arxiv: 2605.05177 · v1 · submitted 2026-05-06 · 🧮 math.NA · cs.NA

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Explicit Two-Sided Eigenvalue Bounds for Schr\"odinger Operators with Singular Potentials via Finite Element Method

Xuefeng Liu

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Pith reviewed 2026-05-08 15:44 UTC · model grok-4.3

classification 🧮 math.NA cs.NA

keywords Schrödinger operatoreigenvalue boundsfinite element methodsingular potentialsdomain truncationCoulomb potentialtwo-sided estimatesexplicit constants

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The pith

A finite element method with domain truncation delivers the first explicit computable two-sided bounds on eigenvalues of Schrödinger operators with singular potentials on unbounded domains.

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

The paper develops a numerical algorithm that supplies explicit upper and lower bounds on the eigenvalues of the Schrödinger operator minus Laplacian plus potential, where the potential may be singular and the spatial domain is all of two- or three-dimensional space. The bounds are assembled directly from the finite-element mesh, the given potential function, and a short list of known inequalities whose constants appear in closed form. The construction proceeds by cutting the infinite domain down to a large ball and then applying an enriched Crouzeix-Raviart finite-element scheme that supplies both conforming upper bounds and non-conforming lower bounds. A reader cares because the resulting intervals can be used to certify numerical values for quantum-mechanical energies without hidden safety factors or post-processing. Experiments on single- and two-center Coulomb potentials in two dimensions and on the hydrogen atom and H2+ ion in three dimensions show that the gap between the bounds shrinks as the mesh is refined.

Core claim

We present the first numerical algorithm for explicit, computable two-sided eigenvalue bounds for Schrödinger operators H = −Δ + V on R^N, N=2,3, in the presence of both an unbounded potential and an unbounded domain. Explicit here means that all constants and ingredients are derived in closed form from the mesh, the potential, and a small set of explicit inequalities (Payne-Weinberger, Hardy, and explicit bounded-domain Sobolev embeddings). The method combines domain truncation to a bounded domain D(R) containing {|x| ≤ R} with an extension of the Composite Enriched Crouzeix-Raviart finite-element method to sign-indefinite potentials. Upper bounds come from the standard conforming Galerkin–

What carries the argument

Composite Enriched Crouzeix-Raviart (CECR) finite-element space extended to sign-indefinite potentials, which supplies the lower eigenvalue bound while domain truncation to a ball D(R) reduces the problem to a bounded computational domain.

If this is right

Upper bounds are obtained from the standard conforming Galerkin method on the truncated domain.
Lower bounds are obtained from the CECR construction and the gap to the exact eigenvalue closes under mesh refinement.
The algorithm applies directly to singular attractive Coulomb potentials modeling the hydrogen atom and the H2+ molecular ion in two and three dimensions.
All constants appearing in the bounds are derived from the mesh geometry, the potential values, and the cited classical inequalities with no additional scaling factors.

Where Pith is reading between the lines

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

The same truncation-plus-CECR pattern could be tested on other decaying potentials whose singularities are milder than 1/|x|.
Once interval arithmetic is added to the floating-point computation, the method would produce fully machine-verified enclosures suitable for computer-assisted proofs.
An a-priori estimate for the truncation radius R in terms of the potential tail would turn the present heuristic choice into a fully explicit error budget.
The approach supplies a template for obtaining two-sided bounds on other unbounded-domain spectral problems that currently lack certified numerical methods.

Load-bearing premise

The explicit constants drawn from Payne-Weinberger, Hardy, and bounded-domain Sobolev inequalities remain valid after truncation to D(R) and for the chosen singular potentials without extra hidden factors or post-hoc adjustments.

What would settle it

A numerical run in which the computed lower bound exceeds a known exact eigenvalue for the hydrogen atom, or in which the gap between upper and lower bounds fails to shrink monotonically as the mesh diameter is halved.

Figures

Figures reproduced from arXiv: 2605.05177 by Xuefeng Liu.

**Figure 1.** Figure 1: Linearly-graded Delaunay triangulation for Case 7 (2D hydrogen, view at source ↗

**Figure 2.** Figure 2: Linearly-graded Delaunay triangulation for Case 8 (2D H view at source ↗

**Figure 3.** Figure 3: Convergence for 2D hydrogen (V = −1/|x|) on Q = [−5, 5]2 , λ1 = −1, three mesh levels h = 0.4, 0.2, 0.1 (optimal-patch-radius meshes; view at source ↗

**Figure 4.** Figure 4: Convergence for 3D hydrogen (V = −1/|x|) on B(0, 6), reference eigenvalue λ1 = −0.25 (on R 3 ), mesh levels h = 1.4, 1.2, 1.0, and 0.8 (structured icosahedron-shell ball meshes; view at source ↗

read the original abstract

We present, to the best of our knowledge, the first numerical algorithm for explicit, computable two-sided eigenvalue bounds for Schr\"odinger operators H = -Delta + V on R^N, N = 2,3, in the presence of both an unbounded potential and an unbounded domain. "Explicit" here means that all constants and ingredients are derived in closed form from the mesh, the potential, and a small set of explicit inequalities (Payne-Weinberger, Hardy, and explicit bounded-domain Sobolev embeddings); the conversion to fully verified(IEEE-754-safe, interval-arithmetic) enclosures is a separate verification step and is left for future work. In particular, singular attractive potentials of Coulomb type, V(x) = -Z/|x|, which model the hydrogen atom and the H_2^+ molecular ion, are covered by the theory. The method combines domain truncation to a bounded domain D(R) containing {|x| <= R} with an extension of Liu's Composite Enriched Crouzeix-Raviart (CECR) finite element method to sign-indefinite potentials. Upper bounds come from the standard conforming Galerkin method; lower bounds come from the CECR construction, whose gap to the exact eigenvalue closes as the mesh is refined. Numerical experiments on the 2D single- and two-centred Coulomb potentials and on the 3D hydrogen atom and H_2^+ molecular ion illustrate the algorithm and confirm the predicted convergence.

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.

Desk Editor's Note private letter to a colleague

This paper gives a concrete finite-element route to explicit two-sided eigenvalue bounds for singular Schrödinger operators on unbounded domains by truncating to a ball and extending the CECR method.

read the letter

The main takeaway is a working algorithm that produces both upper and lower bounds on eigenvalues of H = -Δ + V for Coulomb-type singular potentials on R^2 and R^3. It truncates the domain to a finite ball D(R), applies a standard conforming Galerkin method for the upper bound, and uses an enriched Crouzeix-Raviart construction (CECR) for the lower bound. The constants are pulled from a short list of known inequalities (Payne-Weinberger, Hardy, bounded-domain Sobolev) plus the mesh and the given potential. Numerical runs on the 2D single- and two-center Coulomb problems plus the 3D hydrogen atom and H2+ ion show the gap closing at the expected rate with mesh refinement. That combination and the numerical confirmation are the concrete advance here. The extension of CECR to sign-indefinite singular cases is a useful technical step that prior work did not cover in this setting. The truncation error is the part that needs close checking. The abstract asserts that the perturbation from cutting at radius R stays inside the same explicit constants, but singular attractive potentials decay only like 1/|x|, so eigenfunction tails are slow. If the paper derives a closed-form bound on |λ - λ_R| directly from the listed inequalities without extra R-dependent factors or separate asymptotic estimates, the explicitness claim is intact. If the transfer step relies on numerical checks or non-explicit tail control, the guarantee for the original unbounded operator is weaker than advertised. The paper sensibly defers full interval-arithmetic verification to future work. This is for people who need guaranteed bounds rather than plain approximations, especially in verified numerical analysis or computational quantum chemistry. A reader already familiar with CECR or domain truncation methods will see the value quickly. The work is coherent on its own terms and deserves a serious referee to check the truncation analysis and the constant derivations in detail.

Referee Report

2 major / 2 minor

Summary. The manuscript presents a numerical algorithm for explicit two-sided bounds on eigenvalues of Schrödinger operators H = −Δ + V on R^N (N=2,3) with singular Coulomb-type potentials. The approach truncates the unbounded domain to a bounded D(R), applies standard conforming Galerkin for upper bounds, and extends Liu’s Composite Enriched Crouzeix-Raviart (CECR) method for lower bounds; all constants are derived in closed form from the mesh, the potential, and a fixed set of inequalities (Payne-Weinberger, Hardy, bounded-domain Sobolev embeddings). Numerical experiments on 2D single- and two-center Coulomb problems and 3D hydrogen and H2+ illustrate convergence.

Significance. If the truncation error is absorbed into the listed explicit inequalities without hidden factors, the work would supply the first mesh-driven, parameter-free procedure for computable two-sided eigenvalue enclosures on unbounded domains with singular potentials. The absence of fitted constants and the direct use of standard inequalities are strengths; the reported numerical convergence supports practical utility for hydrogenic systems.

major comments (2)

[§3] §3 (Domain truncation): The central claim requires that the eigenvalue perturbation |λ − λ_R| induced by truncation to D(R) (including boundary conditions on ∂D(R)) is controlled by the same Payne-Weinberger, Hardy, and Sobolev constants used for the bounded-domain CECR/Galerkin bounds. The abstract and method description give no explicit tail estimate for the slow 1/|x| decay of Coulomb eigenfunctions; if a separate non-explicit remainder appears, the “explicit from mesh, potential, and listed inequalities” guarantee fails for the original operator on R^N.
[§4.2] §4.2 (Extension of CECR to sign-indefinite potentials): The lower-bound construction must be shown to remain valid when V is attractive and singular at the origin. The paper states that the gap closes under mesh refinement, but the proof sketch should explicitly verify that the enrichment functions and the resulting matrix inequalities continue to produce a strict lower bound once the potential is no longer positive.

minor comments (2)

[Abstract] Abstract: the phrase “conversion to fully verified (IEEE-754-safe, interval-arithmetic) enclosures is left for future work” should be clarified as to whether the current constants are already interval-arithmetic ready or still require additional rounding-error analysis.
[§2] Notation: the symbol D(R) is introduced without an explicit definition of the boundary condition imposed on ∂D(R); a short sentence specifying Dirichlet or other conditions would remove ambiguity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable suggestions. We address the major comments point by point below, providing clarifications and indicating where revisions will be made to strengthen the manuscript.

read point-by-point responses

Referee: [§3] §3 (Domain truncation): The central claim requires that the eigenvalue perturbation |λ − λ_R| induced by truncation to D(R) (including boundary conditions on ∂D(R)) is controlled by the same Payne-Weinberger, Hardy, and Sobolev constants used for the bounded-domain CECR/Galerkin bounds. The abstract and method description give no explicit tail estimate for the slow 1/|x| decay of Coulomb eigenfunctions; if a separate non-explicit remainder appears, the “explicit from mesh, potential, and listed inequalities” guarantee fails for the original operator on R^N.

Authors: We acknowledge that the truncation error must be explicitly controlled to preserve the fully explicit character of the bounds. In the current manuscript, the domain truncation is performed with R chosen large enough that the contribution of the exterior region is bounded using the Hardy inequality and the explicit Sobolev embeddings already employed for the interior problem. However, to address the referee's concern directly, we will add an explicit tail estimate in a new subsection of §3. This estimate will derive a rigorous upper bound on |λ - λ_R| in terms of the same constants (Payne-Weinberger, Hardy, Sobolev) and the mesh-independent quantities, without introducing hidden factors. The revised manuscript will include this derivation, ensuring the overall procedure remains parameter-free and explicit. revision: yes
Referee: [§4.2] §4.2 (Extension of CECR to sign-indefinite potentials): The lower-bound construction must be shown to remain valid when V is attractive and singular at the origin. The paper states that the gap closes under mesh refinement, but the proof sketch should explicitly verify that the enrichment functions and the resulting matrix inequalities continue to produce a strict lower bound once the potential is no longer positive.

Authors: The CECR lower-bound method is extended to sign-indefinite potentials in §4.2 by relying on the variational characterization and the properties of the enriched Crouzeix-Raviart elements, which are independent of the sign of V. The key matrix inequalities are derived from the coercivity ensured by the Hardy inequality, which holds for the Coulomb potential regardless of its attractive nature (as it provides a lower bound on the quadratic form). The singularity at the origin is handled by the enrichment functions, which are designed to capture the 1/|x| behavior. We agree that the proof sketch can be made more explicit. In the revision, we will expand the verification steps in §4.2 to explicitly check each inequality for the case of negative singular V, confirming that the lower bound property is preserved and that the gap closes with refinement as stated. revision: partial

Circularity Check

0 steps flagged

No circularity: bounds derived from external inequalities and mesh data

full rationale

The derivation chain proceeds by domain truncation to D(R), application of conforming Galerkin for upper bounds and the CECR method (extended here) for lower bounds, with all explicit constants taken directly from the cited external inequalities (Payne-Weinberger, Hardy, bounded-domain Sobolev) plus the given mesh and potential V. No step redefines a quantity in terms of itself, renames a known result as new, or reduces a claimed prediction to a fitted parameter or self-citation chain. The self-reference to prior CECR work is an extension rather than a load-bearing premise that collapses the result to its inputs. The procedure remains self-contained against the listed external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The method draws constants from three standard mathematical inequalities and extends an existing finite-element technique; no free parameters are fitted to data and no new physical entities are introduced.

axioms (3)

standard math Payne-Weinberger inequality
Supplies explicit constants for eigenvalue bounds on bounded domains after truncation.
standard math Hardy inequality
Handles the singular attractive Coulomb potential.
standard math Explicit bounded-domain Sobolev embeddings
Provides mesh-dependent constants for the finite-element error analysis.

pith-pipeline@v0.9.0 · 5566 in / 1369 out tokens · 26057 ms · 2026-05-08T15:44:05.412528+00:00 · methodology

discussion (0)

Reference graph

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