97-357 De la breteche R.

Preuve de la conjecture de Lieb-Thirring dans le cas des potentiels quadratiques strictement convexes. (32K, plain) Jun 19, 97

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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.

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