Optimal Lieb-Thirring type inequalities for Schrödinger and Jacobi operators with complex potentials

We prove optimal Lieb-Thirring type inequalities for Schrödinger and Jacobi operators with complex potentials. Our results bound eigenvalue power sums (Riesz means) by the Lᵖ norm of the potential, where in contrast to the self-adjoint case, each term needs to be weighted by a function of the ratio of the distance of the eigenvalue to the essential spectrum and the distance to the endpoint (s) thereof. Our Lieb-Thirring type bounds only hold for integrable weight functions. To prove optimality, we establish divergence estimates for non-integrable weight functions. The divergence rates exhibit a logarithmic or even polynomial gain compared to semiclassical methods (Weyl asymptotics) for real potentials.