On Lieb-Thirring inequalities for multidimensional Schrödinger operators with complex potentials
Sabine Bögli, Sukrid Petpradittha, František Štampach
TL;DR
The paper resolves an open question on extending Lieb–Thirring-type bounds to Schrödinger operators with complex potentials by constructing a higher-dimensional counterexample. By reducing the eigenvalue problem to a radial equation and deriving a characteristic equation involving Bessel functions, the authors locate a family of complex eigenvalues for $H_h$ and show a logarithmic divergence in the relevant spectral sum as $h\to\infty$. This proves that the naive tau-free generalization fails in dimensions $d\ge 2$, clarifying the limits of such inequalities. The work situates the result within the broader landscape of non-selfadjoint spectral theory, while also connecting to known positive results in cases with stronger assumptions on $p$ (e.g., $p\ge d/2+1$). The methodology hinges on precise asymptotics of Bessel functions in a coupled large-parameter and complex regime, coupled with a delicate zero-counting analysis of the characteristic equation.
Abstract
We solve the open problem by Demuth, Hansmann, and Katriel announced in [Integr. Equ. Oper. Theory 75 (2013), 1-5] by a counter-example construction. The problem concerns a possible generalisation of the Lieb-Thirring inequality for Schrödinger operators in to the case of complex-valued potentials. A counter-example has already been found for the one-dimensional case by the first and third authors in [J. Spectr. Theory 11 (2021), 1391-1413]. Here we generalise the counter-example to higher dimensions.