The Laplacian with complex magnetic fields

TL;DR

This paper investigates the spectral properties of the two-dimensional magnetic Laplacian with complex magnetic fields, establishing conditions for operator sectoriality, resolvent compactness, and constructing semiclassical pseudomodes using a WKB approach.

Contribution

It introduces a framework for analyzing complex magnetic fields in the magnetic Laplacian, including sectoriality and pseudomode construction, which are novel contributions.

Findings

Operator is m-sectorial under certain conditions.

Resolvent is compact for specific complex magnetic fields.

Semiclassical pseudomodes exist for non-critical complex magnetic fields.

Abstract

We study the two-dimensional magnetic Laplacian when the magnetic field is allowed to be complex-valued. Under the assumption that the imaginary part of the magnetic potential is relatively form-bounded with respect to the real part of the magnetic Laplacian, we introduce the operator as an m-sectorial operator. In two dimensions, sufficient conditions are established to guarantee that the resolvent is compact. In the case of non-critical complex magnetic fields, a WKB approach is used to construct semiclassical pseudomodes, which do not exist when the magnetic field is real-valued.