Martinez A., Yajima K.
On the Fundamental Solution of Semiclassical Schr\"odinger Equations at
Resonant Times
(50K, LATeX 2e)

ABSTRACT.  We consider perturbations  of the semiclassical harmonic oscillator of
the form
${\ds P=-\frac{h^2}2\Delta + \frac{x^2}2 + h^{\delta}W(x)}$, $x \in {\bf
R}^m$,
with $W(x)\sim \la x\ra^{2-\mu}$ as $\vert x\vert \rightarrow +\infty$
and $\delta ,\mu \in (0,1)$, and we investigate  the fundamental
solution $E(t,x,y)$ of the corresponding time-dependent Schr\"odinger
equation.  We prove that at resonant times $t=n\pi$ ($n\in {\bf Z}$) it
admits a semiclassical asymptotics of the form:
$E(n\pi ,x,y) \sim h^{-m(1+\nu)/2}a_0e^{iS( x,y)/h}$ with
$a_0\not=0$ and $\nu = \delta /(1-\mu)$, under the conditions  $x\not=
(-1)^ny$ and $\nu <1$.