Optimal Lieb-Thirring type inequalities for Schrödinger and Jacobi operators with complex potentials
Sabine Bögli, Sukrid Petpradittha
TL;DR
This work extends Lieb–Thirring type inequalities to Schrödinger and Jacobi operators with complex potentials by introducing a weighted eigenvalue sum bound expressed through ${\rm Ratio}(V,f)$, where the weight $f$ depends on the log ratio of the distance to the essential spectrum. It proves that integrable weight functions yield finite, optimal bounds in terms of the $L^p$ norm of the potential, while non‑integrable weights lead to explicit divergence rates, including logarithmic and polynomial growth relative to semiclassical expectations. The authors develop uniform spectral asymptotics for strong‑coupling models, derive lower bounds that establish optimality, and extend the framework to Jacobi operators, obtaining nonselfadjoint LT bounds that generalize and sharpen known results. The results reveal sharp thresholds between boundable and divergent spectral sums under complex perturbations, and they address open questions concerning the rate of divergence and the dependence on cone parameters, with implications for non‑selfadjoint spectral theory and semiclassical analysis.
Abstract
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 $L^p$ 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.