Clotilde Fermanian Kammerer
Prof. Dr. Clotilde Fermanian Kammerer

Technische Universität München
John-von-Neumann-Professor
Zentrum Mathematik, M8
Boltzmannstraße 3
85748 Garching bei München

Raum: 02.08.039
mail clotilde.fermanianematu-pec.fr

Homepage: Clotilde Fermanian Kammerer

Semiclassical analysis of Schrödinger operators

Preview: Our main goal will be to analyze the dynamics of an electron in a crystal in the presence of an external potential with slow variations. It is described by a wave function that solves a semi-classical Schrödinger equation, where the small parameter involved is the ratio between the mean spacing of the lattice and the characteristic length scale of variation of the external potential. Then Effective Mass Theory shows that, under suitable assumptions on the initial data, the wave function can be approximated in the semi-classical limit by a solution of a simpler Schrödinger equation, the Effective Mass Equation, which is independent of the semi-classical parameter. As it is classical in the context of solid state physics, we shall use Floquet-Bloch decomposition which relies on the spectral theory of periodic Schrödinger operators. We shall also use the framework of semi-classical analysis, in particular the theory of pseudo-differential operators and of Wigner measures. We shall explain how the Wigner measure approach together with Floquet-Bloch decomposition can be used to derive Effective Mass Equations and how it allows to treat degenerate situations which have not yet been studied mathematically by introducing Effective Mass Equations of Heisenberg types and involving operator-valued two-scale Wigner measures.

Contents:

  • 1. Chapter:
    Introduction
    Presentation of the equations modeling the dynamics of an electron in a crystal and of the effective mass theory, discussion of examples, statement of the main theorems.
  • 2. Chapter:
    Floquet–Bloch decomposition
    Presentation of Floquet-Bloch theory. Bloch energies, Bloch waves and Bloch modes. Analysis of the high frequency behavior of Bloch band projectors. Introduction of the dispersive equations satisfied by Bloch modes.
  • 3. Chapter:
    Pseudo-differential operators
    We introduce position and momentum variables, together with the phase space. Definition of observables and pseudo-differential operators. Boundedness of pseudo-differential operators in energy spaces. Symbolic calculus and weak Garding inequality.
  • 4. Chapter:
    Wigner measures
    Definition and main properties of Wigner measures. Application to families of solutions to dispersive equations arising in the context of effective mass theory, localization of their Wigner measures.
  • 5. Chapter:
    Two-scale Wigner measures
    Definition and main properties. Application to the analysis of the lack of dispersion of Pde-s. Scalar equations with isolated critical points and special case of a set of degenerate critical points.
  • 6. Chapter:
    Application to effective mass theory for isolated Bloch bands
    A priori estimates for the solutions of the Bloch equations. Bloch decomposition of the Wigner measure. Proof of the main theorems.
  • 7. Chapter:
    Effective mass theory with intersecting Bloch bands
    Description of model problems. Analysis of smooth crossings. Analysis of crossings with conical intersections.
  • ECTS Credits:
    Prerequisites:
    Basic functional analysis and measure theory.


    Lecture Time:
    Übersicht

    First class:
    Tuesday, 16.10.2018 from 12:15 - 13:45
    Room 02.10.011
     
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