On Lieb-Thirring inequalities for multidimensional Schrödinger operators with complex potentials

2025/10/02 by Bögli, Sabine, Petpradittha, Sukrid, Štampach, František · 1 citation
#35P15 #47A10 #47A75 #81Q12 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2510.02192

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.

Citations

• Lieb-Thirring and Jensen sums for non-self-adjoint Schrödinger operators on the half-line
• Improved Lieb-Thirring type inequalities for non-selfadjoint Schrödinger operators
• On Lieb--Thirring inequalities for one-dimensional non-self-adjoint Jacobi and Schr" odinger operators
• On non-selfadjoint perturbations of infinite band Schrödinger operators and Kato method
• Eigenvalue bounds for Schrödinger operators with complex potentials. III
• On the asymptotics of Bessel functions in the Fresnel regime
• On quantitative bounds on eigenvalues of a complex perturbation of a Dirac operator
• Lieb-Thirring type inequalities for non self-adjoint perturbations of magnetic Schrödinger operators

Cited by

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

Related

• Improved Lieb-Thirring type inequalities for non-selfadjoint Schrödinger operators
• On Lieb--Thirring inequalities for one-dimensional non-self-adjoint Jacobi and Schr" odinger operators
• Optimal Lieb-Thirring type inequalities for Schrödinger and Jacobi operators with complex potentials
• Lieb-Thirring inequalities with improved constants
• Sharp Lieb-Thirring Inequalities in High Dimensions