Eigenmode Convergence Radii

• Eigenmode convergence radii are defined as measures quantifying approximation errors from domain truncations and finite element discretizations in spectral problems.
• They employ exponential error bounds in PML methods and algebraic rates in periodic finite element settings to guide optimal choices of truncation radii and mesh sizes.
• These principles are crucial for applications in computational quantum mechanics and wave propagation, leveraging operator theory and fractional Sobolev norms to ensure spectral accuracy.

The concept of eigenmode convergence radii centers on rigorous characterizations of eigenvalue and eigenfunction approximation errors incurred by truncations (such as domain radii or mesh boundaries) and discretizations (e.g., finite element mesh size) in spectral problems. The convergence radius quantifies, for a given truncation or embedding of the problem domain, the corresponding rate and accuracy with which discrete eigenmodes approach those of the continuous operator. Its role is fundamental in computational spectral theory, particularly for partial differential operators such as Dirac-type and resonance operators, where both geometric truncation and numerical discretization perturb the spectrum.

1. Frameworks for Eigenmode Convergence Radii

Eigenmode convergence radii arise in the spectral approximation of differential operators, frequently in two principal contexts:

• Periodic domains with finite element discretization: For operators like the Hodge–Dirac operator on a flat torus, eigenmode approximations are studied within subspaces corresponding to polynomial differential forms, parameterized by mesh radius $h$ and polynomial degree $p$ (Christiansen, 2015).
• Exterior resonance problems with complex scaling (Perfectly Matched Layers, PML): Eigenvalue and eigenfunction errors are characterized with respect to the truncation radius $R$ of the computational domain and scaling profiles α̃(r), for scalar resonance operators (Halla, 2020).

Both contexts deploy Galerkin-type projections and operator-theoretic compactness arguments. In periodic discretizations, the convergence radius pertains almost entirely to the mesh parameter $h$, whereas in exterior domain problems, it is governed by the truncation radius $R$ and the absorption profile's asymptotic behavior.

2. Error Bounds and Convergence Rates

Explicit quantitative bounds are established for the rate at which discrete eigenmodes (pairs of eigenvalues and normalized eigenfunctions) converge to their continuous analogues. In finite element settings over periodic domains, the rates are governed by fractional Sobolev norms and mesh size $h$:

• For a simple eigenpair $(\lambda, u)$ with $u \in H^{p+1}\Lambda^*(S)$, the following error estimates are proven (Christiansen, 2015):

$$\|u - u_h\|_{s_1, s_2} \leq C h^{p + s_1} |u|_{H^{p+1}}, \quad |\lambda - \lambda_h| \leq C h^{2(p + s_1)} |u|_{H^{p+1}}^2,$$

where $s_1 \in (0, 1/2)$, $p$0, and $p$1.

For radial domain truncations with PML, eigenvalue and eigenfunction errors depend exponentially on the truncation radius $p$2:

• For a simple eigenvalue $p$3 and corresponding discrete eigenvalue $p$4 (Halla, 2020):

$p$5

where $p$6, and $p$7.

Eigenfunction errors follow analogous exponential bounds:

$p$8

3. Minimal Convergence Radii and Accuracy Thresholds

A central quantitative result is the characterization of minimal truncation radii required to achieve a target eigenmode approximation error. Setting the desired accuracy $p$9, for exponential error bounds as in PML methods, the minimal radius $R$0 satisfies:

$R$1

where $R$2 is determined by the profile's asymptotic absorption plateau.

For simultaneous truncation and discretization, the total error is additive in $R$3 and mesh size $R$4:

$R$5

with respective minimal choices $R$6 and $R$7 (Halla, 2020).

4. Dependence on Profile and Discretization Parameters

Convergence radii and error rates display dependence on both the geometric truncation and the functional form of the scaling profile α̃(r) employed in exterior domain truncations:

• Affine (linear) profile: $R$8, with plateau $R$9.
• Smooth bounded profile: α̃(r) raising to a constant, converges to similar exponential rates determined by the plateau value.
• Power-law profile: Asymptotic limit mirrors bounded profiles.
• Unbounded (exact PML): Truncationless profiles eliminate the exponential tail error, reducing convergence radii for eigenmodes to mesh parameter $h$0; only the discretization error (algebraic in $h$1) remains (Halla, 2020).

This underscores that, for bounded profiles, the convergence rate in $h$2 is strictly exponential, dominated by the plateau value. For unbounded ("exact") profiles, truncation error vanishes and only discretization governs convergence.

5. Operator-Theoretic and Analytical Foundations

Underlying all convergence radius analyses are abstract operator-theoretic frameworks involving:

• Fredholm holomorphic perturbation theory: The error in eigenvalue approximation for simple poles scales as the square of the approximation error in the Galerkin subspace, i.e., $h$3, and $h$4.
• Gap estimates and discrete compactness: Convergence of subspace gaps (e.g., $h$5) guarantees discrete compactness and convergence of unperturbed eigenmodes (Christiansen, 2015).
• Weak T-coercivity and T-compatible approximations: Ensures stability of operator approximations under truncation and discretization, central to the radial complex scaling framework (Halla, 2020).

Proof techniques often involve mollified interpolators, smoothed projections, discrete commutator and interpolation arguments, discrete Gagliardo–Nirenberg inequalities, and Aubin–Nitsche duality.

6. Fractional Sobolev Spaces and Norms

A key structural choice in eigenmode error analysis is the utilization of fractional Sobolev spaces. For $h$6, $h$7 is defined by the Slobodetskij seminorm:

$h$8

with norm $h$9 (Christiansen, 2015).

Error bounds are controlled via graph-type norms:

$R$0

where $R$1 is chosen close to $R$2 to approach optimal eigenmode convergence rates:

• Eigenvector-error $R$3
• Eigenvalue-error $R$4

7. Implications and Applications

Eigenmode convergence radii provide practitioners with precise, quantitative guidelines for selecting truncation radii and discretization parameters in computational spectral problems. In domains employing PML-type absorbing layers, the convergence-in-radius is exponentially governed by the asymptotic absorption plateau, with minimal layer thickness scaling logarithmically with target accuracy. For finite element spectral computations, the mesh size h directly dictates algebraic convergence orders determined by polynomial degree p and fractional smoothness s. Profiles with unbounded absorption can, in principle, eliminate truncation error, relegating convergence control to discretization schemes. These results are essential for eigenmode approximation in computational quantum mechanics, photonics, and wave propagation analysis, as well as for the design of numerical algorithms with guaranteed spectral accuracy (Christiansen, 2015, Halla, 2020).