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Classical and semi-classical analysis of magnetic fields in two dimensions

Abstract : This manuscript is devoted to classical mechanics and quantum mechanics, especially in the presence of magnetic field. In classical mechanics, we use Hamiltonian dynamics to describe the motion of a charged particle in a domain affected by the magnetic field. We are interested in two classical physical problems: the confinement and the scattering problem. In the quantum case, we study the spectral problem of the magnetic Laplacian at the semi-classical level, in two-dimensional domains: on a compact Riemmanian manifold with boundary and on ℝ ². Under the assumption that the magnetic field has a unique positive and non-degenerate minimum, we can describe the eigenfunctions by WKB methods. Thanks to the spectral theorem, we estimated efficiently the true eigenfunctions and the approximate eigenfunctions locally near the minimum point of the magnetic field. On ℝ ², with the additional assumption that the magnetic field is radially symmetric, we can show that the eigenfunctions of the magnetic Laplacian decay exponentially at infinity and at a rate controlled by the phase function created in WKB procedure. Furthermore, the eigenfunctions are very well approximated in an exponentially weighted space.
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Submitted on : Friday, January 10, 2020 - 2:07:08 PM
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Duc Tho Nguyen. Classical and semi-classical analysis of magnetic fields in two dimensions. Mathematical Physics [math-ph]. Université Rennes 1, 2019. English. ⟨NNT : 2019REN1S045⟩. ⟨tel-02434902⟩

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