Complex magnetic fields: An improved Hardy-Laptev-Weidl inequality and quasi-self-adjointness

TL;DR

This paper improves Hardy-type inequalities by incorporating complex magnetic fields and explores the properties of non-self-adjoint momenta, with implications for PT-symmetric quantum mechanics.

Contribution

It introduces an enhanced Hardy inequality for complex magnetic fields and analyzes the basis properties of associated non-self-adjoint momentum operators.

Findings

Improved Hardy inequality for complex magnetic fields.

Derived formulas for similarity transforms to self-adjoint operators.

Studied basis properties of non-self-adjoint momenta.

Abstract

We show that allowing magnetic fields to be complex-valued leads to an improvement in the magnetic Hardy-type inequality due to Laptev and Weidl. The proof is based on the study of momenta on the circle with complex magnetic fields, which is of independent interest in the context of PT-symmetric and quasi-Hermitian quantum mechanics. We study basis properties of the non-self-adjoint momenta and derive closed formulae for the similarity transforms relating them to self-adjoint operators.