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A spectral-element method for computing many eigenvalues and their asymptotics of the Schr\"odinger operator with Robin boundary condition

Weizhu Bao, Fukeng Huang

arXiv:2609.12369v1math.NAcs.NAmath-phmath.MP

TL;DR

Accurately computing many high-index eigenvalues of the Schrödinger operator with Robin boundary conditions is challenging, especially on complex domains. The paper develops spectral and spectral-element methods and uses their computed eigenvalues to study Robin-to-Neumann gaps. Its computations produce thousands of accurate 2D eigenvalues and support a unified conjecture on cumulative gap averages.

  • Problem

    Accurate, high-resolution computation of many Robin eigenvalues is limited, yet thousands of eigenvalues are needed to study their asymptotics and Robin-to-Neumann gaps.

    Source: §1 · p. 4

  • Method

    The paper uses spectral methods on simple geometries and spectral-element methods on complex geometries to compute many eigenvalues.

    Source: §1 · p. 4, §6 · pp. 16–17

  • Results

    With about 25,000 degrees of freedom in 2D, the methods obtain at least 5,000 eigenvalues with relative error below 10^-8; the paper also formulates a unified conjecture on cumulative RtN-gap averages.

    Source: §4.2 · p. 12, §6 · p. 17

  • Takeaways & Limitations

    The computations support systematic study of RtN gaps and their convergence rates across dimensions, potentials, and geometries.

    Source: §6 · p. 17

    The stated scope uses connected bounded domains with piecewise smooth boundary and real-valued potentials in L∞(Ω); the introduction assumes V(x)≥0.

    Source: §1 · pp. 1–2

Abstract

from arXiv · show

We propose spectral and spectral-element methods for accurately computing many eigenvalues of the Schr\"odinger operator with Robin boundary condition on simple and complex geometries, respectively. Due to their spectral-type accuracy in approximating those eigenfunctions corresponding to high-index eigenvalues, which are usually highly oscillatory, the proposed approaches have excellent resolution in computing thousands of eigenvalues accurately and efficiently with the resolution property that the number of eigenvalues with reasonable accuracy is proportional to the number of degrees of freedom. Using a standard HPC node with 128 GB of memory, we can obtain numerically more than 5,000 reliable eigenvalues with relative errors below $10^{-8}$ for different complex domains in two dimensions (2D). Based on these computed eigenvalues, we first confirm some theoretical results on the Robin-to-Neumann (RtN) gaps of the Laplacian operator in 2D, which were recently studied by Rudnick \textit{et al.} [\textit{Comm. Math. Phys.} 388 (2021)]. Then we systematically study the RtN gaps of the Schr\"odinger operator and their convergence rates. Based on our extensive numerical results, we formulate a unified conjecture on the cumulative averages of the RtN gaps of the Schr\"odinger operator.

1. Introduction

The paper addresses the scarcity of accurate methods for computing many high-index Robin eigenvalues, needed to study their asymptotics and Robin-to-Neumann gaps. It develops spectral methods for simple geometries and spectral-element methods for complex ones, using extensive computations to formulate a conjecture on cumulative gap averages.

1. Introduction

  • For the Laplacian in 2D, Rudnick et al. [31] studied Robin-to-Neumann gaps and cumulative averages, connecting the latter to domain area and boundary length.The cited result is stated for connected bounded domains with piecewise smooth boundary.Source: §1 · p. 3
  • Existing Robin-boundary numerical work is comparatively sparse and focuses mainly on analysis or relatively small numbers of low-index eigenpairs.The cited literature includes more extensive numerical and theoretical development for Dirichlet and Neumann conditions.Source: §1 · pp. 3–4
  • High-index eigenvalues require thousands of accurate computations, while low-order finite elements or finite differences resolve only a limited spectral range at fixed degrees of freedom.Spectral methods offer resolution proportional to degrees of freedom on simple geometries, while spectral elements extend high-order approximation to complicated geometries.Source: §1 · p. 4
  • The proposed methods target more than 5,000 reliable eigenvalues with relative errors below 10^-8 on different complex 2D domains, enabling numerical study of Robin-to-Neumann gaps and their asymptotics.The stated computation uses a standard HPC node with 128 GB of memory.Source: §1 · p. 4
  • The paper formulates a unified conjecture on cumulative averages of Schrödinger-operator Robin-to-Neumann gaps for connected bounded domains with piecewise smooth boundary and V in L∞(Ω).It also states a separate result for an interval in one dimension, though the supplied passage does not include that result's formula.Source: §1 · p. 4

2. Scaling property and the variational formulation

This section rescales the eigenproblem to a unit-diameter domain and gives its weak formulation. The scaling transfers eigenvalues, eigenfunctions, and Robin parameters between the original and rescaled problems; the variational form supports spectral bounds and eigenvalue characterization.

2.1. Scaling property

  • Rescaling by the domain diameter R maps coordinates to a unit-diameter domain and transforms the potential, eigenfunction, eigenvalue, and Robin parameter.The stated transformations are y=x/R, Ṽ(y)=R²V(Ry), λ̃=R²λ, and σ̃=Rσ.Source: §2.1 · p. 5
  • An eigenpair on the rescaled domain maps back to an eigenpair on the original domain with λ=λ̃/R² and u(x)=v(x/R), and the converse also holds.The Robin parameter transforms as σ=σ̃/R.Source: §2.1 · p. 5
  • The scaling lemma transfers the ordered eigenvalue sequence between the two domains while preserving the corresponding parameter scaling.The original and rescaled sequences are related by λ=λ̃/R² and σ=σ̃/R.Source: §2.1 · p. 5

2.2. A variational (weak) formulation

  • Testing the strong eigenproblem against a function in H1(Ω), integrating over Ω, and integrating by parts yields the weak formulation.The variational problem seeks a nonzero u in H1(Ω).Source: §2.2 · p. 5
  • For σ≥0, the Robin form satisfies aσ(v,v)≥a0(v,v); for σ<0, the trace inequality bounds the negative boundary contribution using an arbitrarily chosen ε>0.The text then notes ε can be chosen sufficiently small to control that contribution by the gradient term.Source: §2.2 · p. 5
  • When V is in L∞(Ω), the smallest eigenvalue of the weak problem is bounded below and all its eigenvalues are discrete.The passage invokes the Rayleigh quotient in establishing the lower bound.Source: §2.2 · p. 5
  • The Rayleigh-quotient framework characterizes the next eigenvalue using orthonormal eigenfunctions and a comparison subspace of dimension n+1.The setup begins with the first n orthonormal eigenfunctions and their span.Source: §2.2 · p. 6

3. Spectral and spectral-element methods

The section formulates a Galerkin matrix eigenproblem, then chooses spectral bases for simple geometries and mapped spectral elements for complex ones.

3.1. A Galerkin approximation.

  • The Galerkin approximation represents the eigenfunction in a conforming finite-dimensional space and yields a generalized matrix eigenvalue problem with stiffness and mass matrices.The matrices are assembled from the bilinear forms evaluated on the chosen basis functions.Source: §3.1 · p. 6

3.2. Spectral methods for simple geometries.

  • For simple geometries, spectral methods use domain-appropriate bases; on an interval, Legendre polynomials construct a subspace with spectral accuracy.The same approach extends to rectangles, disks, cubes, balls, and cylinders using standard spectral bases; practical integrals may use numerical quadrature.Source: §3.2 · p. 7

3.3. Spectral-element methods for complex geometries.

  • For complex geometries, the method partitions a planar domain into curvilinear quadrilateral elements and maps each element to a reference square with tensor-product Gauss-Lobatto-Legendre nodes.The discussion presents the 2D method and says extension to 3D is straightforward, with details omitted.Source: §3.3 · pp. 7–8
  • Mapped local integrals incorporate Jacobian and metric factors, while Robin contributions are added on boundary edges and element matrices are assembled with continuity across neighbors.The resulting algebraic eigenproblem has the same form as the Galerkin problem, with a basis adapted to geometry through element-wise mappings.Source: §3.3 · p. 8

4. Accuracy test

The accuracy tests evaluate spectral and spectral-element approximations across one- and two-dimensional settings. Both methods resolve a substantial fraction of the spectrum below 10^-8 relative error, with higher polynomial degree and fewer elements generally improving fixed-DOF results.

4. Accuracy test

  • Accuracy is assessed using relative eigenvalue errors, with sufficiently large numerical reference solutions used when analytic eigenvalues are unavailable.The one-dimensional tests include analytic eigenvalues for zero potential; non-analytic cases use a numerical reference with Nex=5000.Source: §4 · p. 8, §4.1 · p. 8

4.1. Accuracy in 1D.

4.2. Accuracy in 2D.

  • At fixed DOFs on the disk, increasing polynomial degree and decreasing the number of elements generally yields more reliable eigenvalues.This comparison varies element count and polynomial degree while keeping total DOFs fixed.Source: §4.2 · p. 10, Figure 7 · p. 11, Figure 6 · p. 11
  • In 2D, spectral methods resolve about 0.4N eigenvalues and spectral-element methods about 0.25N below 10^-8 relative error; at N≈25,000, at least 5,000 are reliable.The tests include square and disk domains, including a nonzero potential.Source: Figure 4 · p. 10, Figure 5 · p. 10, Figure 8 · p. 11

5. Numerical studies on the Robin-to-Neumann (RtN) gaps

The numerical studies examine Robin-to-Neumann gaps and cumulative-average convergence in 1D, 2D, and 3D, using enough degrees of freedom to obtain reliable eigenvalues. Across these dimensions, the reported results support the stated RtN-gap relations.

5.1. The RtN gaps in 1D.

  • In 1D, the spectral method studies gaps for V(x) = x^2 and different σ values, with additional tests using other potentials.The other tested potentials include 1 + sin(x) and 1 − cos(x).Source: §5.1 · p. 13

5. Numerical studies on the Robin-to-Neumann (RtN) gaps

  • The studies conclude relation (1.12) in 1D, 2D, and 3D, while the 1D study also concludes relation (1.13).The evidence combines the reported conclusions from all three dimensions.Source: §5.1 · p. 13, §5.2 · p. 15, §5.3 · p. 16

5.2. The RtN gaps in 2D.

  • In 2D, spectral elements test several complex domains, including a triangle, L-shaped domain, quarter stadium, and mushroom, with quadratic and zero potentials.The reported convergence studies also vary σ and potential in the quarter-stadium domain.Source: §5.2 · pp. 14–15, Figure 14 · p. 15, Figure 15 · p. 15

5.3. The RtN gaps in 3D.

  • In 3D, the zero-potential problem is studied on a cube and unit ball, with eigenvalues computed semi-analytically from separated transcendental equations.The studies compare gaps and cumulative-average convergence across the two domains and different σ values.Source: §5.3 · p. 16, Figure 16 · p. 16, Figure 17 · p. 16

6. Conclusion

The paper presents spectral and spectral-element methods for computing many Robin-boundary Schrödinger eigenvalues on simple and complex geometries. Using the resulting eigenvalues, it studies RtN gaps across dimensions and geometries and formulates a unified conjecture on their cumulative averages.

6. Conclusion

  • The proposed spectral and spectral-element methods target accurate computation of many eigenvalues on simple and complex geometries, respectively.The framework combines geometrically flexible finite-element partitions with spectral-method accuracy and p-version finite elements.Source: §6 · pp. 16–17
  • The framework obtains thousands of reliable eigenvalues under a fixed number of degrees of freedom.Its accuracy and resolution support computation of many eigenvalues within that fixed-DOF setting.Source: §6 · p. 17
  • Using the computed eigenvalues, the paper studies RtN gaps across dimensions, potentials, and geometries and formulates a unified conjecture on cumulative averages.Source: §6 · p. 17
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