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

David Krejcirik, Tho Nguyen Duc, Nicolas Raymond

TL;DR

The paper analyzes the spectral properties of the magnetic Laplacian with a constant complex magnetic field in the plane, revealing stark differences from the real-field case. By introducing a weak core framework and exploiting unitary symmetries, it shows that the spectrum is the entire complex plane for any nonreal field, with complex Landau levels emerging as rotated eigenvalues unless the field is purely imaginary, where they vanish and the spectrum becomes purely continuous. The results hinge on a fibered analysis after a partial Fourier transform and a careful construction of Weyl sequences, plus explicit eigenfunctions built from Hermite functions. The work highlights gauge-dependence and non-self-adjoint effects absent in the real-field theory, providing a comprehensive classification of the spectrum and its essential components across all possible complex fields. These findings have implications for non-self-adjoint quantum models and underline fundamental spectral differences driven by complex magnetic fields.

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.

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

TL;DR

The paper analyzes the spectral properties of the magnetic Laplacian with a constant complex magnetic field in the plane, revealing stark differences from the real-field case. By introducing a weak core framework and exploiting unitary symmetries, it shows that the spectrum is the entire complex plane for any nonreal field, with complex Landau levels emerging as rotated eigenvalues unless the field is purely imaginary, where they vanish and the spectrum becomes purely continuous. The results hinge on a fibered analysis after a partial Fourier transform and a careful construction of Weyl sequences, plus explicit eigenfunctions built from Hermite functions. The work highlights gauge-dependence and non-self-adjoint effects absent in the real-field theory, providing a comprehensive classification of the spectrum and its essential components across all possible complex fields. These findings have implications for non-self-adjoint quantum models and underline fundamental spectral differences driven by complex magnetic fields.

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.

Paper Structure

This paper contains 12 sections, 12 theorems, 87 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Motivation
  3. Main results
  4. General notations
  5. Structure of the paper
  6. Definition of the magnetic Laplacian
  7. A weak core result
  8. Spectral reducibility
  9. When the magnetic field is non-real and non-imaginary
  10. Weyl sequence construction
  11. Complex Landau levels
  12. When the magnetic field is purely imaginary

Key Result

Proposition 1.1

$\mathscr{L}_{\textbf{b}}$ is a closed, densely defined operator and its adjoint is given by where Consequently, $\mathscr{L}_{\textbf{b}}$ is self-adjoint if and only if $\textbf{b}$ is real.

Figures (1)

  • Figure 1: The spectrum of the operator $\mathscr{L}_{\textbf{b}}$ for various values of $\textbf{b}$. The continuous spectrum is depicted in cyan, while the point spectrum is shown in magenta.

Theorems & Definitions (22)

  • Proposition 1.1
  • Theorem 1.2
  • Proposition 2.1
  • proof
  • proof : Proof of Proposition \ref{['Prop Adjoint']}
  • Proposition 2.2
  • proof
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • ...and 12 more