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A note on the magnetic Steklov operator on functions

Tirumala Chakradhar, Katie Gittins, Georges Habib, Norbert Peyerimhoff

TL;DR

This work develops a comprehensive spectral theory for the magnetic Steklov problem on compact manifolds with boundary. By relating the magnetic Steklov operator $T^ta$ to the magnetic Laplacian on the boundary and examining gauge equivalence, it establishes gauge-characterizations, a Cheeger-type lower bound, and Colbois–Savo-type upper bounds for the first eigenvalue, along with full-spectrum computations on the Euclidean disk and 4-ball. It also proves a spectral comparison between $_k^ta(M)$ and the square root of the boundary Laplacian spectrum via magnetic Pohozaev and Reilly identities, providing quantitative bounds under tubular-geometry assumptions. The results illuminate how magnetic potentials shape Steklov spectra, yielding explicit formulas, asymptotics, and a unified comparison framework for magnetic boundary problems.

Abstract

We consider the magnetic Steklov eigenvalue problem on compact Riemannian manifolds with boundary for generic magnetic potentials and establish various results concerning the spectrum. We provide equivalent characterizations of magnetic Steklov operators which are unitarily equivalent to the classical Steklov operator and study bounds for the smallest eigenvalue. We prove a Cheeger-Jammes type lower bound for the first eigenvalue by introducing magnetic Cheeger constants. We also obtain an analogue of an upper bound for the first magnetic Neumann eigenvalue due to Colbois, El Soufi, Ilias and Savo. In addition, we compute the full spectrum in the case of the Euclidean $2$-ball and $4$-ball for a particular choice of magnetic potential given by Killing vector fields, and discuss the behavior. Finally, we establish a comparison result for the magnetic Steklov operator associated with the manifold and the square root of the magnetic Laplacian on the boundary, which generalizes the uniform geometric upper bounds for the difference of the corresponding eigenvalues in the non-magnetic case due to Colbois, Girouard and Hassannezhad.

A note on the magnetic Steklov operator on functions

TL;DR

This work develops a comprehensive spectral theory for the magnetic Steklov problem on compact manifolds with boundary. By relating the magnetic Steklov operator to the magnetic Laplacian on the boundary and examining gauge equivalence, it establishes gauge-characterizations, a Cheeger-type lower bound, and Colbois–Savo-type upper bounds for the first eigenvalue, along with full-spectrum computations on the Euclidean disk and 4-ball. It also proves a spectral comparison between and the square root of the boundary Laplacian spectrum via magnetic Pohozaev and Reilly identities, providing quantitative bounds under tubular-geometry assumptions. The results illuminate how magnetic potentials shape Steklov spectra, yielding explicit formulas, asymptotics, and a unified comparison framework for magnetic boundary problems.

Abstract

We consider the magnetic Steklov eigenvalue problem on compact Riemannian manifolds with boundary for generic magnetic potentials and establish various results concerning the spectrum. We provide equivalent characterizations of magnetic Steklov operators which are unitarily equivalent to the classical Steklov operator and study bounds for the smallest eigenvalue. We prove a Cheeger-Jammes type lower bound for the first eigenvalue by introducing magnetic Cheeger constants. We also obtain an analogue of an upper bound for the first magnetic Neumann eigenvalue due to Colbois, El Soufi, Ilias and Savo. In addition, we compute the full spectrum in the case of the Euclidean -ball and -ball for a particular choice of magnetic potential given by Killing vector fields, and discuss the behavior. Finally, we establish a comparison result for the magnetic Steklov operator associated with the manifold and the square root of the magnetic Laplacian on the boundary, which generalizes the uniform geometric upper bounds for the difference of the corresponding eigenvalues in the non-magnetic case due to Colbois, Girouard and Hassannezhad.

Paper Structure

This paper contains 14 sections, 17 theorems, 139 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Results
  3. Characterizations of magnetic potentials which can be gauged away
  4. Lower and upper bounds for the smallest magnetic Steklov eigenvalue
  5. Full spectrum considerations for the magnetic Steklov operator
  6. Proof of the Maximum Principle and Theorem \ref{['thm:shikegawa']}
  7. Proofs of the lower and upper magnetic Steklov eigenvalue bounds
  8. Proof of Theorem 2.5 (lower bound)
  9. Examples of magnetic frustration constants
  10. Proof of Theorem 2.7 (upper bound)
  11. Examples: Computation of the full magnetic Steklov spectrum
  12. Magnetic Steklov spectrum for the 2-dimensional Euclidean ball
  13. Magnetic Steklov spectrum for the 4-dimensional Euclidean ball
  14. Comparison of magnetic Steklov and magnetic Laplacian eigenvalues

Key Result

Proposition 1.1

Let $(M,g)$ be a compact Riemannian manifold with smooth boundary $\partial M$, $\eta \in \Omega^1(M)$ and $f \in C^\infty(M)$ satisfying $\Delta^\eta f = 0$. Then the function $|f| \ge 0$ assumes its maximum at a boundary point in $\partial M$.

Figures (2)

  • Figure 1: Eigenvalues of the magnetic Steklov operator $T^{t \eta}$ on $\partial \mathbb{B}^2$ as functions in $t$ (left) and of the magnetic Laplacian $\Delta^{t \eta_0}$ on $\mathbb{S}^1$ as functions in $t$ (right).
  • Figure 2: Eigenvalues of the magnetic Steklov operator $T^{t \eta}$ on $\partial \mathbb{B}^4$ as functions in $t$.

Theorems & Definitions (40)

  • Proposition 1.1: magnetic Maximum Principle
  • Theorem 2.1: Equivalent characterizations of functions in $\mathfrak{B}_M$
  • Remark 2.2
  • Remark 2.3
  • Definition 2.4
  • Theorem 2.5: Cheeger type inequality
  • Remark 2.6
  • Theorem 2.7: cf. CSIS-21 for the Neumann eigenvalue version
  • Example 2.8: Magnetic spectra for the disk
  • Theorem 2.9: Magnetic version of CGH:20
  • ...and 30 more