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Computing Spectral Properties of Differential Operators on Surfaces

Gustav Conradie, Matthew J. Colbrook, Daniel Fortunato

arXiv:2609.37764v1math.NAcs.NA

TL;DR

Surface spectra can include continuous components, while discretization may produce spurious or missing eigenvalues or fail to capture continuous spectra. SurfSpec applies resolvent-based methods through surface PDE solves to compute spectra and spectral measures with error control; its examples demonstrate accuracy and practicality, and residuals help identify spurious eigenvalues.

  • Problem

    Discretizing both an infinite-dimensional operator and its surface can produce spurious or missing eigenvalues and fail to capture continuous spectra, despite the importance of surface spectra in applications.

    Source: §1 · pp. 1–2

  • Method

    SurfSpec extends solve-then-discretize methods to surfaces, using resolvent contour integrals and rational-kernel convolutions with high-order surface PDE solves.

    Source: §1 · p. 2

  • Results

    Numerical examples demonstrate accuracy and practicality, and residuals identify a spurious Laplace–Beltrami eigenvalue that disappears after slight discretization improvements.

    Source: §1.1 · p. 2, §3.3.1 · p. 9

  • Takeaways & Limitations

    SurfSpec provides error-controlled computation of high-frequency eigenmodes, spectral measures, and nonlinear eigenvalue problems on surfaces with convergence guarantees.

    Source: §1 · p. 2

    For the Poisson-kernel approximation, small ϵ requires large discretizations and its error is O(ϵ log(1/ϵ)), limiting practicality.

    Source: §4 · p. 16

Abstract

from arXiv · show

Computing spectra of differential operators on surfaces is crucial in many applications, ranging from medical image analysis to nanoscale quantum structures. However, discretization of the infinite-dimensional operator and the underlying surface can lead to inaccurate and misleading results. This paper introduces the SurfSpec suite of algorithms to compute eigenvalues and spectral measures of operators on surfaces with error control and convergence guarantees. We combine methods based on complex contour integration and residuals to compute discrete spectra and high-frequency eigenmodes, including of nonlinear eigenvalue problems. We then compute convolutions of spectral measures with rational kernels to deal with continuous spectra. In both cases, the key tool is the operator resolvent applied to functions, which is computed using high-order methods that solve surface PDEs. We illustrate our algorithms on a variety of differential operators and surfaces, including a triceratops.

1. Introduction.

Surface spectra may include continuous components, and discretization can produce spurious or missing eigenvalues or fail to capture continuous spectra. SurfSpec delays discretization by applying resolvent-based methods to functions, with error control and convergence guarantees for spectral computations on surfaces.

1. Introduction.

  • Discretization can introduce spurious or missing eigenvalues and fail to capture continuous spectra, problems that may persist as discretizations are refined.Surface spectral problems also require discretizing the underlying surface.Source: §1 · p. 2
  • The paper extends a solve-then-discretize approach to surfaces, discretizing functions acted on by the operator rather than the operator itself.This approach is motivated by the greater tractability of controlling function-discretization errors.Source: §1 · p. 2
  • Resolvent contour integrals project onto eigenspaces, while convolution with high-order rational kernels computes spectral measures.Applying the resolvent to functions allows adaptive discretization by shift and a posteriori error bounds from the resolvent norm.Source: §1 · p. 2
  • The paper introduces a high-order fast direct surface PDE solver to compute the resolvent action by solving shifted linear systems.The introduction identifies this as the second key tool.Source: §1 · p. 2
  • SurfSpec combines routines for surface-operator spectra, spectral measures, and nonlinear eigenvalue problems with error control and convergence guarantees.The guarantees combine results in with convergence of the high-order solver.Source: §1 · p. 2
  • The introduction previews a surface PDE solver, contour-integral eigenvalue methods including nonlinear problems, and resolvent-based spectral-measure methods.Each section includes numerical examples demonstrating accuracy and practicality.Source: §1.1 · p. 2

2. Solving surface PDEs.

The surface PDE solver represents geometry and functions locally on an unstructured quadrilateral mesh, then assembles discrete surface operators using spectral collocation. Surface derivatives and elliptic PDEs are expressed using the surface metric and tangential derivatives.

2. Solving surface PDEs.

  • The surface metric is formed from local parametrization derivatives, and its inverse defines tangential derivatives used for surface differential operators.The surface gradient, divergence, and Laplacian are examples of quantities defined from these operators.Source: §2 · p. 3
  • The solver treats general second-order variable-coefficient elliptic PDEs on surfaces, with boundary conditions available for surfaces with boundary.The text notes that the methods can be extended to higher-order PDEs.Source: §2 · p. 3
  • A fast direct solver uses spectral collocation on unstructured quadrilateral meshes, sampling each element at tensor-product second-kind Chebyshev nodes.Each element has (p + 1)^2 nodes; refinement can increase the element count or local order.Source: §2 · pp. 3–4, Figure 1 · p. 4
  • The method splits the PDE solution into homogeneous and particular components and discretizes the two problems separately on each element.This produces 2N systems of size (p+1)^2 × (p+1)^2 before the solution is computed over the whole surface.Source: §2 · p. 4

1. Local factorization.

Local factorization constructs element-level solution operators and Dirichlet-to-Neumann maps, alongside discrete particular solutions and their fluxes.

1. Local factorization.

  • On each mesh element, local solves map Dirichlet boundary data to homogeneous solutions and provide the corresponding Neumann data through Dirichlet-to-Neumann maps.The local stage also computes discrete particular solutions and their associated fluxes.Source: §1 · p. 4

2. Global factorization.

Global factorization merges adjacent elements upward, enforcing continuity of the solution and its binormal derivative across interfaces.

2. Global factorization.

  • An upward pass merges adjacent elements into parent solution operators, particular solutions, Dirichlet-to-Neumann maps, and particular fluxes.Each merge enforces continuity of the solution and its binormal derivative at the interface.Source: §2 · p. 4

3. Solve.

SurfFEAST computes discrete surface spectra through resolvent contour integrals, with shifted PDE solves and iterative reduction to a finite-dimensional eigenproblem. Its surface solver reconstructs solutions recursively and has stated operation and memory costs.

3. Solve.

  • The downward pass recursively reconstructs child-element interface solutions, then combines local solution operators and particular solutions to recover the full solution.Source: §3 · p. 4
  • For fixed p, local factorization costs O(N), global factorization O(N^3/2), and solving O(N log N); memory also scales as O(N log N).Reusing factorizations lets the solver handle different boundary data without recomputing them.Source: §3 · pp. 4–5
  • At each contour node, SurfFEAST solves a shifted linear system, then uses a truncated SVD and Rayleigh–Ritz projection to update eigenvalue and eigenfunction approximations.The iteration stops when the residual satisfies the specified tolerance.Source: §3 · p. 5
  • The open-source MATLAB package surfacefun implements the surface PDE solver and supplies high-order tools for function computations on surfaces.Source: §3 · p. 5
  • Contour integrals involving the resolvent provide the core strategy for computing discrete spectra of surface differential operators, with error control for linear and nonlinear problems.The section frames the method around evaluating these integrals.Source: §3 · p. 5

3. Differential eigenvalue problems.

This section develops and demonstrates contour-based methods for surface differential eigenvalue problems, using residuals, pseudospectral analysis, and convergence checks to assess computed spectra. It covers Laplace–Beltrami, Schrödinger, and nonlinear eigenvalue problems, including a delayed diffusion example.

3.1. SurfFEAST for linear differential eigenvalue problems.

  • SurfFEAST extends FEAST [70] to surface differential operators, using a Riesz projector to recover eigenfunctions for eigenvalues inside a contour.It applies the projector to oversampled band-limited random test functions, approximates its contour integral using shifted surface PDE solves, then uses Rayleigh–Ritz to reduce the problem to a finite-dimensional eigenproblem without directly discretizing the operator. Residual-based iteration improves accuracy.Source: §3 · p. 5, §3.1 · pp. 5, 7

3.2. Error control.

  • For self-adjoint operators, residual bounds support adaptive shifted-system solves and help reject spurious eigenvalues; pseudospectral inclusions provide verification for nonnormal operators.For nonlinear eigenproblems, residuals do not directly bound distance to the spectrum, so pseudospectral analysis is used instead.Source: §3.2 · p. 7, §3.4 · p. 14

3.3. Example: Laplace–Beltrami and Schrödinger operators.

  • For Laplace–Beltrami operators, Weyl’s law gives eigenvalue-count asymptotics and helps estimate contour rank; the examples also include surface Schrödinger operators with harmonic-oscillator potential.The experiments use Dirichlet conditions on open surfaces and compare counts with the leading Weyl term and a conjectured boundary correction.Source: §3.2 · p. 8, §3.3 · p. 8

3.3.1. A worked example: a triceratops.

  • On the triceratops, 1 Laplace–Beltrami and 2 Schrödinger eigenvalues among the first 1000 were spurious, but residuals identified them and parameter adjustments removed them.With elliptical contours, 15 shifted systems per contour were used; quadrature convergence was rapid, residuals decreased with surface refinement, and higher-frequency modes were harder to resolve.Source: §3.3.1 · pp. 9–10, Figure 3 · p. 10

3.3.2. Further examples.

  • Across six surfaces, computed eigenvalue counts agree strongly with Weyl approximations, with relative error decreasing at higher frequencies and faster on open surfaces when the second term is included.For the Möbius strip Schrödinger operator, including that term also decreases error; the sphere and hemisphere counts coincide with the plotted approximations to machine precision.Source: §3.3.2 · p. 11, Figure 4 · p. 12, Figure 5 · p. 13

3.4. Nonlinear differential eigenvalue problems.

  • SurfBeyn reduces nonlinear surface eigenvalue problems to finite-dimensional eigenproblems through contour integrals, under conditions ensuring a discrete spectrum inside the contour.It approximates contour moments using test functions and quadrature, then applies an SVD and a generalized eigenvalue solve; residuals are checked using pseudospectral inclusions.Source: §3.4 · pp. 11–14

3.4.1. Example.

  • A delayed diffusion equation is Laplace-transformed into the nonlinear eigenvalue problem T(λ) = Δ + (a(x) − λ + e^(−λ)b(x))I.The example uses Dirichlet boundary conditions on a Möbius strip, with specified spatially varying coefficients a(x) and b(x).Source: §3.4.1 · p. 14
  • Contour localization followed by isolated circular contours computes eigenvalues with small residuals, though accuracy drops for larger eigenvalues and eigenfunctions oscillate more as eigenvalue magnitude increases.Table 1 lists eigenvalues and corresponding residuals used for pseudospectral inclusions.Source: §3.4.1 · p. 14, Table 1 · p. 15
  • Convergence requires about 25 quadrature nodes, more than SurfFEAST without symmetry exploitation, while residuals vary linearly with element count on a log-log plot, suggesting polynomial convergence.Source: §3.4.1 · pp. 14–15

4. Spectral measures on surfaces.

The section develops resolvent-based smoothing methods for computing spectral measures, including continuous spectra, and applies them to internal waves and an unbounded-surface Schrödinger operator. Higher-order kernels improve convergence and can reduce solve costs, with example results illustrating convergence and localization.

4. Spectral measures on surfaces.

  • When an operator has continuous spectrum, the spectral measure captures information not represented by discrete eigenvalue delta functions; its smoothed approximations converge weakly as ϵ decreases.With sufficient local regularity, the approximation converges to the Radon–Nikodym derivative; the Poisson-kernel rate is O(ϵ log(1/ϵ)), and small ϵ requires larger, more expensive discretizations.Source: §4 · pp. 15–16

4.1. High-order smoothing kernels.

  • Higher-order rational kernels approximate spectral measures through shifted resolvent solves, converging weakly and, under sufficient regularity, to the density at rate O(ϵ^m log(1/ϵ)).The kernel integrates to one, has vanishing moments through order m−1, and decays as O(1/(1+|x|)^(m+1)); larger ϵ can reduce discretization size and speed solves.Source: §4.1 · pp. 16–17
  • In practical implementation, SurfMeas computes spectral measures of operators on surfaces by solving shifted linear systems (L − (λ − ϵaⱼ)I)gⱼ = f and taking inner products of functions on surfaces.For linear pencils A − λB with A and B self-adjoint and B positive and invertible, the approach studies B⁻¹A. Under suitable assumptions, L = B⁻¹A|D(A)∩D(B¹ᐟ²) is self-adjoint; f is assumed in D(A)∩D(B¹ᐟ²).Source: §4.1 · p. 17, §4.2.1 · p. 17

4.2. Examples.

  • The examples cover bounded and unbounded surfaces, including a linear-pencil internal-wave problem whose spectrum is [0, 1] but whose spectral type is unknown.For internal waves, smoothed projection-density functions become increasingly oscillatory and localized as ϵ decreases; they converge weakly but not in HB.Source: §4.2 · p. 17, §4.2.1 · pp. 17–18

4.2.1. Internal waves.

  • For internal waves on the torus, sixth-order smoothed measures appear to converge to a density, suggesting an absolutely continuous spectrum; higher-order kernels improve convergence.The spectrum is [0, 1], but its spectral type is unknown; the density convergence is therefore presented as a suggestion.Source: §4.2.1 · pp. 17–18, Figure 8 · p. 18

4.2.2. Schr¨odinger operator on unbounded surface.

  • For the unbounded-surface Schrödinger example, the spectrum is [0, +∞), and larger α gives slower spectral-measure decay while the smoothed measure remains well localized to R≥0.The infinite-surface problem is reformulated as coupled bounded-region and unbounded flat-region problems; the latter is solved using potential theory.Source: §4.2.2 · p. 19, §4.2.1 · p. 19, Figure 10 · p. 19

5. Conclusions.

SurfSpec may support applications in spectral geometry and the computation of evolutionary PDEs on manifolds. The paper also identifies possible uses in shape optimization, functional calculus, and control.

5. Conclusions.

  • SurfSpec could help test and formulate conjectures in shape optimization and spectral theory when combined with numerical optimization algorithms.The paper names spectral shape analysis and data representation as additional application areas.Source: §5 · p. 19
  • The high-order-kernel spectral-measure framework can extend to vector-valued functional calculus on manifolds, which can solve linear evolutionary PDEs.More complicated evolutionary PDEs could be addressed by combining this approach with splitting methods.Source: §5 · p. 20
  • The contour-integral methods could also be used for system realization and control of dynamics on surfaces.Source: §5 · p. 20
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