2013, 2013 (special) : 791-796. doi: 10.3934/proc.2013.2013.791
[18]

Kamel Hamdache, Djamila Hamroun. Macroscopic limit of the kinetic Bloch equation. Kinetic & Related Models, 2021, 14 (3) : 541-570. doi: 10.3934/krm.2021015
[19]

Sista Sivaji Ganesh, Vivek Tewary. Bloch wave approach to almost periodic homogenization and approximations of effective coefficients. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021119
[20]

Vivek Tewary. Combined effects of homogenization and singular perturbations: A bloch wave approach. Networks & Heterogeneous Media, 2021, 16 (3) : 427-458. doi: 10.3934/nhm.2021012

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (51)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences