Magnetic rings

TL;DR

This paper investigates the spectral and functional properties of magnetic Laplace operators on the circle, deriving sharp inequalities relevant to quantum mechanics and mathematical physics.

Contribution

It establishes new sharp inequalities for magnetic Laplace operators on the circle and Euclidean space, advancing understanding of their spectral properties.

Findings

Proved a Hardy-type inequality in two-dimensional Euclidean space.

Derived a sharp interpolation inequality on the circle.

Established a sharp Keller-Lieb-Thirring inequality.

Abstract

We study functional and spectral properties of perturbations of the magnetic Laplace operator on the circle. This operator appears when considering the restriction to the unit circle of a two-dimensional Schr{\"o}dinger operator with the Bohm-Aharonov vector potential. We prove a Hardy-type inequality on the two-dimensional Euclidean space and, on the circle, a sharp interpolation inequality and a sharp Keller-Lieb-Thirring inequality.