## Proof of the Lieb-Thirring conjecture in the case of strictly convex quadratic potential. (Preuve de la conjecture de Lieb-Thirring dans le cas des potentiels quadratiques strictement convexes.)(French)Zbl 0938.35148

The author proves the Lieb-Thirring conjecture in the particular case of positive definite quadratic potentials. More precisely, if $$V(x)=\sum k_i^2x_i^2$$ with $$k_i>0$$, it is shown that for any convex function $$\varphi$$ on $${\mathbb R}$$ and $$E\in {\mathbb R}$$ one has: $\sum_{m\in{\mathbb N}^n}\varphi (E-\sum_i (2m_i+1)k_i)\leq (2\pi)^{-n}\int \varphi (E-\xi^2 -V(x)) dx d\xi .$ The proof is based on a reduction to the one-dimensional case and elementary properties of convex functions.

### MSC:

 35Q40 PDEs in connection with quantum mechanics 35P20 Asymptotic distributions of eigenvalues in context of PDEs 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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