De la breteche R.
Preuve de la conjecture de Lieb-Thirring  dans le cas des potentiels
quadratiques strictement convexes.
(32K, plain)

ABSTRACT.  We consider the Schr\"odinger operator $P_V(h)=-h^2\Delta +V$ where
 $V\in C^0(\r^n)$ such that $\lim
_{|x|\to+\infty}V(x)=+\infty $. For every $\phi$ continuous convex
with a support
in $\r^+$, we state the following inequality
$${\rm Tr}\big(\phi (E-P_V(h))\big)\leq {h^{-n}\over
(2\pi )^n}\int_{\r ^n}\int_{\r ^n}\phi (E-\xi^2-V(x))\d x\d
\xi   $$  for all $E$ real
and $h\in\r^+$ when $V$ is strictly convex and
quadratic. When
$\phi_\gamma=\max\{ 0,t\}^\gamma$ $\gamma\geq 1$ and $n\geq 3$, the
inequality  is the
Lieb--Thirring's conjecture.