OAI.SharpLiebThirring.sharp_lieb_thirring

The theorem states that the full sharp Lieb–Thirring claim MainClaim holds. Here the form domain is H¹(ℝ;ℂ), given as an L² function with an L² weak derivative; for a real potential W, a weak eigenfunction at negative energy −k² (k>0) is u in H¹ with ∫ conj(v′)u′ − ∫ W conj(v)u = −k² ∫ conj(v)u for every v in H¹. negativeMoment(γ,W) is the supremum, in [0,∞], over all finite L²-orthonormal families of such eigenfunctions with parameters k_i of the sum of (k_i)^(2γ), so it is the sum over negative eigenvalues with multiplicity and is not assumed finite. oneStateMoment keeps a single eigenfunction. A potential W is Admissible for γ if W ≥ 0 almost everywhere and W lies in L^(γ+1/2); potentialMass(γ,W) = ∫ W^(γ+1/2). The semiclassical constant is Γ(γ+1)/(2√π Γ(γ+3/2)) and the sharp constant is 2((γ−1/2)/(γ+1/2))^(γ−1/2) times it. optimalConstant and oneStateConstant are the suprema of negativeMoment/potentialMass and oneStateMoment/potentialMass over admissible W with positive mass. The equality potential is, with r=(γ−1/2)⁻¹, (r+1)·sech²(r x). The claim says: for every γ with 1/2<γ<3/2, (1) every admissible W satisfies negativeMoment(γ,W) ≤ sharpConstant(γ)·potentialMass(γ,W); (2) optimalConstant(γ) equals the sharp constant; (3) oneStateConstant(γ) also equals the sharp constant; (4) the equality potential is admissible with positive mass; and (5) for it, negativeMoment equals the sharp constant times its potential mass. No finite spectral cutoff is imposed.