Optimal Lieb-Thirring type inequalities for Schr\"odinger and Jacobi operators with complex potentials

Sabine Bögli; Sukrid Petpradittha

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arXiv · 2510.02288

Optimal Lieb-Thirring type inequalities for Schr\"odinger and Jacobi operators with complex potentials

Sabine Bögli·Sukrid Petpradittha

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Abstract

We prove optimal Lieb-Thirring type inequalities for Schr\"odinger 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.

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BibTeX RIS

Sabine Bögli, Sukrid Petpradittha. 2025-10-02. Optimal Lieb-Thirring type inequalities for Schr\"odinger and Jacobi operators with complex potentials. https://arxiv.org/abs/2510.02288

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