Pith. sign in

REVIEW 2 major objections 5 minor 35 references

The paper claims that, after removing constant adjoint modes, the first Gribov horizon on a torus is located exactly when −1 enters the spectrum of a dimensionless Birman-Schwinger operator K_A, a criterion that preserves inertia rather tha

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-02 02:10 UTC pith:HSZ6UFMT

load-bearing objection Exact congruence criterion is solid; the advertised analytic scaffolding has gaps that should be fixed or relabeled before publication. the 2 major comments →

arxiv 2607.14425 v2 pith:HSZ6UFMT submitted 2026-07-15 hep-th

Birman-Schwinger Formulation of the Faddeev-Popov Zero-Mode Problem

Daniel G. Tedesco This is my paper
classification hep-th MSC 81T1335P2047B10 PACS 11.15.-q
keywords Birman-Schwinger operatorFaddeev-Popov zero modesGribov horizonLandau gaugeCwikel estimatesweak Schatten idealsghost propagatorMathieu recurrence
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper recasts the Landau-gauge Faddeev-Popov zero-mode problem on a torus as a spectral question about a dimensionless self-adjoint operator K_A. After removing constant adjoint modes, a zero mode of the Faddeev-Popov operator occurs exactly when −1 enters the spectrum of K_A; the factorization is a congruence, so it preserves signs and nullity rather than eigenvalues. This makes the first horizon crossing a fixed, volume-independent spectral threshold. The paper also proves quantitative compactness estimates for K_A (weak Schatten ideal S^{d,∞}, with singular-value decay n^{−1/d}) at minimal regularity A∈L^d for d≥3, handles d=2 via the Lorentz class L^{2,1}, and expresses the ghost dressing function as the diagonal resolvent of 1+K_A. A periodic SU(2) background reduces the condition to a Mathieu recurrence, giving exact critical amplitudes and volume laws.

Core claim

On T^d, with constant adjoint scalar modes removed, the paper establishes the exact equivalence 0∈spec(M) ⇔ −1∈spec(K_A), via the congruence M = H_0^{1/2}(1+K_A)H_0^{1/2}. Because this congruence preserves inertia (n_± and n_0), horizon crossings are located by the first appearance of −1 in the spectrum of K_A, without equating the numerical spectra of M and 1+K_A. The companion analytic claim is Theorem 1: for d≥3 and transverse A∈L^d, K_A lies in S^{d,∞} with s_n(K_A) ≤ C_d κ_N g ∥A∥_{L^d} n^{−1/d}. The two-dimensional endpoint is characterized on both sides: the plain L^2 estimate fails, while the sharp result holds in L^{2,1}. The same formulation gives the fixed-background ghost dressin

What carries the argument

The central object is the Birman-Schwinger operator K_A = H_0^{−1/2} V_A H_0^{−1/2}, where V_A = −g ad(A_i)∂_i is the first-order background perturbation. K_A is dimensionless and self-adjoint when A is transverse. The load-bearing identity is the congruence M = H_0^{1/2}(1+K_A)H_0^{1/2}, which converts the zero-mode equation into a spectral condition on a bounded operator and preserves the positive, negative, and null dimensions of the quadratic form. Quantitative control comes from reducing K_A to a matrix-valued Cwikel-type operator and applying periodic Cwikel-type estimates at coefficient regularity A∈L^d, with the L^{2,1} endpoint at d=2.

Load-bearing premise

The quantitative singular-value bounds rest on transferring a local scalar Cwikel-type estimate to periodic matrix-valued multipliers 'without change' and on a quoted Lorentz-space endpoint whose per-layer estimate is asserted rather than derived; if either fails, the bounds lose support, while the elementary congruence criterion would survive.

What would settle it

Compute s_n(K_A) on T^d for a periodic transverse matrix-valued A∈L^d with d≥3 and check the predicted n^{−1/d} decay; slower decay or noncompactness would refute Theorem 1. For d=2, take f∈L^{2,1} with finite Lorentz norm, for example a smoothed version of the annulus family with ||f||_{L^{2,1}} finite, and test whether M_fΛ has finite S^{2,∞} norm; if the norm diverges, Proposition 2 fails.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Horizon crossings are governed by a fixed threshold: the first Gribov horizon occurs exactly when −1 becomes an eigenvalue of K_A, independent of volume normalization.
  • The exact ghost dressing at fixed background is the diagonal resolvent d(k)=⟨k|(1+K_A)^{−1}|k⟩; along amplitude scaling the dressing is meromorphic with simple poles whose residues are spectral overlaps of the critical mode.
  • For d≥3, K_A is compact with quantified singular-value decay n^{−1/d} controlled by ∥A∥_{L^d}, so the analytic framework holds at critical regularity.
  • In d=2 the plain L^2 estimate fails; the sharp class is L^{2,1}, which still contains every bounded and smooth background.
  • The periodic SU(2) family yields a solvable sector: the threshold satisfies the Mathieu condition a_0(2a_crit)=−4, the no-pole/Born estimate gives a_np²=2(Q²+m²), and the critical amplitude falls as 1/L with action scaling L^{d−4}.

Where Pith is reading between the lines

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

  • A lattice test should diagonalize the normalized operator directly: equation (42) implies that the observed drift of Faddeev-Popov eigenvalues toward zero could arise from the free infrared factor H_0^{−1} rather than from K_A approaching −1, so the two mechanisms are numerically distinguishable.
  • The linearity of V_A in A gives K_{aA}=aK_A with fixed eigenvectors; this suggests using the residue R(k)=|⟨k|η_min⟩|² of the ghost dressing as a direct probe of which momenta feel an approaching horizon.
  • The entropy-action competition in the periodic model becomes volume-independent at d=4; one could test whether this marginality persists for multi-mode or SU(3) backgrounds as a heuristic for the special role of four spacetime dimensions.
  • Because the congruence preserves only inertia, not spectra, approximate methods such as Born/no-pole and Feshbach truncations can be compared to the exact criterion only through the extremal eigenvalue of K_A; this frames a systematic truncation hierarchy in the solvable sector.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper develops a Birman-Schwinger formulation of the Landau-gauge Faddeev-Popov operator on a torus after removal of constant adjoint modes. The central object is K_A=H_0^{-1/2}V_A H_0^{-1/2}, whose spectral value -1 is shown, via a congruence rather than a similarity, to be equivalent to a Faddeev-Popov zero mode. The paper claims a quantitative weak-Schatten bound for K_A in d>=3, a borderline positive result for d=2 in L^{2,1}, an expression for the fixed-background ghost dressing as the diagonal resolvent of (1+K_A)^{-1}, a solvable periodic SU(2) background reducing to a Mathieu recurrence, and volume scaling laws for the critical amplitude and the number of near-critical modes. The exact congruence criterion and the explicit solvable sector are cleanly presented, but the advertised analytic scaffolding for the quantitative bounds relies on two unproved technical transfers.

Significance. If the quantitative estimates are fully established, the paper would provide a useful normalized spectral criterion for the first Gribov horizon and a concrete bridge to the ghost dressing function. The exact congruence (3)/(11), the inertia identity (12), and the explicit Mathieu characterization (47) are valuable and independently checkable. The paper is also appropriately cautious in separating fixed-background statements from ensemble statements, and it explicitly labels the entropy model as approximate. The main risk is not the central equivalence but the spectral-theoretic scaffolding: Theorem 1 and Proposition 2 are presented as settled while their proofs contain gaps that are load-bearing for the n^{-1/d} and n^{-1/2} decay claims.

major comments (2)
  1. [§4.1, Theorem 1 (Eq. (22))] Step four of §4.1 asserts that the periodic Cwikel–Birman–Solomyak theorem transfers from R^d to T^d 'without change' for matrix-valued multipliers, and that this delivers (21). This is the step that converts the scalar L^d estimate into the S^{d,\infty} bound for K_A, so it is load-bearing for (22). No argument or precise theorem statement is supplied for the torus transfer, nor for the matrix-valued case beyond the FLS reduction in (20). Please provide a self-contained proof or an exact statement with reference for \|M_F\Lambda\|_{S^{d,\infty}}\le C_d\||F|\|_{L^d} on T^d, or explicitly mark Theorem 1 as conditional.
  2. [§4.2, Proposition 2 and Eq. (28)] The endpoint L^{2,1} estimate is presented as proved, but the proof is only a sketch. The per-layer bound (28) uses p_j=2+1/\log(2^j/\sigma_j), but no specification is given for the regime where \log_+(2^j/\sigma_j) is nonpositive; the dyadic/Yano summation is delegated to refs. [25,26]; and the notation N_{M_{f_j}\Lambda}(\sigma_j) is undefined. Since the Conclusion explicitly relies on Proposition 2 as settled analytic scaffolding, this is a genuine gap. The proof should be completed, or the proposition should be stated with an exact theorem-and-version citation covering this form. As written, the endpoint is asserted rather than demonstrated.
minor comments (5)
  1. [Abstract] 'The resulting normalized operator, is dimensionless' contains a stray comma after 'operator'.
  2. [§4.2, Eq. (28)] Define the counting function N_T(\sigma)=\#\{n: s_n(T)>\sigma\} before first use.
  3. [§5, Fig. 1] The fitted log-log slopes (-1.99, 1.00, 0.76, 0.00) are quoted without error bars or grid-convergence analysis. Since these figures are used as verifications of (33) and (55), please include resolution/uncertainty information or state explicitly that they are illustrative.
  4. [§7, Eq. (47)] State the Mathieu-equation convention used for a_0(q), since conventions for characteristic values differ and 'a' is also used for the background amplitude.
  5. [Conclusion] The phrase 'settled in every dimension except the two-dimensional endpoint' is confusing because Proposition 2 addresses the two-dimensional endpoint. Rephrase to distinguish the endpoint S^{2,\infty} estimate from the unproved endpoint Weyl law in (33).

Circularity Check

0 steps flagged

No circular derivation; the Birman-Schwinger criterion is a self-contained identity, while the cited analytic estimates are external and the numerical fits are post-hoc checks.

full rationale

The central claim, Prop. 1 / eq. (11), follows from the exact congruence M = H0^{1/2}(1+K_A)H0^{1/2} with K_A = H0^{-1/2} V_A H0^{-1/2}; this is an algebraic identity, not an assumption containing the conclusion, and the inertia/nullity transfer (12) is standard Sylvester-law form theory. No parameter is fitted and then called a prediction: the exponents in Figs. 1 and 3 (e.g., -1.99 vs -2, and 1.00/0.76/0.00 vs d(4-d)/4) are checked against independently derived formulas. The paper contains no self-citations by the author; all load-bearing analytic input (Cwikel, Birman-Solomyak, Frank-Lieb-Seiringer, Orland-Semenoff) is external. The genuine limitations are analytic, not circular: §4.1 step four asserts the periodic CBS transfer 'without change' for matrix-valued multipliers; Prop. 2 delegates the endpoint Yano-type summation to Solomyak [25,26] with the per-layer estimate (28) sketched rather than fully derived; and the paper itself disclaims a proved endpoint Weyl law for (33) ('not, at this stage, a proved endpoint Weyl law for coefficients of critical L^2 regularity') and labels (55) 'an entropy estimate, stated as an approximation and not as a theorem.' If those external estimates fail, Theorem 1 and Prop. 2 lose support, but that is a correctness/rigor risk about borrowed theorems, not a reduction of a prediction to its own input. The congruence criterion (11) is elementary and survives independently of those estimates. Score 1 reflects the mild structural oversell in the Conclusion of calling Prop. 2 'settled' when its proof is delegated, without any circularity.

Axiom & Free-Parameter Ledger

2 free parameters · 6 axioms · 0 invented entities

The paper's contribution is measured against what it imports: two external analytic estimates (periodic CBS for d≥3; Solomyak's L^{2,1} endpoint for d=2), the Frank-Lieb-Seiringer matrix-to-scalar reduction, one self-declared asymptotic (33), and standard Gribov domain assumptions (transversality, constant-mode removal, positive-definiteness of the reduced FP operator). No free parameter is fitted to make a derivation work; the two listed free parameters are post-hoc diagnostic fits verifying derived exponents, and the entropy-model constant c drops out of the leading scaling. Zero invented physical entities: K_A is a definitional operator, not a new particle, force, or conserved quantity.

free parameters (2)
  • fitted log-log slopes of the counting and volume laws = −1.99 (d=2, Fig. 1); 1.00, 0.76, 0.00 (d=2,3,4, Fig. 3)
    Fitted to numerically computed spectra of the periodic background to verify the derived exponents −2 and d(4−d)/4 (eqs. 33, 55). Verification diagnostics only; no claim's constants are set by these fits.
  • per-mode threshold constant c in the entropy model (Section 8) = unspecified (drops out of the leading exponent)
    Introduced inside the 'approximation and not a theorem' entropy estimate Z_hor(L); after rescaling u=(βL^{d−4})^{1/4}n the exponent d(4−d)/4 is independent of c, so the scaling claim survives its ambiguity.
axioms (6)
  • standard math Periodic matrix-valued Cwikel-Birman-Solomyak estimate: for d≥3, ∥M_fΛ∥_{S^{d,∞}} ≤ C_d∥f∥_{L^d}, given the ℓ^{d,∞} symbol condition {p: |p|^{−1} > t} = O(t^{−d})
    Invoked in Section 4.1 step 4 (eq. 21); the paper asserts the hypotheses are local and transfer 'from R^d to T^d without change'. This is the load-bearing analytic input for Theorem 1.
  • standard math L^{2,1} two-dimensional endpoint: M_fΛ ∈ S^{2,∞} with ∥M_fΛ∥_{S^{2,∞}} ≤ C∥f∥_{L^{2,1}} (Prop. 2), via off-endpoint Kato-Seiler-Simon bounds and Solomyak's Yano extrapolation
    Section 4.2: the proof is a sketch — the per-layer estimate (28) is asserted and the endpoint summation is delegated to Solomyak [25,26]. The statement matches the known L^{2,1} endpoint of Cwikel's theorem, but it is not fully demonstrated in-text.
  • standard math Frank-Lieb-Seiringer matrix-to-scalar reduction for weak Schatten ideals, with color factor r^{1/d} (eq. 20)
    Section 4.1 step 3: pointwise inequality F(x)*F(x) ⪯ |F(x)|²1_r lifted to operator level and combined with Weyl monotonicity of eigenvalues; this step carries the N²−1 color multiplicity into Theorem 1.
  • domain assumption Principal-symbol semiclassical counting for the order-(−1) operator K_A (eqs. 31-33): negative eigenvalues counted by symbol phase-space volume
    Section 5: the paper explicitly labels (33) 'a principal-symbol asymptotic...not, at this stage, a proved endpoint Weyl law'. The crossing law used for Fig. 1 and the factor-2 singular-value remark depend on this unproved asymptotic.
  • domain assumption Exact Landau transversality of the background (q_iA_i(q)=0) is a spectral hypothesis making K_A self-adjoint
    Section 3: transversality replaces k_i by the symmetric (p_i+k_i)/2 and forces Hermiticity of kernel (10); without it, the spectrum is not real and inertia (12) and the spectral representation (35) collapse. The paper flags the discretization failure mode in Section 3.
  • domain assumption First Gribov region is the set of transverse backgrounds with positive-definite reduced FP operator; constant adjoint modes are removed as the kernel of H_0
    Sections 1-2: standard Gribov-Zwanziger setup; the mean-zero reduction is what makes H_0 positive and the equivalence (11) meaningful on the quotient.

pith-pipeline@v1.3.0-alltime-deepseek · 18897 in / 46905 out tokens · 377846 ms · 2026-08-02T02:10:09.304145+00:00 · methodology

0 comments
read the original abstract

After removing the constant adjoint modes associated with global gauge rotations, we formulate the Landau-gauge Faddeev-Popov zero-mode problem in Birman-Schwinger form. The resulting normalized operator, is dimensionless and self-adjoint for transverse backgrounds, and the first Gribov horizon is identified with the appearance of the spectral value $-1$. Because this reduction is a congruence rather than a similarity transformation, it preserves the inertia and nullity relevant to horizon crossings without identifying the numerical spectra of the two operators. We study these analytic properties on a periodic domain, using matrix-valued Cwikel estimates to control its singular-value behavior at the regularity scale selected by the first-order Faddeev-Popov interaction, with separate attention to the two-dimensional endpoint. The same formulation expresses the fixed-background ghost Green function through the diagonal resolvent, providing a common framework in which the exact spectral condition can be compared with the Born expansion and Gribov's no-pole construction while remaining distinct from statements involving the functional average over gauge fields. As an explicit application, we consider a periodic transverse $SU(2)$ background for which the zero-mode equation reduces to a Mathieu-type recurrence and admits a systematic analysis through finite-channel and Feshbach reductions. This solvable sector also permits an examination of the volume dependence of horizon-touching configurations, without assigning them a statistical weight in the Yang-Mills measure.

Figures

Figures reproduced from arXiv: 2607.14425 by Daniel G. Tedesco.

Figure 1
Figure 1. Figure 1: Two-dimensional crossing law (33) for A3 y = a cos(Qx) on T 2 . Left: the counting function n−(ε) against the parameter-free prediction 1 8π ε −2 R |gA| 2 (dashed), with fitted slope −1.99 versus −2. Right: the compensated ratio n−(ε) ε 2/ R |gA| 2 , level at 1/(8π) ≈ 0.0398 (dashed) until the mesh coarsens at large ε. Equation (33) is a principal-symbol asymptotic, valid for the smooth backgrounds to whic… view at source ↗
Figure 2
Figure 2. Figure 2: Ghost dressing as the diagonal resolvent of [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Entropy-action competition within the periodic family. Left: the critical-direction count [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

35 extracted references · 5 canonical work pages · 1 internal anchor

  1. [1]

    Quantization of non-Abelian gauge theories,

    V. N. Gribov, “Quantization of non-Abelian gauge theories,” Nucl. Phys. B139, 1-19 (1978), doi:10.1016/0550-3213(78)90175-X

    doi:10.1016/0550-3213(78)90175-x 1978

  2. [2]

    Local and renormalizable action from the Gribov horizon,

    D. Zwanziger, “Local and renormalizable action from the Gribov horizon,” Nucl. Phys. B323, 513-544 (1989), doi:10.1016/0550-3213(89)90122-3

    doi:10.1016/0550-3213(89)90122-3 1989

  3. [3]

    The Gribov problem and QCD dynamics,

    N. Vandersickel and D. Zwanziger, “The Gribov problem and QCD dynamics,” Phys. Rep. 520, 175-251 (2012), arXiv:1202.1491, doi:10.1016/j.physrep.2012.07.003. 18

    Pith/arXiv arXiv 2012

  4. [4]

    An all-order proof of the equivalence between Gribov’s no-pole and Zwanziger’s horizon conditions,

    M. A. L. Capri, D. Dudal, M. S. Guimaraes, L. F. Palhares, and S. P. Sorella, “An all-order proof of the equivalence between Gribov’s no-pole and Zwanziger’s horizon conditions,” Phys. Lett. B719, 448-453 (2013), arXiv:1212.2419, doi:10.1016/j.physletb.2013.01.039

    Pith/arXiv arXiv 2013

  5. [5]

    The Gribov parameter and the dimension two gluon condensate in Euclidean Yang-Mills theories in the Landau gauge,

    D. Dudal, R. F. Sobreiro, S. P. Sorella, and H. Verschelde, “The Gribov parameter and the dimension two gluon condensate in Euclidean Yang-Mills theories in the Landau gauge,” Phys. Rev. D72, 014016 (2005), arXiv:hep-th/0502183, doi:10.1103/PhysRevD.72.014016

    Pith/arXiv arXiv 2005

  6. [6]

    Local and BRST-invariant Yang-Mills theory within the Gribov hori- zon,

    M. A. L. Capriet al., “Local and BRST-invariant Yang-Mills theory within the Gribov hori- zon,” Phys. Rev. D94, 025035 (2016), arXiv:1605.02610, doi:10.1103/PhysRevD.94.025035

    Pith/arXiv arXiv 2016

  7. [7]

    The universal character of Zwanziger’s horizon function in Euclidean Yang-Mills theories,

    M. A. L. Capriet al., “The universal character of Zwanziger’s horizon function in Euclidean Yang-Mills theories,” Phys. Lett. B781, 48-54 (2018), arXiv:1802.04582, doi:10.1016/j.physletb.2018.03.058

    Pith/arXiv arXiv 2018

  8. [8]

    Infrared massive gluon propagator from a BRST-invariant Gribov horizon in a family of covariant gauges,

    B. W. Mintz, L. F. Palhares, G. Peruzzo, and S. P. Sorella, “Infrared massive gluon propagator from a BRST-invariant Gribov horizon in a family of covariant gauges,” Phys. Rev. D99, 034002 (2019), arXiv:1812.03166, doi:10.1103/PhysRevD.99.034002

    Pith/arXiv arXiv 2019

  9. [9]

    The BRST- invariant vacuum state of the Gribov-Zwanziger theory,

    D. Dudal, C. P. Felix, L. F. Palhares, F. Rondeau, and D. Vercauteren, “The BRST- invariant vacuum state of the Gribov-Zwanziger theory,” Eur. Phys. J. C79, 731 (2019), arXiv:1901.11264, doi:10.1140/epjc/s10052-019-7235-0

    Pith/arXiv arXiv 2019

  10. [10]

    Toward background field indepen- dence within the Gribov horizon,

    I. F. Justo, A. D. Pereira, and R. F. Sobreiro, “Toward background field indepen- dence within the Gribov horizon,” Phys. Rev. D106, 025015 (2022), arXiv:2206.04103, doi:10.1103/PhysRevD.106.025015

    Pith/arXiv arXiv 2022

  11. [11]

    A few remarks on the zero modes of the Faddeev-Popov operator in the Landau and maximal Abelian gauges,

    M. S. Guimaraes and S. P. Sorella, “A few remarks on the zero modes of the Faddeev-Popov operator in the Landau and maximal Abelian gauges,” J. Math. Phys.52, 092302 (2011), arXiv:1106.3944, doi:10.1063/1.3641892

    Pith/arXiv arXiv 2011

  12. [12]

    On the zero modes of the Faddeev-Popov operator in the Landau gauge,

    R. R. Landim, L. C. Q. Vilar, O. S. Ventura, and V. E. R. Lemes, “On the zero modes of the Faddeev-Popov operator in the Landau gauge,” J. Math. Phys.55, 022901 (2014), arXiv:1212.4098, doi:10.1063/1.4865424

    Pith/arXiv arXiv 2014

  13. [13]

    Spectral properties of the Landau gauge Faddeev-Popov operator in lattice gluodynamics,

    A. Sternbeck, E.-M. Ilgenfritz, and M. Müller-Preussker, “Spectral properties of the Landau gauge Faddeev-Popov operator in lattice gluodynamics,” Phys. Rev. D73, 014502 (2006), arXiv:hep-lat/0510109, doi:10.1103/PhysRevD.73.014502

    Pith/arXiv arXiv 2006

  14. [14]

    Towards the in- frared limit in SU(3) Landau gauge lattice gluodynamics,

    A. Sternbeck, E.-M. Ilgenfritz, M. Müller-Preussker, and A. Schiller, “Towards the in- frared limit in SU(3) Landau gauge lattice gluodynamics,” Phys. Rev. D72, 014507 (2005), arXiv:hep-lat/0506007, doi:10.1103/PhysRevD.72.014507

    Pith/arXiv arXiv 2005

  15. [15]

    Faddeev-Popov spectra at the Gribov horizon,

    J. Greensite, “Faddeev-Popov spectra at the Gribov horizon,” Phys. Rev. D81, 114011 (2010), arXiv:1001.0784, doi:10.1103/PhysRevD.81.114011

    Pith/arXiv arXiv 2010

  16. [16]

    Crossing the Gribov horizon: An unconventional study of geometric properties of gauge-configuration space in Landau gauge,

    A. Cucchieri and T. Mendes, “Crossing the Gribov horizon: An unconventional study of geometric properties of gauge-configuration space in Landau gauge,” PoS(LATTICE2013)367 (2014), arXiv:1311.4699, doi:10.22323/1.187.0367

    Pith/arXiv arXiv 2014

  17. [17]

    Bloch waves in minimal Landau gauge and the infinite- volume limit of lattice gauge theory,

    A. Cucchieri and T. Mendes, “Bloch waves in minimal Landau gauge and the infinite- volume limit of lattice gauge theory,” Phys. Rev. Lett.118, 192002 (2017), arXiv:1612.01279, doi:10.1103/PhysRevLett.118.192002. 19

    Pith/arXiv arXiv 2017

  18. [18]

    Deconfinement in pure gauge SU(3) Yang-Mills theory: The ghost propagator,

    O. Oliveira, V. Paiva, and P. J. Silva, “Deconfinement in pure gauge SU(3) Yang-Mills theory: The ghost propagator,” EPJ Web Conf.274, 05008 (2022), arXiv:2301.01229, doi:10.1051/epjconf/202227405008

    Pith/arXiv arXiv 2022

  19. [19]

    Mass generation in Landau- gauge Yang-Mills theory,

    G. Eichmann, J. M. Pawlowski, and J. M. Silva, “Mass generation in Landau- gauge Yang-Mills theory,” Phys. Rev. D104, 114016 (2021), arXiv:2107.05352, doi:10.1103/PhysRevD.104.114016

    Pith/arXiv arXiv 2021

  20. [20]

    On the spectrum of singular boundary-value problems,

    M. Sh. Birman, “On the spectrum of singular boundary-value problems,” Mat. Sb. (N.S.) 55(97), no. 2, 125-174 (1961)

    1961

  21. [21]

    On the bound states of a given potential,

    J. Schwinger, “On the bound states of a given potential,” Proc. Natl. Acad. Sci. U.S.A.47, 122-129 (1961), doi:10.1073/pnas.47.1.122

    doi:10.1073/pnas.47.1.122 1961

  22. [22]

    Simon,Trace Ideals and Their Applications, 2nd ed

    B. Simon,Trace Ideals and Their Applications, 2nd ed. (American Mathematical Society, Providence, RI, 2005), doi:10.1090/surv/120

    doi:10.1090/surv/120 2005

  23. [23]

    The generalized Birman-Schwinger principle,

    J. Behrndt, A. F. M. ter Elst, and F. Gesztesy, “The generalized Birman-Schwinger principle,” Trans. Am. Math. Soc.375, 799-845 (2022), arXiv:2005.01195, doi:10.1090/tran/8401

    Pith/arXiv arXiv 2022

  24. [24]

    Weak type estimates for singular values and the number of bound states of Schrödinger operators,

    M. Cwikel, “Weak type estimates for singular values and the number of bound states of Schrödinger operators,” Ann. Math.106, 93-100 (1977), doi:10.2307/1971160

    work page doi:10.2307/1971160 1977

  25. [25]

    Estimates for the number of negative eigenvalues of the Schrödinger operator and its generalizations,

    M. Sh. Birman and M. Z. Solomyak, “Estimates for the number of negative eigenvalues of the Schrödinger operator and its generalizations,” inEstimates and Asymptotics for Discrete Spec- tra of Integral and Differential Equations, Adv. Soviet Math.7, 1-55 (American Mathematical Society, Providence, RI, 1991), doi:10.1090/advsov/007/01

    work page doi:10.1090/advsov/007/01 1991

  26. [26]

    Number of bound states of Schrödinger opera- tors with matrix-valued potentials,

    R. L. Frank, E. H. Lieb, and R. Seiringer, “Number of bound states of Schrödinger opera- tors with matrix-valued potentials,” Lett. Math. Phys.82, 107-116 (2007), arXiv:0710.1877, doi:10.1007/s11005-007-0211-x

    Pith/arXiv arXiv 2007

  27. [27]

    Mémoire sur le mouvement vibratoire d’une membrane de forme elliptique,

    É. Mathieu, “Mémoire sur le mouvement vibratoire d’une membrane de forme elliptique,” J. Math. Pures Appl., 2e série,13, 137-203 (1868)

  28. [28]

    N. W. McLachlan,Theory and Application of Mathieu Functions(Oxford University Press, London, 1947)

    1947

  29. [29]

    Meixner and F

    J. Meixner and F. W. Schäfke,Mathieusche Funktionen und Sphäroidfunktionen: Mit Anwendungen auf physikalische und technische Probleme(Springer-Verlag, Berlin, 1954), doi:10.1007/978-3-662-00941-3

    work page doi:10.1007/978-3-662-00941-3 1954

  30. [30]

    Abramowitz and I

    M. Abramowitz and I. A. Stegun (eds.),Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, NBS Applied Mathematics Series55(U.S. Government Printing Office, Washington, D.C., 1964)

    1964

  31. [31]

    Infrared instability of the vacuum state of gauge theories and asymptotic freedom,

    G. K. Savvidy, “Infrared instability of the vacuum state of gauge theories and asymptotic freedom,” Phys. Lett. B71, 133-134 (1977), doi:10.1016/0370-2693(77)90759-6

    doi:10.1016/0370-2693(77)90759-6 1977

  32. [32]

    Vacuum polarization in uniform non-Abelian gauge fields,

    L. S. Brown and W. I. Weisberger, “Vacuum polarization in uniform non-Abelian gauge fields,” Nucl. Phys. B157, 285-326 (1979), doi:10.1016/0550-3213(79)90508-X

    work page doi:10.1016/0550-3213(79)90508-x 1979

  33. [33]

    The Metric on the Space of Yang-Mills Configurations

    P. Orland, “The metric on the space of Yang-Mills configurations,” Report No. NBI-HE-96-35, arXiv:hep-th/9607134 (1996; revised 1997), doi:10.48550/arXiv.hep-th/9607134. 20

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.hep-th/9607134 1996

  34. [34]

    Extremal curves in(2 + 1)-dimensional Yang-Mills theory,

    P. Orland and G. W. Semenoff, “Extremal curves in(2 + 1)-dimensional Yang-Mills theory,” Nucl. Phys. B576, 627-654 (2000), arXiv:hep-th/9912009, doi:10.1016/S0550-3213(00)00134- 6

    Pith/arXiv arXiv 2000

  35. [35]

    Gauge-invariant coordinates on gauge-theory orbit space,

    P. Orland, “Gauge-invariant coordinates on gauge-theory orbit space,” Phys. Rev. D70, 045014 (2004), arXiv:hep-th/0402003, doi:10.1103/PhysRevD.70.045014. 21

    Pith/arXiv arXiv 2004