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.