Effective Mass Theorems with Bloch Modes Crossings View Full Text


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Article Info

DATE

2022-07-13

AUTHORS

Victor Chabu, Clotilde Fermanian Kammerer, Fabricio Macià

ABSTRACT

We study a Schrödinger equation modeling the dynamics of an electron in a crystal in the asymptotic regime of small wave-length comparable to the characteristic scale of the crystal. Using Floquet Bloch decomposition, we obtain a description of the limit of time averaged energy densities. We make a rather general assumption assuming that the initial data are uniformly bounded in a high order Sobolev spaces and that the crossings between Bloch modes are at worst conical. We show that despite the singularity they create, conical crossing do not trap the energy and do not prevent dispersion. We also investigate the interactions between modes that can occurred when there are some degenerate crossings between Bloch bands. More... »

PAGES

1339-1400

References to SciGraph publications

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  • 2011-09-22. Quantum Transport in Crystals: Effective Mass Theorem and K·P Hamiltonians in COMMUNICATIONS IN MATHEMATICAL PHYSICS
  • 2015. High-Frequency Dynamics for the Schrödinger Equation, with Applications to Dispersion and Observability in NONLINEAR OPTICAL AND ATOMIC SYSTEMS
  • 2018-08-18. Wavepackets in Inhomogeneous Periodic Media: Propagation Through a One-Dimensional Band Crossing in COMMUNICATIONS IN MATHEMATICAL PHYSICS
  • 2003-06. A Landau-Zener Formula for Non-Degenerated Involutive Codimension 3 Crossings in ANNALES HENRI POINCARÉ
  • 2003-10-10. Effective Dynamics for Bloch Electrons: Peierls Substitution and Beyond in COMMUNICATIONS IN MATHEMATICAL PHYSICS
  • 1978-12. Theory of Bloch waves in JOURNAL D'ANALYSE MATHÉMATIQUE
  • 1946-12. Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe in ACTA MATHEMATICA
  • 2019-01-28. Semiclassical Analysis of Dispersion Phenomena in ANALYSIS AND PARTIAL DIFFERENTIAL EQUATIONS: PERSPECTIVES FROM DEVELOPING COUNTRIES
  • 2011-04-20. The Schrödinger Flow in a Compact Manifold: High-frequency Dynamics and Dispersion in MODERN ASPECTS OF THE THEORY OF PARTIAL DIFFERENTIAL EQUATIONS
  • 2005-03-30. Homogenization of the Schrödinger Equation and Effective Mass Theorems in COMMUNICATIONS IN MATHEMATICAL PHYSICS
  • 2017-05-13. An Egorov Theorem for Avoided Crossings of Eigenvalue Surfaces in COMMUNICATIONS IN MATHEMATICAL PHYSICS
  • 1975-10. The spectrum of Hill's equation in INVENTIONES MATHEMATICAE
  • 2013-04-03. Wigner Measure Propagation and Conical Singularity for General Initial Data in ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
  • 2016-05-30. Wigner measures and observability for the Schrödinger equation on the disk in INVENTIONES MATHEMATICAE
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    http://scigraph.springernature.com/pub.10.1007/s00205-022-01803-2

    DOI

    http://dx.doi.org/10.1007/s00205-022-01803-2

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