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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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References:

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