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arxiv: 2604.21074 · v1 · submitted 2026-04-22 · 🧮 math.NA · cs.NA

Old and new Schr\"odinger eigenvalue localisation

Carsten Carstensen , Tim Stiebert This is my paper

Pith reviewed 2026-05-09 22:55 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords Schrödinger eigenvalue problemguaranteed lower boundsnonconforming finite elementsadaptive mesh refinementeigenvalue localizationfinite element methodnumerical analysis
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The pith

A fine-tuned extra-stabilised nonconforming scheme provides guaranteed lower bounds for Schrödinger eigenvalues without mesh-size dependence.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper adapts nonconforming finite element methods to compute guaranteed lower bounds for eigenvalues of the Schrödinger operator with general potentials. By adding extra stabilization, the new scheme avoids the mesh-size parameters that appeared in earlier post-processing approaches. This makes the bounds compatible with adaptive mesh refinement and allows efficient two-sided eigenvalue estimates when paired with conforming upper bounds. Such reliable bounds matter for understanding spectral properties in quantum models without discretization uncertainty.

Core claim

By adapting the nonconforming guaranteed lower bound technique from the harmonic eigenvalue problem and introducing extra stabilization, the method computes direct guaranteed lower bounds for Schrödinger eigenvalues with piecewise constant potentials that remain valid under adaptive refinement and do not depend on maximal mesh size parameters, as verified through numerical comparisons showing superiority over previous schemes.

What carries the argument

The extra-stabilised nonconforming finite element discretization that enables direct computation of guaranteed lower eigenvalue bounds (GLB) while preserving the bound property.

If this is right

  • Two-sided eigenvalue control becomes available from a single computation with reduced cost compared to separate conforming schemes for upper bounds.
  • Adaptive mesh-refinement can be applied directly without reintroducing mesh-size dependence in the bounds.
  • The bounds apply unconditionally to general and piecewise constant potentials for spectral gaps.
  • Numerical benchmarks show competitive accuracy with less computational effort than additional lowest-order conforming schemes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The direct GLB approach could extend to time-dependent or nonlinear eigenvalue problems sharing similar elliptic structure.
  • Combining this with higher-order elements might further reduce degrees of freedom needed for target accuracy.
  • Reliable spectral gaps from these bounds could support more trustworthy uncertainty quantification in quantum chemistry models.

Load-bearing premise

The adaptation of the nonconforming GLB method to Schrödinger problems with piecewise constant potentials preserves the guaranteed lower bound property without additional hidden constants.

What would settle it

A numerical computation on the harmonic oscillator or similar exactly solvable case where the reported lower bound exceeds the known exact eigenvalue would disprove the guarantee.

Figures

Figures reproduced from arXiv: 2604.21074 by Carsten Carstensen, Tim Stiebert.

Figure 1
Figure 1. Figure 1: Initial triangulations of the square (left) and L-shaped domain (right) in [PITH_FULL_IMAGE:figures/full_fig_p018_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Methods for GLB/GUB in the computational benchmarks of Section [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Convergence history plot for λ1 ´ GLBp1q (left) and GUBp1q ´ λ1 (right) on uniform meshes of Ω “ p´8, 8q 2 for V1. 101 102 103 104 105 106 103 102 101 100 101 1 1 |T | 101 102 103 104 105 106 103 102 101 100 101 1 1 |T | [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Convergence history plot for λ20´GLBp20q (left) and GUBp20q´λ20 (right) on uniform meshes of Ω “ p´8, 8q 2 for V1. Excited states on Ω “ p´8, 8q 2 [PITH_FULL_IMAGE:figures/full_fig_p020_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Convergence history plot for λ1 ´ GLBp1q (left) and GUBp1q ´ λ1 (right) on uniform (dashed) and adaptive (solid) meshes of Ω “ p´8, 8q 2 zr0, 8q 2 for V1. −8 −4 0 4 8 −8 −4 0 4 8 −8 −4 0 4 8 −8 −4 0 4 8 [PITH_FULL_IMAGE:figures/full_fig_p021_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Adaptive triangulations Tℓ on level ℓ “ 23 (left) and ℓ “ 25 (right) for λ1 with V1 on Ω “ p´8, 8q 2 zr0, 8q 2 with |T23| “ 412 and |T25| “ 1274 in Subsection 7.2. 7.3 Lattice potential The lattice potential V2pxq :“ p|x| 2 {2 ´ 16q` ` t30 ` 10 sinpπx1{2qsinpπx2{2qu in [PITH_FULL_IMAGE:figures/full_fig_p021_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Convergence history plot for λ1 ´ GLBp1q (left) and GUBp1q ´ λ1 (right) on uniform meshes of Ω “ p´8, 8q 2 for V2. history plot of the GLB from [PITH_FULL_IMAGE:figures/full_fig_p022_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Projection onto piecewise constants of V1, V2, V3 (first row from left to right) and CR approximations of the corresponding ground states (second row). 22 [PITH_FULL_IMAGE:figures/full_fig_p022_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Convergence history plot for λ1 ´ GLBp1q (left) and GUBp1q ´ λ1 (right) on uniform (dashed) and adaptive (solid) triangulations of the unit square under the disordered potential V3 in Subsection 7.4. 0 1 0 1 0 1 0 1 [PITH_FULL_IMAGE:figures/full_fig_p023_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Initial (left) and adaptive (right) triangulations of the unit square Ω [PITH_FULL_IMAGE:figures/full_fig_p023_10.png] view at source ↗
read the original abstract

Unconditional guaranteed lower and upper eigenvalue bounds are mandatory for the understanding of the Schr\"odinger eigenvalue spectrum and its spectral gaps. While upper eigenvalue bounds are naturally induced by conforming discretisations, guaranteed lower eigenvalue bounds (GLB) are less immediate. This paper clarifies the adaptation of nonconforming GLB from the harmonic eigenvalue problem and discusses their comparison for general and piecewise constant potentials. A fine-tuned extra-stabilised scheme is proposed and found superior in numerical comparisons. This new direct calculation of GLB is compatible with adaptive mesh-refinement and successfully circumvents the appearance of maximal mesh-size parameters in former GLB based on post-processing. Computational benchmarks also investigate guaranteed upper eigenvalue bounds (GUB) for two-sided eigenvalue control by conforming test functions associated to the underlying nonconforming computations. A numerical comparison with GUB from additional lowest-order conforming finite element schemes shows competitive accuracy with less computational cost.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript adapts nonconforming finite element methods, originally developed for the harmonic eigenvalue problem, to compute guaranteed lower bounds (GLB) on eigenvalues of the Schrödinger operator for general and piecewise-constant potentials. It proposes a fine-tuned extra-stabilised nonconforming scheme enabling direct GLB evaluation that is compatible with adaptive mesh refinement and avoids the maximal mesh-size parameters arising in prior post-processing approaches. Numerical benchmarks demonstrate the scheme's superiority, while also investigating guaranteed upper bounds (GUB) obtained from conforming test functions associated with the nonconforming solutions, yielding competitive accuracy at reduced cost compared to standard lowest-order conforming FEM.

Significance. If the adaptation rigorously preserves the unconditional GLB property with constants independent of h_max and without hidden dependence on the potential, the work would provide a practical and theoretically sound tool for reliable two-sided eigenvalue control in quantum-mechanical spectral problems. The emphasis on direct computation, adaptive compatibility, and numerical efficiency over post-processed methods represents a clear incremental advance, supported by the reported computational comparisons.

major comments (2)
  1. [Section on the extra-stabilised nonconforming scheme and associated theorem] The central claim that the extra-stabilised nonconforming scheme yields a mesh-size-independent GLB for Schrödinger operators with piecewise-constant potentials rests on the adaptation of the variational form. The treatment of the L2 inner product with V and the choice of stabilisation parameters must be shown explicitly to introduce no hidden constants depending on ||V||_∞ or local mesh ratios; otherwise the circumvention of post-processing parameters fails. This analytic verification is load-bearing for the abstract's assertion of unconditional guarantees.
  2. [Numerical benchmarks and error analysis sections] The numerical superiority and GUB comparisons are presented, but the error analysis or theorem establishing that the discrete eigenvalue remains a strict lower bound without reintroducing h-dependent remainders (as raised in the skeptic note) needs to be stated with all constants tracked explicitly to confirm independence from maximal mesh size.
minor comments (2)
  1. [Method description] Notation for the stabilisation parameters and the precise definition of the discrete bilinear form could be clarified with an explicit equation to aid reproducibility of the fine-tuned scheme.
  2. [Figures and captions] Figure captions would benefit from indicating the specific benchmark potentials and mesh types used in the comparisons to make the superiority claims easier to interpret at a glance.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments, which help clarify the presentation of the unconditional GLB property. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [Section on the extra-stabilised nonconforming scheme and associated theorem] The central claim that the extra-stabilised nonconforming scheme yields a mesh-size-independent GLB for Schrödinger operators with piecewise-constant potentials rests on the adaptation of the variational form. The treatment of the L2 inner product with V and the choice of stabilisation parameters must be shown explicitly to introduce no hidden constants depending on ||V||_∞ or local mesh ratios; otherwise the circumvention of post-processing parameters fails. This analytic verification is load-bearing for the abstract's assertion of unconditional guarantees.

    Authors: We appreciate the referee's focus on this key point. The proof of the GLB property (Theorem 3.2) adapts the nonconforming form by incorporating the potential term directly. For piecewise-constant V the L2 inner product is evaluated exactly elementwise, and the stabilisation parameter is selected as a fixed multiple independent of h. The analysis relies on a discrete Poincaré inequality whose constant depends only on shape regularity and the domain, with no additional factors from local mesh ratios or ||V||_∞ entering the lower-bound constant. Nevertheless, to make the independence fully explicit we will revise the proof section to track every constant in the estimates, confirming that none depend on h_max or local ratios beyond the already-controlled shape-regularity assumption. This will strengthen the verification of the unconditional guarantee. revision: yes

  2. Referee: [Numerical benchmarks and error analysis sections] The numerical superiority and GUB comparisons are presented, but the error analysis or theorem establishing that the discrete eigenvalue remains a strict lower bound without reintroducing h-dependent remainders (as raised in the skeptic note) needs to be stated with all constants tracked explicitly to confirm independence from maximal mesh size.

    Authors: We agree that explicit constant tracking improves clarity. The error analysis in Section 4 shows that the extra stabilisation removes the post-processing step that previously introduced h_max, so the discrete eigenvalue remains a strict lower bound with no reintroduced h-dependent remainders. The skeptic note is already addressed by the direct variational argument. Following the referee's request we will expand the error-analysis subsection to list all appearing constants explicitly and state their dependence solely on fixed quantities (shape regularity, potential bounds, and domain), thereby confirming independence from maximal mesh size. Minor clarifications will also be added to the GUB comparison paragraphs. revision: yes

Circularity Check

0 steps flagged

No significant circularity in adaptation of nonconforming GLB methods

full rationale

The paper adapts established nonconforming variational techniques for guaranteed lower bounds from the pure Laplacian (harmonic) eigenvalue problem to Schrödinger operators with general or piecewise-constant potentials. The load-bearing analytic step is the verification that the discrete eigenvalue remains a strict lower bound after incorporating the potential term while preserving mesh-size independence and compatibility with adaptive refinement. This rests on direct variational arguments and numerical validation rather than any self-definitional reduction, fitted parameter renamed as prediction, or load-bearing self-citation chain. Prior results on the harmonic case serve as independent external support (externally falsifiable via standard FEM theory), and the new scheme is presented as a fine-tuned extension whose superiority is checked computationally. No equation or claim reduces to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on standard finite element theory for eigenvalue problems and the transfer of nonconforming lower bound techniques; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Nonconforming finite element spaces and stabilization terms from the harmonic eigenvalue problem extend directly to the Schrödinger operator with general or piecewise constant potentials while preserving the guaranteed lower bound property.
    The abstract states that the adaptation clarifies and discusses comparison for general and piecewise constant potentials.

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