New Bounds on the Lieb-thirring Constants

are known as Lieb-Thirring bounds and hold true with finite constants Lγ,d if and only if γ ≥ 1/2 for d = 1, γ > 0 for d = 2 and γ ≥ 0 for d ≥ 3. Here and in the following, A± = (|A| ± A)/2 denote the positive and negative parts of a self-adjoint operator A. The case γ > (1 − d/2)+ was shown by Lieb and Thirring in [21]. The critical case γ = 0, d ≥ 3 is known as the Cwikel-Lieb-Rozenblum inequality, see [8, 19, 22] and also [18, 7]. The remaining case γ = 1/2, d = 1 was verified in [25]. It is known that as soon as V ∈ L(R) and the constant Lγ,d is finite, then we have Weyl’s asymptotic formula