Abrupt changes in the spectra of the Laplacian with constant complex magnetic field

TL;DR

This paper investigates how the spectrum of the Laplacian changes under complex magnetic fields, revealing a transition from discrete Landau levels to a spectrum covering the entire complex plane, depending on the magnetic field's nature.

Contribution

It provides a detailed analysis of spectral behavior of the Laplacian with complex magnetic fields, highlighting the abrupt spectral changes and the conditions for Landau levels to persist or vanish.

Findings

Spectrum covers entire complex plane with nonzero imaginary magnetic component.

Landau levels persist under real magnetic fields, rotate in the complex plane.

Landau levels disappear when the magnetic field is purely imaginary.

Abstract

We analyze the spectrum of the Laplace operator, subject to homogeneous complex magnetic fields in the plane. For real magnetic fields, it is well-known that the spectrum consists of isolated eigenvalues of infinite multiplicities (Landau levels). We demonstrate that when the magnetic field has a nonzero imaginary component, the spectrum expands to cover the entire complex plane. Additionally, we show that the Landau levels (appropriately rotated and now embedded in the complex plane) persists, unless the magnetic field is purely imaginary in which case they disappear and the spectrum becomes purely continuous.