Magnetic Steklov Operator

• Magnetic Steklov operator is a generalization of the Dirichlet-to-Neumann map on domains with magnetic potentials, relating boundary data to magnetic harmonic extensions.
• Its variational formulation reveals a self-adjoint, elliptic operator whose spectrum encodes rich geometric, magnetic, and topological information.
• Applications span electromagnetics, inverse spectral problems, and quantum models, with explicit analyses in settings like Aharonov–Bohm potentials and rotational magnetic fields.

The magnetic Steklov operator generalizes the classical Dirichlet-to-Neumann map (Steklov operator) to accommodate the presence of a magnetic vector potential on a Riemannian manifold or domain with boundary. This operator connects the theory of elliptic boundary problems, spectral geometry, and the analysis of quantum systems influenced by magnetic fields. Its paper encompasses diverse settings, including scalar functions, differential forms, and Maxwell theory for vector fields, with applications ranging from inverse spectral theory to electromagnetics and geometric analysis.

1. General Formulation and Paradigms

Let $(M, g)$ denote a smooth, compact Riemannian manifold (possibly with boundary $\partial M$). A magnetic potential $A\in \Omega^1(M;\mathbb R)$ (or, more generally, $\eta \in \Omega^1(M;\mathbb R)$ for forms) modifies the exterior derivative $d$ into the magnetic differential $d^A := d + iA$ and, for $k$-forms, $d^\eta := d + i\eta \wedge$. For functions, the magnetic Laplacian is given by: $$\Delta^A f = \delta^A d^A f = \Delta^M f - 2iA(\nabla f) + (i\delta^M A + |A|^2)f.$$ On $k$-forms, it is: $$\Delta^\eta = d^\eta\delta^\eta + \delta^\eta d^\eta.$$

The magnetic Steklov boundary value problem on $(M, g, A)$ seeks nontrivial $u$ (function or $k$-form) and real $\sigma$ such that: $$\begin{cases} \Delta^A u\, (= 0)\ \ \text{in } M; \ \partial_\nu^A u = \sigma u \text{ on } \partial M, \end{cases}$$ where $\partial_\nu^A u$ is the magnetic normal derivative, i.e.\ $\nabla_\nu u + iA(\nu)u$ for functions, or $-\nu\lrcorner (d^\eta \hat\omega)$ for $k$-forms.

The magnetic Steklov operator $T^A$ maps boundary data $f$ to the magnetic normal derivative of its unique (magnetic-)harmonic extension: $T^A(f) = -\nu\lrcorner d^A \hat{f}|_{\partial M}.$ It is a self-adjoint, order-one elliptic pseudodifferential operator with spectrum $$0 \leq \sigma_1^A \leq \sigma_2^A \leq \cdots \rightarrow \infty$$ (Chakradhar et al., 9 Oct 2024, Chakradhar et al., 10 Nov 2025).

For vector-valued settings, as in electromagnetics, the magnetic Steklov (or Stekloff) operator applies to solutions of Maxwell's equations, relating tangential traces of electric or magnetic fields on the boundary via a curl-curl operator (Halla, 2019, Lamberti et al., 2020, Ferraresso et al., 2022).

2. Variational Principles and Spectral Properties

The magnetic Steklov eigenvalues admit Rayleigh--Ritz and min-max formulations analogous to the classical case, but with magnetic covariants: $$\sigma_1^A = \inf_{0 \neq u \in C^\infty(M;\mathbb{C})} \frac{\int_M |d^A u|^2\, d\mathrm{vol}_M}{\int_{\partial M} |u|^2\, d\mathrm{vol}_{\partial M}},$$ and

$$\sigma_k^A = \min_{\substack{V \subset H^1_A \ \dim V = k}} \max_{0 \neq u \in V} \frac{\int_M |d^A u|^2}{\int_{\partial M}|u|^2}.$$

For magnetic Schrödinger operators with electric potential $q$, the appropriate operator is $$L_{g,A,q} u = -(\nabla + iA) \cdot (\nabla + iA) u + q u$$ with analogous variational characterization (Liu et al., 2021, Chakradhar et al., 10 Nov 2025).

Fundamental properties include:

• $T^A$ is self-adjoint and elliptic of order 1.
• The spectrum is real, discrete, unbounded above, with finite-multiplicity eigenvalues.
• The kernel of $T^{[k],\eta}$ (for forms) is the magnetic absolute cohomology $\mathcal H^{k,\eta}_{\mathrm{abs}}(M)$ (Chakradhar et al., 10 Nov 2025).

Gauge transformations relate Steklov spectra for potentials $A$ differing by an exact form; $A \in \mathfrak{B}_M$ (gauge-equivalent to zero) implies $T^A$ is unitarily equivalent to the classical Steklov operator (Chakradhar et al., 9 Oct 2024).

3. Explicit Model Problems and Computations

Aharonov–Bohm Potentials: Planar Disks

For a disk $D_R \subset \mathbb R^2$ with Aharonov-Bohm ($\alpha \notin \mathbb Z$) potential $A(x) = \alpha d\theta$, one has (Colbois et al., 2022):

• Eigenfunctions: $u(r,\theta) = r^{|k-\alpha|} e^{ik\theta}$, $k\in \mathbb Z$.
• Eigenvalues: $\sigma_k(D_R,A,\alpha) = |k - \alpha| / R$.

The first eigenvalue is strictly positive if and only if $\alpha\notin \mathbb Z$. Sharp isoperimetric inequalities generalizing those of Brock and Weinstock hold for the lowest eigenvalue under area or perimeter constraint, with extremality by the disk centered at the pole.

Balls and Spheres with Rotational Magnetic Potentials

• For the $n=2$ and $n=4$ balls equipped with rotational magnetic Killing fields ($A = r^2 d\theta$ in 2D, Hopf field in 4D), all eigenvalues can be computed in terms of generalized Laguerre polynomials (Chakradhar et al., 9 Oct 2024, Chakradhar et al., 10 Nov 2025).
• In $2$D:

$$\sigma_1(t\eta) = t\coth\left(\frac t2\right),\quad \sigma_k^\pm(t) = \frac{...}{...} \ \text{(see explicit formula)}$$

• In $4$D: Closed-form expressions involve higher-order Laguerre polynomials.

Electromagnetic/Maxwell Setting

The magnetic Steklov operator is extended to vector field problems, notably to the curl-curl operator for Maxwell equations in $\mathbb R^3$ cavities (Ferraresso et al., 2022, Lamberti et al., 2020, Halla, 2019):

• Weak form leads to a generalized eigenproblem for a pair of compact, self-adjoint operators in $L^2$ or $H(\text{curl})$.
• In the unit ball, explicit spectral branches for the Steklov eigenvalues $\lambda_l^{(1)}$, $\lambda_l^{(2)}$ are given in terms of spherical Bessel functions, with asymptotics $\lambda_l^{(i)} \sim -l$ as $l \to \infty$.

Strong- and Weak-Field Asymptotics

For exterior domains under uniform and Aharonov–Bohm-type flux, the lowest magnetic Steklov eigenvalue exhibits:

• Strong field $b\to\infty$: $$\Lambda_1 \sim \alpha_1 b^{1/2} + \alpha_2 + \alpha_3(\Phi) b^{-1/2}$$, with flux effects encoded in $\alpha_3(\Phi)$ (Helffer et al., 25 Aug 2025).
• Weak field $b\to 0$: leading order is linear in the (fractional) flux $\nu$, $\Lambda_1 \sim |\nu|b^{1/2}$ for nontrivial flux.

4. Geometric Inequalities and Spectral Invariants

Significant isoperimetric inequalities are satisfied by the lowest magnetic Steklov eigenvalue, generalizing classical results:

• If $|\Omega|=|D_R|$, then $$\sigma_1(\Omega,A,\alpha) \leq \sigma_1(D_R,A,\alpha)$$, with equality only for disks (Colbois et al., 2022).
• For simply connected $\Omega$ of fixed perimeter, the same bound holds.

Spectral invariants derived from the heat-trace expansion of the magnetic Steklov operator encode rich geometric and physical information (Liu et al., 2021, Helffer et al., 11 Jul 2024):

• The leading coefficients $a_0, a_1, a_2, a_3$ in the short-time asymptotic expansion of $\operatorname{Tr} e^{-t T^A}$ recover the boundary volume, mean curvature, and curvature invariants, with magnetic corrections first entering at $a_2$ and in the logarithmic term $b_1$.
• The tangential component of the magnetic field enters explicitly into these invariants (notably $a_2$ and $b_1$), confirming sensitivity of the spectrum to magnetic flux and geometry.

5. Beyond Functions: Differential Forms and Maxwell Steklov Theory

The magnetic Steklov framework extends naturally to differential forms $k\geq1$. For forms, the operator $T^{[k],\eta}$ exhibits new phenomena not present in the scalar case (Chakradhar et al., 10 Nov 2025):

• The eigenvalue problem is well-posed, and the spectrum is discrete and non-negative, but the diamagnetic inequality fails: the first Steklov eigenvalue can decrease under (certain) magnetic perturbations, contrary to the functional case.
• Explicit spectral computations for balls and spheres show that the lowest nonzero Steklov form eigenvalues can be strictly below their non-magnetic analogues for small $t>0$, and approach the magnetic flux $t$ for large $t$.

In the electromagnetics context, the vector Steklov problem (based on Maxwell's equations and suitable impedance/logical boundary conditions) is central both in mathematical analysis and in applied inverse scattering (Halla, 2019, Lamberti et al., 2020):

• The associated Steklov spectra are discrete (or have essential point-spectrum at zero for certain original problems), and their eigenfunctions form bases for trace or energy spaces.
• Spectral branches can be characterized by block operator and Schur complement analysis.

6. Applications and Inverse Spectral Geometry

The magnetic Steklov operator carries detailed geometric, topological, and physical information:

• Its spectral asymptotics recover the lengths of boundary components, magnetic holonomy/flux along boundaries, and integrated curvature and field parameters (Cekić et al., 11 Oct 2024, Liu et al., 2021).
• For planar or annular domains, sharp upper bounds for normalized eigenvalues are achieved for "maximizing geometries" (e.g., critical catenoids, free-boundary minimal surfaces in balls), often associated with physical models in geometry and physics (Provenzano et al., 2023).
• Inverse spectral results show that, under non-degeneracy, the magnetic Steklov spectrum may determine both geometric and magnetic data; however, explicit counterexamples exist: combinatorial phenomena can cause the spectrum to "not see" the number of boundary components or distinguish between certain magnetic flux configurations (Cekić et al., 11 Oct 2024).

These properties render the magnetic Steklov spectrum a sensitive tool for inverse problems and a testbed for quantum, geometric, and topological phenomena in spectral theory.

7. Outstanding Phenomena, Limitations, and Open Problems

• Obstruction to Diamagnetic Inequality for Forms: For differential forms, monotonicity of the lowest Steklov eigenvalue with respect to magnetic field strength fails (explicit counterexamples in even-dimensional balls with rotational magnetic potential) (Chakradhar et al., 10 Nov 2025).
• Spectral Non-Uniqueness and Covering Systems: Inverse spectral problems in the presence of a magnetic field can exhibit non-uniqueness—distinct boundary data and fluxes producing identical asymptotic spectra—due to arithmetic covering system phenomena (Cekić et al., 11 Oct 2024).
• Nonlocal and Logarithmic Corrections in Spectral Invariants: Higher-order and logarithmic terms in the heat trace detect more delicate, often nonlocal, aspects of the magnetic field (e.g., boundary-normal components), marking a clear departure from the non-magnetic setting (Helffer et al., 11 Jul 2024).
• Computational and Variational Challenges for Maxwell Problems: For electromagnetic (Maxwell) Steklov problems, variational formulations are well-understood, but numerical approximations require subtle space decompositions and Galerkin strategies with mesh-commuting boundary traces for spectral convergence (Halla, 2019).

The comprehensive paper of the magnetic Steklov operator thus interlaces spectral geometry, PDEs, and mathematical physics, with rich interactions between gauge theory, global analysis, and inverse problems.