Universal Monotonicity of Eigenvalue Moments and Sharp Lieb-thirring Inequalities

for some constant Lσ,d ≥ L σ,d and are widely discussed in the literature (see e.g. [3, 9, 11]). A longstanding question is when (1.4) holds with Lσ,d = L cl σ,d. The most general result is due to Laptev and Weidl [10] who proved that Lσ,d = L cl σ,d for all σ ≥ 32 and d ≥ 1. Their proof is based on a dimensional reduction of Schrödinger operators with operator valued potentials which allows them to make use of the bound for σ = 3 2 , d = 1 which has been first proven by Lieb and Thirring [12]. For simplified proof see also [2]. On the other hand, by analyzing the spectra of harmonic oscillators Helffer and Robert have shown that Lσ,d > L cl σ,d for σ < 1