Complex Magnetic Fields: An Improved Hardy--Laptev--Weidl Inequality and Quasi-Self-Adjointness

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.

The minimally anisotropic metric operator in quasi-Hermitian quantum mechanics

“…While the condition on the Bari basis property is automatically satisfied only in finitedimensional Hilbert spaces, it also holds, for example, for a class of Schrödinger operators on a bounded interval with complex Robin boundary conditions [11]. On the other hand, there exist quasi-self-adjoint operators without the Bari property [12].…”

“…While the aforementioned examples are eligible for the proposed minimization scheme, alternative choices might be necessary for other models (e.g. [12]).…”

Pseudospectra of the Damped Wave Equation with Unbounded Damping

“…It might appear that the dependence of the rates in (3.7) as well as the conditions on β, determining the curves along which we have a decay in (3.7), are just a limitation of the method. However, several examples and results for Schrödinger operators, see [2,1,11,6,9], suggest that this dependence is fundamental and indeed reflecting the regularity of coefficients and their behavior at infinity. A more detailed discussion can be found in the introduction in [9].…”

“…However, several examples and results for Schrödinger operators, see [2,1,11,6,9], suggest that this dependence is fundamental and indeed reflecting the regularity of coefficients and their behavior at infinity. A more detailed discussion can be found in the introduction in [9].…”