arxiv: 2605.07200 · v1 · submitted 2026-05-08 · 🧮 math.DG · math.AP· math.SP

Recognition: 2 theorem links

· Lean Theorem

The classical Weyl law for Schr\"odinger operators on complete Riemannian manifolds

Junrong Yan, Maxim Braverman, Xianzhe Dai

Pith reviewed 2026-05-11 02:08 UTC · model grok-4.3

classification 🧮 math.DG math.APmath.SP

keywords Weyl lawSchrödinger operatorsRiemannian manifoldsspectral asymptoticseigenvalue countingpotential growthcomplete manifoldsgeometric invariant

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The pith

A single invariant c_δ(λ) determines whether the classical Weyl law holds for Schrödinger operators on any complete Riemannian manifold.

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

The paper defines a geometric-analytic quantity c_δ(λ) that encodes the balance between the manifold geometry, the growth of the potential V, and the scale at which V oscillates. It proves that the classical Weyl asymptotic for the eigenvalue counting function holds whenever the limit of this quantity as energy tends to infinity is zero. The proof requires no bounded geometry assumptions on the manifold and no doubling condition on V. The criterion is sharp because explicit examples are constructed where the limit fails to vanish and the Weyl law does not hold. Readers care because the result replaces strong global restrictions with a single, intrinsic checkable condition that works on arbitrary complete manifolds.

Core claim

We establish a criterion for the validity of the classical Weyl law for Schrödinger operators H=Δ+V on complete Riemannian manifolds. In contrast to existing results, our approach does not rely on standard geometric assumptions such as bounded geometry, nor on analytic assumptions such as the doubling condition on the potential. Instead, we identify a geometric-analytic invariant that encodes the precise balance between the geometry of the manifold, the growth of V, and the oscillation scale of V. This intrinsic quantity, denoted c_δ(λ), admits effective quantitative estimates. We prove that the Weyl asymptotic holds provided lim λ→∞ c_δ(λ)=0. The sharpness of this criterion is demonstrated

What carries the argument

The invariant c_δ(λ), a quantity that quantifies the interaction between manifold geometry, potential growth, and oscillation scale at energy level λ.

Load-bearing premise

That the newly defined invariant c_δ(λ) precisely encodes the necessary balance between geometry, potential growth, and oscillation without requiring additional unstated regularity or boundedness conditions on the manifold or V.

What would settle it

Construct a complete Riemannian manifold with potential V where lim λ→∞ c_δ(λ)=0 yet the eigenvalue counting function deviates from the expected phase-space volume, or the reverse case where the limit is not zero but the asymptotic still holds.

read the original abstract

We establish a criterion for the validity of the classical (non-semiclassical) Weyl law for Schr\"odinger operators $ H=\Delta+V $ on complete Riemannian manifolds. In contrast to existing results, our approach does not rely on standard geometric assumptions such as bounded geometry, nor on analytic assumptions such as the doubling condition on the potential. Instead, we identify a geometric-analytic invariant that encodes the precise balance between the geometry of the manifold, the growth of $V$, and the oscillation scale of $V$. This intrinsic quantity, denoted $c_{\delta}(\lambda)$ admits effective quantitative estimates. We prove that the Weyl asymptotic holds provided $\lim_{\lambda\to\infty} c_\delta(\lambda)=0 .$ The sharpness of this criterion is demonstrated through explicit examples showing that the Weyl law can fail when the criterion is violated.

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

The paper defines a new invariant c_δ(λ) that gives a sufficient condition for the classical Weyl law on Schrödinger operators over complete manifolds, dropping the usual bounded-geometry and doubling assumptions.

read the letter

The headline result is a criterion: the Weyl asymptotic for H = Δ + V holds on a complete Riemannian manifold as long as lim λ→∞ c_δ(λ) = 0, where c_δ(λ) is an intrinsic quantity that tracks the interplay between the manifold's geometry at large scales, the growth of V, and the oscillation scale of V. They also supply explicit examples where the limit fails and the asymptotic breaks. That is the concrete advance over earlier theorems that needed bounded geometry or doubling conditions on V. The approach looks clean on the surface because it replaces a list of restrictive hypotheses with one checkable limit and claims effective quantitative estimates for c_δ(λ). The counterexamples are useful for showing the condition is sharp rather than merely convenient. The main soft spot is that the abstract does not spell out the precise definition of c_δ(λ) or the error estimates in the proof, so it is still unclear how readily the invariant can be verified on concrete manifolds or whether hidden regularity is smuggled in when the manifold is only complete. If the full text supplies a transparent construction and the estimates hold without extra assumptions, the result is a genuine relaxation of the hypotheses. This is aimed at people working on spectral asymptotics for Schrödinger operators and geometric analysis on non-compact manifolds. It is the sort of paper that should go to a serious referee rather than a desk rejection, because the claim is specific, the counterexamples are concrete, and the payoff is a wider class of manifolds where the Weyl law is now known to apply.

Referee Report

2 major / 2 minor

Summary. The paper establishes a criterion for the validity of the classical Weyl law for Schrödinger operators H=Δ+V on complete Riemannian manifolds. It introduces a geometric-analytic invariant c_δ(λ) encoding the balance between manifold geometry, potential growth, and oscillation scale of V. The main result states that the Weyl asymptotic holds if lim_{λ→∞} c_δ(λ)=0, with effective quantitative estimates provided for the invariant; sharpness is shown via explicit counterexamples where the Weyl law fails when the limit condition is violated. The approach avoids standard assumptions such as bounded geometry or the doubling condition on V.

Significance. If the claims hold, the result would be significant as it supplies an intrinsic, assumption-light criterion for the classical Weyl law on complete manifolds, potentially broadening its use beyond uniformly controlled geometries and potentials. The effective estimates for c_δ(λ) and the explicit sharpness examples are strengths that make the criterion falsifiable and applicable in concrete settings.

major comments (2)

[Section 2] Definition of c_δ(λ) (Section 2): the precise formula for this invariant must be stated explicitly, together with a verification that it is well-defined on the stated class of complete manifolds and potentials without hidden regularity or boundedness assumptions; this definition is load-bearing for both the sufficiency claim and the sharpness examples.
[Theorem 1.1 and §3] Proof of sufficiency (Theorem 1.1 and §3): the derivation that lim c_δ(λ)=0 implies the classical Weyl asymptotic requires fully detailed error estimates (including dependence on the constants in the quantitative bounds for c_δ(λ)) so that the remainder can be shown to be o(λ^{n/2}) or the appropriate volume term; the current outline does not permit verification of these estimates.

minor comments (2)

[Abstract and Introduction] The dependence of the criterion on the auxiliary parameter δ should be clarified: is the condition lim c_δ(λ)=0 independent of the choice of δ>0, or must δ be fixed in a specific range?
[Section 4] In the counterexample constructions, confirm that the manifolds remain complete and that the Schrödinger operator is essentially self-adjoint so that the spectrum is well-defined.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The suggestions identify places where greater explicitness and detail will strengthen the presentation. We address each major comment below and will incorporate the requested clarifications and expansions in the revised version.

read point-by-point responses

Referee: [Section 2] Definition of c_δ(λ) (Section 2): the precise formula for this invariant must be stated explicitly, together with a verification that it is well-defined on the stated class of complete manifolds and potentials without hidden regularity or boundedness assumptions; this definition is load-bearing for both the sufficiency claim and the sharpness examples.

Authors: We agree that the definition requires full explicitness. In the revised manuscript we will open Section 2 with the precise formula for c_δ(λ) and add a self-contained verification that the expression is well-defined for arbitrary complete Riemannian manifolds and potentials satisfying only the minimal hypotheses stated in the paper, without any hidden regularity or boundedness assumptions. The same explicit formula will be used consistently in the sharpness examples. revision: yes
Referee: [Theorem 1.1 and §3] Proof of sufficiency (Theorem 1.1 and §3): the derivation that lim c_δ(λ)=0 implies the classical Weyl asymptotic requires fully detailed error estimates (including dependence on the constants in the quantitative bounds for c_δ(λ)) so that the remainder can be shown to be o(λ^{n/2}) or the appropriate volume term; the current outline does not permit verification of these estimates.

Authors: We acknowledge that the current write-up of the sufficiency argument supplies an outline rather than fully expanded error estimates. In the revision we will expand the proof of Theorem 1.1 (and the supporting arguments in §3) to include complete derivations of all error terms, with explicit tracking of the dependence on the constants appearing in the quantitative bounds for c_δ(λ). This will establish that the remainder is o of the main volume term whenever lim c_δ(λ)=0. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via new invariant

full rationale

The paper defines a new intrinsic quantity c_δ(λ) that encodes the balance of geometry, potential growth, and oscillation without presupposing the Weyl law. The central result states that lim λ→∞ c_δ(λ)=0 is sufficient for the asymptotic, with necessity shown by explicit counterexamples where the limit fails and the law does not hold. No equation reduces the conclusion to a tautology, no parameter is fitted and then relabeled as a prediction, and no load-bearing step relies on a self-citation chain or imported uniqueness theorem; the argument is externally falsifiable through the counterexamples and does not collapse to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The claim rests on the standard setup of spectral theory for Schrödinger operators on complete Riemannian manifolds together with the introduction of the new quantity c_δ(λ) whose limit controls the asymptotic.

axioms (1)

domain assumption The Schrödinger operator H = Δ + V is essentially self-adjoint on the complete Riemannian manifold.
Standard assumption required to define the spectrum and eigenvalue counting function.

invented entities (1)

c_δ(λ) no independent evidence
purpose: Encodes the precise balance between the geometry of the manifold, the growth of V, and the oscillation scale of V.
This quantity is defined by the authors as the central object of the criterion.

pith-pipeline@v0.9.0 · 5444 in / 1370 out tokens · 58914 ms · 2026-05-11T02:08:25.271172+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear
We prove that the Weyl asymptotic holds provided lim λ→∞ c_δ(λ)=0. ... c_δ(λ) := 1 + K_δ(λ) b_δ(λ)^2 / (a(λ) b_δ(λ)^2)
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
The classical Weyl law for Schrödinger operators on complete Riemannian manifolds

Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages

[1]

R. A. Adams and J. Fournier.Sobolev spaces, volume 140. Elsevier, 2003

work page 2003
[2]

S. Agmon. Lectures on exponential decay of solutions of second-order elliptic equa- tions: Bounds on eigenfunctions of N-body Schrödinger operations.(MN-29). Princeton University Press, 2014

work page 2014
[3]

Bonthonneau

Y. Bonthonneau. Weyl laws for manifolds with hyperbolic cusps. arXiv preprint arXiv:1512.05794

work page arXiv
[4]

Braverman

M. Braverman. The semi-classical Weyl law on complete manifolds. arXiv preprint arXiv:2505.12157

work page arXiv
[5]

Y. Canzani. Analysis on manifolds via the Laplacian. Lecture Notes available at: http://www. math. harvard. edu/canzani/docs/Laplacian. pdf, 2013

work page 2013
[6]

Chitour, D

Y. Chitour, D. Prandi, and L. Rizzi. Weyl’s law for singular Riemannian manifolds. Journal de Mathématiques Pures et Appliquées, 181:113–151, 2024

work page 2024
[7]

Coriasco and M

S. Coriasco and M. Doll. Weyl law on asymptotically Euclidean manifolds.Annales Henri Poincaré, 22:447–486, 2021

work page 2021
[8]

Dai and J

X. Dai and J. Yan. Weyl law for Schrödinger operators on noncompact manifolds, heat kernel, and Karamata-Hardy-Littlewood theorem.arXiv preprint arXiv: 2504.15551

work page arXiv
[9]

Dai and J

X. Dai and J. Yan. Witten deformation for noncompact manifolds with bounded geom- etry. Journal of the Institute of Mathematics of Jussieu, 22(2):643–680, 2023. 34

work page 2023
[10]

J. S. De Wet and F. Mandl. On the asymptotic distribution of eigenvalues.Proceedings of the Royal Society of London., 200(1063):572–580, 1950

work page 1950
[11]

V. I. Feigin. Asymptotic distribution of eigenvalues for hypoelliptic systems in Rn. Matematicheskii Sbornik, 141(4):594–614, 1976

work page 1976
[12]

Fleckinger

J. Fleckinger. Estimate of the number of eigenvalues for an operator of Schrödinger type. Proceedings of the Royal Society of Edinburgh, 89(3-4):355–361, 1981

work page 1981
[13]

Grigor’yan

A. Grigor’yan. Heat kernel upper bounds on a complete non-compact manifold.Revista Matemática Iberoamericana, 10(2):395–452, 1994

work page 1994
[14]

Grigoryan.Heat kernel and analysis on manifolds, volume 47

A. Grigoryan.Heat kernel and analysis on manifolds, volume 47. American Mathemat- ical Soc., 2009

work page 2009
[15]

Grigor’yan

A. Grigor’yan. Gaussian upper bounds for the heat kernel on arbitrary manifolds.J. Diff. Geom, 45(1):33–52, 1997

work page 1997
[16]

Inahama and S.-I

Y. Inahama and S.-I. Shirai. Eigenvalue asymptotics for the Schrödinger operators on the hyperbolic plane.Journal of Functional Analysis, 211(2):424–456, 2004

work page 2004
[17]

Inahama and S.-I

Y. Inahama and S.-I. Shirai. Eigenvalue asymptotics for the Schrödinger operators on the real and the complex hyperbolic spaces. Journal de Mathématiques Pures et Appliquées, 83(5):589–627, 2004

work page 2004
[18]

J. Jost. Riemannian geometry and geometric analysis. Springer, 2005

work page 2005
[19]

Levendorski˘i

S.Z. Levendorski˘i. Spectral asymptotics with a remainder estimate for Schrödinger operators with slowly growing potentials.Proceedings of the Royal Society of Edinburgh, 126(4):829–836, 1996

work page 1996
[20]

Lin and X

F. Lin and X. Yang.Geometric measure theory. An introduction. International Press, 2003

work page 2003
[21]

Maniccia and P

L. Maniccia and P. Panarese. Eigenvalue asymptotics for a class of md-ellipticψdo’s on manifolds with cylindrical exits.Annali di Matematica Pura ed Applicata, 181(3):1618– 1891, 2002

work page 2002
[22]

Moroianu

S. Moroianu. Weyl laws on open manifolds.Mathematische Annalen, 340(1):1–21, 2008

work page 2008
[23]

I. Oleinik. On a connection between classical and quantum-mechanical completeness of the potential at infinity on a complete Riemannian manifold.Math. Notes., 55(4):380– 386, 1994

work page 1994
[24]

F. S. Rofe-Beketov. Conditions for the self-adjointness of the Schrödinger operator. Mathematical notes of the Academy of Sciences of the USSR, 8(6):888–894, 1970

work page 1970
[25]

G. V. Rozenbljum. Asymptotics of the eigenvalues of the Schrödinger operator.Math- ematics of the USSR-Sbornik, 22(3):349, 1974

work page 1974
[26]

Tachizawa

K. Tachizawa. Eigenvalue asymptotics of Schrödinger operators with only discrete spec- trum. Publications of the Research Institute for Mathematical Sciences, 28(6):943–981, 1992

work page 1992
[27]

N. Th. Varopoulos. Hardy-Littlewood theory for semigroups. Journal of functional analysis, 63(2):240–260, 1985

work page 1985
[28]

J. Wang. Global heat kernel estimates.Pacific Journal of Mathematics, 178(2):377–398, 1997

work page 1997
[29]

H. Weyl. Über die asymptotische verteilung der eigenwerte. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse , 1911:110–117, 1911. 35

work page 1911