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

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.