Weyl laws for exponentially small singular values of the $\overline{\partial}$ operator

Johannes Sj\"ostrand; Martin Vogel; Michael Hitrik

REVIEW 1 major objections 6 minor 13 references

Weyl laws for exponentially small singular values of the $\overline{\partial}$ operator

T0 review · 1 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The number of exponentially small singular values of the semiclassical $\bar{\partial}$ operator is governed by a Weyl law whose constant is the integral of $\Delta\varphi$ over the optimal weight's contact set.

desk verdict Solid semiclassical Weyl law for exponentially small singular values of the ∂-bar operator on the torus, with an abstract that overclaims to compact Riemann surfaces. read the letter →

arxiv 2505.07292 v3 pith:CU3PNKK3 submitted 2025-05-12 math.SP math.APmath.CV

Michael Hitrik , Johannes Sjöstrand , Martin Vogel This is my paper · ORCID

classification math.SPmath.APmath.CV MSC 35P2047B0635R3558J50

keywords singularvaluesWeyllawsemiclassicalanalysisbar-partialoperatorexponentialweightsdoubleobstacleproblemfreeboundaryquantumtunneling

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper establishes an asymptotic Weyl law for the exponentially small singular values of the semiclassical $\bar{\partial}$ operator conjugated by an exponential weight on a two-dimensional torus. Under the condition that the Laplacian $\Delta\varphi$ changes sign transversally across its zero set, the number of singular values in $[0,e^{-\tau/h}]$ equals $\frac{1}{2\pi h}\int_{M_+(\psi)}\Delta\varphi\,L(dz)+o(1/h)$, where $\psi$ is the unique solution of a double obstacle problem that squeezes an optimal weight between $\varphi$ and $\varphi-\tau$. The leading constant is the symplectic area of the region where the optimal weight touches $\varphi$, which is also the region where the singular states concentrate. For small $\tau$, the paper refines the constant further, writing the correction as $\tau^{2/3}$ times an integral along the curve where $\Delta\varphi$ vanishes. The result turns a fragile non-self-adjoint spectral count into a deterministic free-boundary computation.

What carries the argument

The load-bearing device is the optimal weight $\psi$, obtained as the unique solution of the double obstacle problem (1.11): $\varphi-\tau\le\psi\le\varphi$ almost everywhere, with $\Delta\psi\ge 0$ above the lower obstacle and $\Delta\psi\le 0$ below the upper obstacle. Because $\psi$ is simultaneously an upper and a lower bound weight, the singular-value count is squeezed between two trace-class computations: one uses weighted $L^2$ estimates to show singular states decay away from $M_+(\psi)$, and the other uses coherent states localized in the interior of $M_+(\psi)$ whose spectral projection reproduces them. The trace of the resulting asymptotic projection onto holomorphic sections gives the factor $\frac{1}{2\pi h}\int\Delta\varphi$. In the thin-band regime the band where $\psi$ is harmonic is analyzed in coordinates around the curve $\gamma$, using an $\varepsilon$-pseudodifferential calculus with operator-valued symbols in the normal variable, which constructs the boundary functions $f_\pm$ and produces the $\tau^{2/3}$ correction. The free-boundary part, proved on general compact manifolds, establishes existence, $C^{1,1}$ regularity, positive measure of both contact sets, and zero volume of the free boundaries, the last fact being what lets the upper and lower bounds meet.

What would settle it

Take a smooth weight on the torus with $\Delta\varphi(y)=y^2$ near $y=0$ so that the non-degeneracy condition fails, compute $N([0,e^{-\tau/h}])$ numerically for small $h$, and compare with $\frac{1}{2\pi h}\int_{M_+(\psi)}\Delta\varphi$; a persistent discrepancy of order $1/h$ would show the condition is load-bearing, while agreement would show it is unnecessary.

Watch

Extended reading notes

Core claim

The central claim is Theorem 4: for a nonconstant smooth weight $\varphi$ on the torus with $0<\tau<\max\varphi-\min\varphi$ and $d\Delta\varphi\neq 0$ on $(\Delta\varphi)^{-1}(0)$, the counting function satisfies $$N([0,$e^{{-\tau/h}}$])=\frac{1}{2\pi h}\int_{M_+(\psi)}\$\Delta$\varphi(z)\,L(dz)+o(1/h),\quad h\to 0^+,$$ where $\psi$ is the unique $C^{1,1}$ solution of the double obstacle problem (1.11) and $M_+(\psi)=\{\psi=\varphi\}$ is its upper contact set. The same $\psi$ is both an upper and a lower bound weight, so the count is trapped between two estimates with the same constant. The paper also proves Theorem 5, which for $\tau=\varepsilon^3\hat\tau$ and connected zero set $\gamma$ gives the sharper expansion $\int_{M_+(\psi)}\Delta\varphi=\int_{M_+}\Delta\varphi-\tau^{2/3}\frac{1}{2}\left(\frac{3}{2}\right)^{2/3}\int_\gamma(\partial_n\Delta\varphi)^{1/3}dx+O(\varepsilon^3)$, making explicit how the sign-change curve controls the subleading term.

Load-bearing premise

Everything depends on the Laplacian of the weight crossing zero cleanly wherever it is zero: only then do the free boundaries of the optimal weight have zero volume and the small-$\tau$ analysis reduces to a single curve.

Editorial extensions

If this is right

The count of exponentially small singular values is of order $1/h$, and its leading constant depends only on the Laplacian of the weight on the contact set, not on the full symbol of the operator.
Because the same optimal weight serves as both upper and lower bound weight, the asymptotic is two-sided: the true count is trapped between two trace computations with the same leading constant.
For small exponential decay rate $\tau$, the subleading correction to the Weyl constant scales like $\tau^{2/3}$ and is determined by the integral of $(\partial_n\Delta\varphi)^{1/3}$ along the curve where $\Delta\varphi$ changes sign.
For the model weight $\varphi(y)=\sin y$, the general theorem recovers the separation-of-variables result, confirming that the free-boundary mechanism is consistent with the known case.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The torus is used for convenience rather than necessity; a natural testable extension is to a general compact Riemann surface, where the same double obstacle problem should yield the same Weyl constant once a suitable strictly subharmonic exhaust function is available.
The $\tau^{2/3}$ exponent suggests a broader tunneling rule: when the Laplacian of a weight changes sign transversally, the subleading tunneling count is organized by $\int_\gamma |\nabla\Delta\varphi|^{1/3}$, so one could look for the same exponent in magnetic Pauli or Schrödinger tunneling problems.
The theorem suggests a numerical pipeline: solve the double obstacle problem once, evaluate the contact integral, and predict the exponentially small spectrum; deviations at finite $h$ would directly measure the $o(1/h)$ remainder.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

You should know two things about this paper. First, it is a serious, technically substantial piece of work that proves the first general Weyl asymptotics for exponentially small singular values of the semiclassical ∂-bar operator on the torus, going well beyond the earlier sin(y) separation-of-variables example. Second, the metadata abstract says "compact Riemann surface" but the body proves everything only on the two-dimensional torus; the authors themselves say the Riemann surface case is coming. That is a real overclaim, but it is cosmetic and easily fixed.

The main result, Theorem 4, holds up on my reading. The upper and lower bounds are matched through Carleman estimates and the double obstacle problem, and the optimal weight is constructed under the explicit non-degeneracy assumption d∆φ ≠ 0 along the zero set of ∆φ. The sharp small-τ correction in Theorem 5, with its τ^{2/3} term, is a genuine new result. The proof structure is coherent: Section 3 does the trace computations, Section 4 does the thin-band analysis, Section 5 gives a self-contained treatment of the double obstacle problem on compact manifolds. The C^{1,1} regularity, the measure-zero free boundaries, and the positivity of the contact sets are all proved in a way that uses the non-degeneracy assumption exactly where it is needed. I found no circularity and no fitted parameters.

The stress-test concern about hypothesis (1.10) does not land as a flaw on my reading. Yes, the theorem relies on that assumption, and without it the free-boundary measure-zero property and the strict positivity on the contact set are not proved. But the paper states (1.10) as an explicit hypothesis and never pretends to cover the degenerate case. That is a limitation, not a hidden error. A more serious soft spot is the abstract overclaim: the arXiv metadata promises compact Riemann surfaces, while the body only proves the torus case. The authors mention in the introduction that the extension is forthcoming, so the correct fix is to align the abstract with the content.

The proofs are long and intricate; I did not independently verify every estimate, but the structure is sound and the cited tools (Bergman kernel asymptotics, two-scale pseudodifferential calculus, penalization for obstacle problems) are appropriate. The paper deserves a serious referee. For a spectral theorist or semiclassical analyst, this is worth reading and citing. My recommendation: send it to peer review, with the abstract corrected.

Who is this for? People working on non-self-adjoint operators, semiclassical analysis, Bergman kernels, or tunneling. If you read it, focus on Sections 3 and 5; Section 4 is heavier but self-contained.

Referee Report

1 major / 6 minor

Summary. The paper studies the semiclassical Cauchy-Riemann operator P = h∂_z + ∂_z φ on exponentially weighted L^2 spaces over the two-dimensional torus, focusing on the number N([0,e^{-τ/h}]) of singular values in an exponentially small interval. Theorems 1 and 2 provide upper and lower bounds on this counting function in terms of auxiliary upper and lower bound weights satisfying suitable (super/sub)harmonicity conditions. Under the non-degeneracy hypothesis dΔφ≠0 on (Δφ)^{-1}(0), Theorem 3 constructs a unique C^{1,1} solution of a double obstacle problem, whose contact sets have positive measure, lie in the regions ±Δφ>0, and have measure-zero free boundaries. Theorem 4 derives a Weyl law with leading term (2πh)^{-1}∫_{M+(ψ)}Δφ dL. Theorem 5 gives a refined expansion when τ=ε^3 bτ is small: the leading term is expressed through the positive region of Δφ minus a thin band around the zero curve, with an explicit τ^{2/3} correction. The proofs combine Hörmander estimates, asymptotic Bergman projections, trace computations, a penalization argument for the obstacle problem, and a two-scale pseudodifferential analysis of the Dirichlet problem in a thin band.

Significance. If the results are correct, Theorem 4 is a genuinely parameter-free Weyl law for exponentially small singular values of a non-self-adjoint operator: the leading constant is determined by the obstacle problem and contains no fitted parameter. The paper is also valuable for its self-contained treatment of double obstacle problems on compact manifolds without boundary, including existence, uniqueness, C^{1,1} regularity, and porosity of the free boundary. The thin-band analysis in Section 4 is a technically substantial construction of near-optimal weights. The upper and lower bound framework of Theorems 1 and 2 is clean and likely to be reusable. The main concern is a gap in the final step of the proof of Theorem 5, discussed below; Theorem 4 itself appears sound under its stated hypotheses.

major comments (1)

[Section 4, equations (4.117)–(4.118)] The final step of the proof of Theorem 5 is not justified as written. Proposition 4.6 and (4.117) are estimates for N([0,e^{-eτ/h}]) with eτ=τ−Cε^N<τ. The text then shows that Vol(Ω_f^+(eτ)) can be replaced by Vol(Ω_f^+(τ)) up to O(ε^{N-1}), and immediately writes 'we get N([0,e^{-τ/h}]) = ...' in (4.118). This replaces the decay rate in the counting interval without argument. Since e^{-eτ/h}=e^{-τ/h}e^{Cε^N/h}, the larger interval is exponentially wider, and a lower bound for the larger interval does not imply the same lower bound for the smaller interval. Moreover, on Ω− the function ψlb defined in (4.92) satisfies ψlb=φ−eτ>φ−τ and Δψlb=Δφ<0, so ψlb is not a lower bound weight for the original τ, and the Lipschitz clause of Theorem 2 cannot be invoked to transfer the lower bound to the interval [0,e^{-τ/h}]. The authors must either prove that the shell (e^{-τ/h}, e^{-eτ/h}] contains only o(1/h) singular values, construct lower bound weights directly for τ, or explicitly restate Theorem 5 with the decay rate eτ and verify that the leading term is unchanged up to the stated error.

minor comments (6)

[Abstract and Section 1.1] The abstract speaks of a compact Riemann surface, while the main body works on the two-dimensional torus and Section 1.6 explicitly lists the extension to general compact Riemann surfaces as future work. The abstract should be aligned with the actual scope.
[Theorem 1 statement] There is a missing closing parenthesis in the displayed statement: 'N([0,e−τ/h]' should read 'N([0,e^{-τ/h}])'.
[Section 1.4, Theorem 5] In the displayed formula (1.13), the notation 'O(ε^{(N+1)/2})' and the subsequent 'o(1)' are not clarified regarding uniformity in ε; the proof later states that the O-term is uniform and the o(1) is not. This distinction should be stated in the theorem itself.
[Section 4, around (4.3)] The phrase '∂y∆φ> 0, when y = 0' is slightly ambiguous because Δφ is a function on the torus; it would be clearer to write ∂_y(Δφ)>0 on γ, where y is the normal coordinate in the almost holomorphic extension.
[Section 5, Proposition 5.6] The constants C0 and C in the statement are said to depend only on M and g, but the proof also uses ∥Δφ−∥_{L∞} and the C² norm of φ−; the statement should mention this dependence explicitly.
[Section 4, final paragraph] The sentence 'writing τ instead of eτ' is ambiguous: if the theorem's τ is redefined to be eτ at the end of the proof, the hypotheses τ=ε^3 bτ with bτ≍1 and the volume formula (1.14) still hold with bτ changed by O(ε^{N-3}), but this change of notation should be stated explicitly rather than left implicit.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Weyl count is compared against a functional of the obstacle solution determined only by φ and τ; no fitted parameter or self-citation chain forces the result.

full rationale

The derivation chain is self-contained and non-circular. Theorems 1 and 2 give upper and lower bounds on the singular-value counting function in terms of arbitrary upper and lower bound weights, with the comparison constant being the symplectic volume of the contact set M+(ψ) on the Lagrangian Λφ. These bounds are proved from Hörmander–Carleman estimates, quasimode constructions, and trace-class asymptotics for Bergman projections; no property of the counting function itself is inserted into the weight ψ. The optimal weight ψ is then obtained in Section 5 as the unique solution of the double obstacle problem (1.11), in which the obstacles are φ and φ−τ only. Thus ψ, its contact sets, and the integral ∫_{M+(ψ)} Δφ are determined entirely by the input φ and the parameter τ, not by the singular values being counted. The non-degeneracy hypothesis (1.10) is explicitly assumed and used to prove the measure-zero free-boundary property and the strict containment of the contact sets; weakening it would invalidate the proof, but that is a limitation of the hypothesis, not a circular dependence. The cited works by the same authors, [SjVo24] and [HiSt22], serve as a model-case remark and as technical tools for Bergman projections, respectively; neither supplies the target Weyl law nor is the reasoning reduced to a self-citation. In particular, the paper's Theorem 4 is not an input to Theorem 3 or to the obstacle-problem construction, and the leading asymptotics are not fitted to the counting function. Therefore no circular step is present.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard semiclassical and elliptic tools, plus the explicit geometric non-degeneracy assumption on the input weight φ. There are no fitted free parameters: the optimal weight ψ is the unique solution of the double obstacle problem determined by φ and τ. No new physical entities are introduced.

assumptions (5)

standard math The asymptotic Bergman projection theorem of [BeBrSj08, RoSjVu20, HiSt22] for weights with Δφ > 0.
Used in Section 3.1 to relate the singular states to Toeplitz operators; these are published results with attribution.
standard math Hörmander's L² estimates for the ∂ operator with subharmonic weights, in the form of Propositions 2.1 and 3.3.
Classical results from [Hö90, Hö94]; the paper extends them to the torus using a Stein-manifold embedding argument.
standard math Semiclassical Weyl law for the elliptic self-adjoint operator P*P, giving N = O(h^{-2}).
Used in Section 3.1, equation (3.46); standard spectral asymptotics cited to [DiSj99, Chapter 9].
standard math Maximum principle and Hopf's lemma on manifolds.
Used in Sections 5.2 and 5.3 to establish properties of the contact sets and free boundaries; classical elliptic theory.
domain assumption φ is non-constant, 0 < τ < max φ - min φ, and dΔφ ≠ 0 along (Δφ)^{-1}(0); for Theorem 5, (Δφ)^{-1}(0) is connected.
These are the standing hypotheses (1.9), (1.10), (1.12) of the main theorems; they define the regime of the results.

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Pith. "Pith review of Weyl laws for exponentially small singular values of the $\overline{\partial}$ operator." pith.science (2026). https://pith.science/paper/CU3PNKK3

@misc{pith2026250507292,
  author       = {Pith},
  title        = {Pith review of: Weyl laws for exponentially small singular values of the $\overline\partial$ operator},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CU3PNKK3}},
  note         = {Machine review of arXiv:2505.07292}
}

abstract

We study the number of exponentially small singular values of the semiclassical $\overline{\partial}$ operator on exponentially weighted $L^2$ spaces on a compact Riemann surface. Accurate upper and lower bounds on the number of such singular values are established in terms of auxiliary notions of upper and lower bound weights. Assuming that the Laplacian of the exponential weight changes sign along a curve, we construct optimal such weights by solving a free boundary problem, which yields Weyl asymptotics for the counting function of the singular values in an interval of the form $[0,\mathrm{e}^{-\tau/h}]$, for $\tau>0$ smaller than the oscillation of the weight. We also provide a precise description of the leading term in the Weyl asymptotics, in the regime of small $\tau > 0$.

Figures

Figures reproduced from arXiv: 2505.07292 by the authors.

**Figure 1.** An illustration of the penalty function gε(t) above (5.8). Remark. The method of penalization is a by now classical approach to proving regularity of obstacle problems going back at least to [BrSt68, LeSt69], see for instance [Ro87, PeShUr12] and the references therein. Here, in (5.10), we used the penalization function fε which (up to a minor modification) was introduced in [LePa23]. What is novel here is that sin… view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

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Fournais, L

[FoMoRa25] S. Fournais, L. Morin, and N. Raymond, Purely magnetic tunneling between radial mag- netic fields, Rev. Mat. Iberoam., to appear (2025). [Fr72] J. Frehse, On the regularity of the solution of a second order variational inequality , Boll. Un. Mat. Ital. 6 (1972), 312–315. [GeMaSj91] C. G´ erard, A. Martinez, and J. Sj¨ ostrand, A mathematical ap...

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Simon, Semiclassical analysis of low lying eigenvalues

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Nonnenmacher and M

[NoVo21] S. Nonnenmacher and M. Vogel, Local eigenvalue statistics of one dimensional random nonselfadjoint pseudodifferential operators, J. Eur. Math. Soc. 23 (2021), 1521–1612. [Ol23] I. Oltman, A probabilistic Weyl-law for Berezin-Toeplitz operators, J. Spectr. Theory 13 (2023), 727–754. [PeShUr12] A. Petrosyan, H. Shahgholian, and N. Uraltseva, Regula...

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[BaTrRaSt21] J.-M. Barbaroux, L. Le Treust, N. Raymond, and E. Stockmeyer, On the semiclassical spectrum of the Dirichlet-Pauli operator , J. Eur. Math. Soc. 23 (2021), 3279—3321. [BcOlVo24] S. Becker, I. Oltman, and M. Vogel, Absence of small magic angles for disordered tunnel- ing potentials in twisted bilayer graphene , to appear in Contemp. Math., htt...

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Rouby, J

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Helffer and A

[HeKa24] B. Helffer and A. Kachmar, Quantum tunneling in deep potential wells and strong magnetic field revisited, Pure Appl. Anal. 6 (2024), 319–352. [HeKaSu24] B. Helffer, A. Kachmar, and M. P. Sundqvist, Flux and symmetry effects on quantum tunneling, Math. Ann. 390 (2024), 5185–5234. [HeKoSu19] B. Helffer, H. Kovaric, and M. P. Sundqvist, On the semic...

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[7] [1995]

Lee and J

[LePa23] K.-A. Lee and J. Park, The regularity theory for the double obstacle problem for fully non- linear operator, Nonlinear Anal. 235 (2023), Paper No. 113332, 24 pp. [LeSt69] H. Lewy and G. Stampacchia, On the regularity of the solution of a variational inequality , Comm. Pure Apl. Math. 22 (1969), 153-188. [Ma02] A. Martinez, An Introduction to semi...

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Ekholm, H

[EkKoPo16] T. Ekholm, H. Kovarik, and F. Portmann, Estimates for the lowest eigenvalue of mag- netic Laplacians, J. Math. Anal. Appl. 439 (2016), 330–346. [EmTr05] M. Embree and L. N. Trefethen, Spectra and pseudospectra. The behavior of nonnormal matrices and operators, Princeton University Press, 2005 [Es11] G. Eskin, Lectures on linear partial differen...

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