Lahmar-Benbernou A., Martinez A.
Semiclassical Asymptotics of the Residues of the Scattering
Matrix for Shape Resonances
(83K, LATeX 2e)
ABSTRACT. The aim of this study is to give complete semiclassical asymptotics of
the residues Res$[S(\lambda ,\omega ,\omega' ),\rho ]$
at some pole $\rho$ of the distributional kernel of the scattering
matrix $S(\lambda )$ corresponding to a
semiclassical two-body
Schr\"{o}dinger operator $P=-h^2\Delta +V$, and considered as a
meromorphic operator-valued function with
respect to the energy $\lambda$. We do it in the case where the pole
$\rho$ considered is a
shape resonance of $P$.
This is a
continuation of \cite{Be2} where an extra geometrical condition was
assumed (namely the absence
of caustics near the energy level Re$\rho$). Here we drop
this assumption
by using an FBI transform which permits to work in the complexified
phase space.
Then we show that some semiclassical WKB expansions
are global, and this allows
us to find out estimates for the residue of the type ${\cal O}(h^N
e^{-2S_0/h})$ where $S_0$
is the Agmon width of the potential barrier, and
$N$ may be arbitrarily large depending on an explicit geometrical
location of the incoming
and outgoing waves $\omega$ and
$\omega'$ one consider. Full asymptotic expansions are obtained under
some
additional generic geometric assumption on the potential $V$.