Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Magnetic Steklov operator on differential forms

Tirumala Chakradhar, Katie Gittins, Georges Habib, Norbert Peyerimhoff

TL;DR

This work extends the Steklov eigenproblem to differential forms by introducing the magnetic Steklov operator $T^{[k],\eta}$ via $\,\eta$-harmonic extensions and the magnetic differential $d^{\eta}$. It establishes well-posedness of the associated boundary value problem and develops the spectral theory, including a min-max characterization and discreteness of the spectrum, while revealing that a Diamagnetic Inequality analogue can fail in this magnetic setting. The authors prove uniqueness and existence of solutions, analyze the operator’s coercivity through magnetic Gaffney inequalities, and provide explicit spectral computations in low-dimensional geometries (2- and 4-dimensional Euclidean balls) to illustrate how the magnetic potential $\eta$ influences the spectrum. They also present exact results on the magnetic Hodge Laplacian for spheres $\mathbb{S}^1$ and $\mathbb{S}^3$, and discuss how boundary spectra relate to magnetic cohomology. Overall, the paper offers a robust theoretical foundation and concrete spectral data for magnetic Steklov problems on forms, bridging geometric analysis with spectral theory under magnetic perturbations.

Abstract

In this paper, we introduce the magnetic Steklov operator on differential forms and show that the underlying boundary value problem is well-posed. Moreover, we show that an analogue of the Diamagnetic Inequality does not always hold for this operator, and we present some spectral computations of magnetic Steklov operators for $2$-dimensional and $4$-dimensional balls in Euclidean space.

Magnetic Steklov operator on differential forms

TL;DR

This work extends the Steklov eigenproblem to differential forms by introducing the magnetic Steklov operator via -harmonic extensions and the magnetic differential . It establishes well-posedness of the associated boundary value problem and develops the spectral theory, including a min-max characterization and discreteness of the spectrum, while revealing that a Diamagnetic Inequality analogue can fail in this magnetic setting. The authors prove uniqueness and existence of solutions, analyze the operator’s coercivity through magnetic Gaffney inequalities, and provide explicit spectral computations in low-dimensional geometries (2- and 4-dimensional Euclidean balls) to illustrate how the magnetic potential influences the spectrum. They also present exact results on the magnetic Hodge Laplacian for spheres and , and discuss how boundary spectra relate to magnetic cohomology. Overall, the paper offers a robust theoretical foundation and concrete spectral data for magnetic Steklov problems on forms, bridging geometric analysis with spectral theory under magnetic perturbations.

Abstract

In this paper, we introduce the magnetic Steklov operator on differential forms and show that the underlying boundary value problem is well-posed. Moreover, we show that an analogue of the Diamagnetic Inequality does not always hold for this operator, and we present some spectral computations of magnetic Steklov operators for -dimensional and -dimensional balls in Euclidean space.

Paper Structure

This paper contains 12 sections, 22 theorems, 157 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Magnetic Steklov operator on differential forms
  3. Uniqueness of a solution
  4. Existence of a solution
  5. Proof of Theorem \ref{['thm:magneticboundprob']}
  6. Obstruction to the Diamagnetic Inequality
  7. Spectral properties of the magnetic Steklov operator on differential forms
  8. Proofs of Theorems \ref{['thm:eigtaylor']} and \ref{['thm:balldiamineqcounter']}
  9. Computation of the spectrum
  10. Case of the magnetic Hodge Laplacian on $1$- and $3$-spheres
  11. Computation of the spectrum of the magnetic Steklov operator on the Euclidean unit ball in $\mathbb{R}^2$
  12. Computation of the spectrum of the magnetic Steklov operator on the Euclidean unit ball in $\mathbb{R}^4$

Key Result

Theorem 1.1

Let $(M^m,g)$ be a compact Riemannian manifold of dimension $m$ with smooth boundary $\partial M$ and let $\eta$ be a smooth, real differential $1$-form on $M$. The boundary value problem has a unique smooth solution for each $\varphi\in \Omega^k(M,\mathbb{C})$ and $\psi\in \Omega^k(M,\mathbb{C})|_{\partial M}$.

Figures (2)

  • Figure 1: Eigenvalues of $1$-forms of the magnetic Hodge Laplacian $\Delta^{t \eta}$ on $\mathbb{S}^3$ as functions over $t \in [0,5]$ (left) and first eigenvalue $\lambda^{t\eta}_{1,1}(\mathbb{S}^3)$ as a function over $t \in [0,12]$ (right).
  • Figure 2: Eigenvalues of the magnetic Steklov operator $T^{[1],t\eta}$ on $1$-forms in $\mathbb{S}^1$.

Theorems & Definitions (44)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3: Magnetic Gaffney inequality, c.f. Sch:95 for the non-magnetic version
  • proof
  • Remark 2.4
  • ...and 34 more