Spectral Computations of Magnetic Steklov Operators

• Magnetic Steklov operators are defined on compact Riemannian manifolds with boundary, linking Dirichlet-to-Neumann maps to magnetic and electric potentials.
• Spectral asymptotics and heat trace invariants elucidate how magnetic holonomy, curvature, and flux affect eigenvalue distributions and boundary invariants.
• Practical computation involves finite element methods and pseudodifferential techniques to tackle inverse spectral problems and electromagnetic applications.

Spectral computations of magnetic Steklov operators concern the study of eigenvalue problems associated with Dirichlet-to-Neumann (DtN) maps in the presence of magnetic fields, primarily on compact Riemannian manifolds with boundary. These operators encode how magnetic and geometric data on the manifold and its boundary influence boundary spectral invariants, and have deep connections with inverse spectral geometry, boundary value problems for Maxwell and Schrödinger operators, and mathematical physics.

1. Definition and Framework for Magnetic Steklov Operators

Let $(M,g)$ be a smooth, compact Riemannian manifold of dimension $n$ with non-empty boundary $\partial M$. Denote by $A \in \Omega^1(M; \mathbb{C})$ a (typically real or purely imaginary) magnetic potential $1$-form and $q \in C^{\infty}(M;\mathbb{C})$ an electric potential. The magnetic (or magnetic Schrödinger) operator acting on scalar functions is

$L_{g,A,q} = d_A^* d_A + q,$

where $d_A = d + A$, and $d_A^*$ is the formal adjoint. The magnetic Steklov problem is formulated as: given boundary data $f \in H^{1/2}(\partial M)$, solve

$$L_{g,A,q}\,u = 0\ \text{in } M,\quad u|_{\partial M} = f,$$

and define the magnetic Dirichlet-to-Neumann (Steklov) operator

$\Lambda_{g,A,q}f = (d_A\,u)(\nu)|_{\partial M},$

where $\nu$ is the outward unit normal to $\partial M$.

For differential $k$-forms, the setup generalizes by introducing the magnetic exterior derivative $d^A = d + i\,A \wedge$, its adjoint $\delta^A = \delta - i\,A \lrcorner$, and the magnetic Hodge Laplacian $\Delta^A = d^A \delta^A + \delta^A d^A$. The magnetic Steklov operator on forms maps boundary data to the magnetic conormal derivative (see (Chakradhar et al., 10 Nov 2025, Cekić et al., 11 Oct 2024)).

In electromagnetics, the analog is given through Steklov-type problems for time-harmonic Maxwell equations, where magnetic boundary conditions prescribe the tangential trace of the curl of the field, leading to a magnetic Steklov eigenproblem in appropriate vector Sobolev spaces (see (Ferraresso et al., 2022, Lamberti et al., 2020)).

2. Spectral Asymptotics and Symbolic Calculus

Spectral computations in the magnetic Steklov setting focus on precise asymptotic expansions of the eigenvalues $\{\sigma_k\}$ of $\Lambda_{g,A,q}$ as $k \to \infty$. On surfaces ($n=2$), with $\partial M = \bigsqcup_{j=1}^m N_j$ (disjoint union of boundary components, each of length $\ell_j$), the spectrum divides into $m$ ladders $S_j$, enumerated by $n \in \mathbb{Z}$: $$\lambda_n^{(j)} \sim \sum_{k=0}^\infty b_k^{(j)}(\ell_j, A, q)\, n^{1-k},\quad n \to +\infty.$$ The leading coefficients are: $$b_0^{(j)}(\pm1) = \frac{2\pi}{\ell_j}, \quad b_1^{(j)}(\pm1) = \pm\frac{2\pi}{\ell_j} \alpha_j, \quad b_2^{(j)}(\pm1) = \pm \frac{i}{4\pi}\oint_{N_j}\iota_\nu dA + \frac{1}{4\pi}\oint_{N_j} q,$$ where $$\alpha_j := p_j + (2\pi i)^{-1}\int_{N_j} A \in [0,1)$$, with $p_j \in \mathbb{Z}$ chosen accordingly.

The derivation proceeds by reduction to a pseudodifferential operator of order $1$ on $\partial M$, pseudodifferential normal form conjugation (Birkhoff normal form for each boundary circle), and application of classical results for spectral asymptotics of circle operators (Rozenbljum–Agranovich theorem).

These expansions show explicitly how magnetic holonomy (via $\alpha_j$) and magnetic flux through the boundary affect the Steklov spectrum, not present in the non-magnetic problem (Cekić et al., 11 Oct 2024, Liu et al., 2021).

3. Heat Trace Asymptotics and Spectral Invariants

The trace of the heat semigroup $\operatorname{Tr}(e^{-t\Lambda_A})$ admits an asymptotic expansion as $t\to0^+$,

$$\operatorname{Tr}(e^{-t\Lambda_A}) \sim \sum_{j=0}^{\infty} a_j\, t^{-(n-1-j)}.$$

The coefficients $a_j$ are computable in terms of curvature and the magnetic/electric potentials. The first four are given by explicit integrals involving geometrical data of the boundary (mean curvature $H$, scalar curvature $R_{\partial M}$), as well as the potentials $q$ and (from $a_4$ onward) derivatives of $A$. For $n \geq 3$, for example,

$$\begin{aligned} a_0 & = C(n) \int_{\partial M} 1\, dS, \ a_1 & = C'(n)\int_{\partial M} H\, dS, \ a_2 & = C''(n) \int_{\partial M} [3(n+1)(n-2)H^2 - (n+1)(n-4)R_{\partial M} - 12q + 12k^2]\, dS, \end{aligned}$$

with explicit constants, and similar formulae for $a_3$ (Liu et al., 2021). Contributions from the magnetic potential $A$ or its curvature $dA$ first enter through $a_4$ and higher.

Algorithmically, these coefficients are determined by expansion of the boundary operator near $\partial M$ in local coordinates, symbol calculus, and recursion using the composition and inversion formulas for pseudodifferential operator symbols.

The spectral invariants $\{a_j\}$ yield, in principle, geometric, topological, and physical information about $M$ and $(A,q)$ from the spectrum, providing tools for inverse spectral problems.

4. Explicit Computations and Model Geometries

Several explicit computations illustrate the dependence of the magnetic Steklov spectrum on magnetic and geometric data:

• Cylinders and Annuli: For $M = S^1 \times [-L, L]$ with $A$ constant, the spectrum consists of two families (ladders) of eigenvalues:

$$\left\{|k - ic|\tanh(|k-ic| L),\ |k-ic|\coth(|k-ic| L):\ k \in \mathbb{Z}\right\},$$

with large-$L$ asymptotics featuring exponential decay terms (Cekić et al., 11 Oct 2024).

• Euclidean Balls: On $B^2$ and $B^4$ in $\mathbb{R}^2, \mathbb{R}^4$, with constant Killing magnetic potentials ($A = t r^2 d\theta$ or Hopf one-forms), the spectrum can be given in terms of special functions (confluent hypergeometric or Laguerre polynomials), with magnetic field strength ($t$) controlling spectral splitting and degeneracy (Chakradhar et al., 9 Oct 2024, Chakradhar et al., 10 Nov 2025).
• Differential Forms: The magnetic Steklov spectrum on $k$-forms displays behavior distinct from the scalar case. Explicit formulas for eigenvalues and eigenforms exist for balls in low dimensions, showing, for instance, the failure of diamagnetic-type inequalities for Steklov eigenvalues on forms (Chakradhar et al., 10 Nov 2025).
• Electromagnetic (Maxwell) Problems: For time-harmonic Maxwell’s equations on the ball, the spectrum is determined by vector spherical harmonics, with eigenvalues given by explicit algebraic expressions involving Bessel functions and their derivatives (Ferraresso et al., 2022, Lamberti et al., 2020).

5. Inverse Spectral Theory and Uniqueness Phenomena

The parametric dependence of Steklov spectra on geometric and magnetic data has significant implications for inverse spectral problems. For instance, on oriented surfaces:

• When $\partial M$ is connected, the full spectrum uniquely determines the boundary length, magnetic holonomy (parallel transport phase $e^{\pm \int A}$), modulus of the integral of the normal part of $dA$, and $\int_{\partial M} q$.
• If boundary components are of unequal lengths and holonomies are generic, the spectrum (modulo exceptional small shifts and deletions) recovers the number $m$ of components, their lengths $\{\ell_j\}$, and their parallel transport phases.
• Nonetheless, there exist non-isometric surfaces with the same polynomial part of the spectrum, particularly when certain resonance conditions ($\alpha_j=1/4$ or $3/4$) are met or $A$ is constant near $\partial M$, so that multiplicities and subtle arithmetic progressions of eigenvalues can mask geometric distinctions (Cekić et al., 11 Oct 2024).

Such results underscore both the power and the intrinsic limitations of spectral geometry in the magnetic case, contrasting with the (sometimes more rigid) non-magnetic Steklov problem.

6. Computational Methods and Practical Implementation

The computational pipeline for spectral approximation of magnetic Steklov operators is well-established:

1. Mesh Generation: Discretize $\Omega$ (the domain) with a tetrahedral mesh (e.g., via Gmsh).
2. Finite Element Spaces: Use Nédélec edge-element spaces to represent (vector-)fields and enforce tangential conformity. For scalar or form problems, appropriate $H^1$ or mixed finite elements are chosen.
3. Matrix Assembly: Assemble stiffness ($\mathbf{K}$) and boundary mass ($\mathbf{M}$) matrices, encoding the variational forms for the PDE and boundary map.
4. Eigenproblem: Formulate the generalized Hermitian-definite eigenproblem $\mathbf{K} u_h = \lambda_h \mathbf{M} u_h$. For large problems, employ ARPACK/SLEPc solvers and H(curl) block-diagonal preconditioners.
5. Post-processing: Normalize and visualize computed eigenfunctions. Validate spectral approximations through convergence studies and comparison with known analytic results (Lamberti et al., 2020).

Accuracy is limited by mesh quality and polynomial order; high-order Nédélec elements and adaptive refinement enhance performance. Practical implementations must handle geometric alignment at the boundary, preconditioning, and orthonormalization with respect to mass matrices.

7. Geometric and Physical Significance, Open Problems

Magnetic Steklov operator spectra encode, through their asymptotics and heat trace invariants, a rich array of geometric (boundary area, curvature), topological (number of boundary components), and physical (magnetic holonomy/flux, electric potentials) information. Their study has illuminated new rigidity and non-uniqueness phenomena, as well as connections to quantum mechanics, inverse boundary problems, and electromagnetic cavity design.

Notable open directions include precise characterization of non-uniqueness in the magnetic inverse spectrum, asymptotic questions for extreme magnetic fields (e.g., logarithmic vs. power-law growth of $\sigma_1^{tA}$ as $t \to \infty$), and extension to more general gauge group settings and non-orientable manifolds. The failure of classical inequalities, such as the diamagnetic bound for forms, signals deeper geometric interplay between topology, gauge, and analysis (Chakradhar et al., 10 Nov 2025, Cekić et al., 11 Oct 2024).