99-478 Bernard Parisse
Construction $BKW$ en fonds de puits, cas particuliers (Schr\"odinger, Dirac avec champ magn\'etique) (67K, LaTeX) Dec 15, 99

Abstract. We study the spectral properties of pseudo-differential operators in the semi-classical limit at energies near a non degenerate minimum of the principal symbol $p$. We give precise asymptotics of the non resonant energy levels for a scalar holomorphic $p$, we get explicit expressions in dimension 2. Precise asymptotics are also derived for the Schr\"odinger and Dirac operator with electro-magnetic field in dimension 2 and 3 (and we give the transport equation of the first term of the $WKB$ expansion of the associated eigenfunction). Then we study the Schr\"odinger and Dirac equation under rotational invariance hypothesis (in dimension 2), and prove that the holomorphic assumption of $p$ can be replaced by $C^\infty $ assumption on the fields. Moreover we prove that there is an associated effective Agmon distance and obtain decay properties of eigenfunctions similar to the case of Schr\"odinger or Dirac operator without magnetic field.

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