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Sloshing, Steklov and corners: Asymptotics of Steklov eigenvalues for curvilinear polygons

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Pith's one-line read The Steklov eigenvalues of a curvilinear polygon are determined to within O(m^{-ε}) by an explicitly computable sequence of quasi-eigenvalues built from the side lengths and angles.

desk verdict A genuinely new sharp Steklov asymptotic for curvilinear polygons, but the exceptional-angle enumeration is sketched rather than proved, and that is exactly where a wrong index shift would break the main theorem. read the letter →

arxiv 1908.06455 v3 pith:5ZY2QWZR submitted 2019-08-18 math.SP math.AP

Michael Levitin , Leonid Parnovski , Iosif Polterovich , David A. Sher This is my paper · ORCID
classification math.SPmath.AP MSC 35P2034B45
keywords Stekloveigenvaluescurvilinearpolygonsquasi-eigenvaluestransfermatricesscatteringPeterssolutionsquantumgraphsexceptionalanglesspectralasymptotics
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 proves that the Steklov eigenvalues of a curvilinear polygon—a planar domain with smooth sides and corners—are approximated, up to an error that tends to zero like a negative power of the index m, by a sequence of quasi-eigenvalues computed only from the side lengths and the interior angles. The result replaces the crude o(m) Weyl law with a sharp asymptotic description that retains detailed geometric information. A surprising corollary is that the arithmetic nature of the angles controls the eigenfunctions: if every angle is π divided by an odd integer, the boundary traces are equidistributed, while if every angle is π divided by an even integer, they concentrate on a single side. The paper also shows that the quasi-eigenvalues are the square roots of eigenvalues of a quantum graph Laplacian on the boundary, and gives explicit trigonometric polynomials whose roots are these quasi-eigenvalues.

What carries the argument

The central object is the transfer matrix product T(α,ℓ,σ) formed by alternating the vertex matrix A(α) = [[csc(π²/2α), -i cot(π²/2α)],[i cot(π²/2α), csc(π²/2α)]] and the side matrix B(ℓ,σ) = diag($e^{{iℓσ}}$, $e^{{-iℓσ}}$) around the boundary. Quasi-eigenvalues are the σ for which T has eigenvalue 1 (non-exceptional case), or for which the exceptional-component condition U X·X⊥ = 0 holds. The proof that these quasi-eigenvalues actually enumerate the Steklov spectrum uses a lift of the transfer matrices to the universal cover of the punctured plane, where their argument is strictly monotone in σ, combined with Dirichlet-Neumann bracketing on auxiliary zigzag domains. The quasimodes themselves are assembled from scattering Peters solutions in sectors, whose existence and decay r = π/α come from the authors' earlier sloshing analysis.

What would settle it

Compute numerically the first several hundred Steklov eigenvalues of a curvilinear triangle with angles (π/3, π/4, 5π/12) and one curved side, then compare them with the roots of the trigonometric polynomial F_P(α,ℓ,σ); if the gap between a true eigenvalue and the nearest quasi-eigenvalue does not shrink like a negative power of m, Theorem 1.4 is false.

Watch

Extended reading notes

Core claim

For any curvilinear polygon P with side lengths ℓ and angles α, the Steklov eigenvalues λ_m satisfy λ_m = σ_m + O($m^{{-ε}}$) for some ε>0 depending only on the angles, where σ_m is an explicitly defined sequence of quasi-eigenvalues. The quasi-eigenvalues are defined through products of 2×2 vertex transfer matrices A(α) and side transfer matrices B(ℓ,σ), equivalently as roots of trigonometric polynomials, and also as the square roots of eigenvalues of a quantum graph Laplacian on the boundary with angle-dependent matching conditions. The same construction gives asymptotic control of eigenfunctions: on each side, the boundary trace of u_m is, up to O($m^{{-ε}}$) in $L^{2}$, a trigonometric function of frequency σ_m. The arithmetic dichotomy is sharp: if all angles are special (π/(2k+1)), traces equidistribute; if all angles are exceptional (π/(2k)), each trace concentrates on one side, with a splitting of exceptional components governing which side.

Load-bearing premise

The construction assumes the sector scattering solutions from the earlier sloshing analysis exist with decay rate r = π/α; if those solutions decay more slowly or do not exist, the polygon quasimodes would not be nearly harmonic and the O($m^{{-ε}}$) error would fail.

Editorial extensions

If this is right

  • Two curvilinear polygons with the same angles and the same side lengths have Steklov spectra that differ only by O(m^{-ε}), so spectra encode the geometry at this resolution.
  • The Weyl law N(λ) = |∂P|/π λ + O(1) and the Riesz mean R(λ) = |∂P|/(2π) λ² + O(λ^{1-ε}) hold for all curvilinear polygons with angles less than π.
  • If all angles are special, the quasi-eigenvalues form periodic arithmetic progressions with double multiplicity, matching the smooth-domain pattern; even special angles can be removed without changing quasi-eigenvalues.
  • If all angles are exceptional, eigenfunctions concentrate on one side, whereas all-special angles force equidistribution on the boundary.
  • Quasi-eigenvalues can be computed explicitly as roots of trigonometric polynomials, giving a practical numerical recipe for the Steklov spectrum of cornered domains.

Reading between the lines

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

  • The quantum-graph formulation suggests an inverse spectral program: since the quasi-eigenvalues are determined by side lengths and angles, a sufficiently long Steklov spectrum should in principle recover the polygon's geometry, as the authors indicate they plan to pursue.
  • The exceptional-angle concentration phenomenon could be tested experimentally in sloshing tanks with wedge-shaped corners: the free-surface eigenfunctions would localize on alternating walls depending on angle parity.
  • The transfer-matrix enumeration machinery may extend to higher-order Weyl corrections for polygons with rational angle ratios, where the lifts on the universal cover become periodic and a full asymptotic expansion might exist.
  • The same construction could be adapted to mixed Dirichlet-Neumann Steklov problems on keyhole domains and other non-simply connected configurations, following the paper's remark on conformal maps.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 3 minor

Summary. The paper studies the asymptotic distribution of Steklov eigenvalues and eigenfunctions on curvilinear polygons, i.e., domains with smooth sides and corners of angle strictly between 0 and π. The main result (Theorem 1.4) asserts that the Steklov eigenvalues λ_m can be approximated, up to an error O(m^{-ε}), by a sequence of quasi-eigenvalues σ_m defined purely in terms of the side lengths and angles via vertex and side transfer matrices. The paper also proves a companion statement (Theorem 1.7) on the trigonometric form of boundary traces of eigenfunctions, and describes qualitatively different behaviour for polygons all of whose angles are 'special' (equidistribution) versus all 'exceptional' (concentration on a subset of sides). The proofs combine scattering Peters solutions in sectors, a quasimode construction, a delicate enumeration argument for quasi-eigenvalues, layer potential estimates for curved boundaries, and numerical illustrations.

Significance. If the results are fully established, this is a major contribution to the spectral geometry of non-smooth domains. The paper gives, for the first time, asymptotics with error tending to zero for Steklov eigenvalues of polygons, and reveals a surprising dependence on arithmetic properties of the angles. The quasi-eigenvalues are defined entirely from the geometry, with no reference to the actual Steklov spectrum, so the argument is not circular. The paper also provides a clean quantum-graph interpretation and, in Section 9, numerical benchmarks that confirm the formulas in concrete examples, including the equilateral triangle, the right isosceles triangle, and regular polygons. The proofs are detailed and mostly self-contained; the exceptional-angle enumeration is the one part that needs further support.

major comments (2)
  1. [5.7, Theorem 5.31 and Proposition 5.32] The proof of the exceptional-case enumeration theorem is incomplete in a way that affects the central claim. After Definition 5.30, the text states that Theorem 5.31 is proved 'similarly to Theorem 2.39' and says 'we outline the main steps ... leave the details to the reader.' Proposition 5.32, which is the base case for two equal straight sides with an exceptional angle, is dismissed with the sentence that the result follows 'by explicitly computing the total loss of quasi-eigenvalues' using [LPPS17]; however, no computation is shown. Since Theorem 5.35 (exceptional polygons) and hence Theorem 1.4 for any polygon with an exceptional angle rely on this enumeration, a parity or sign error in the omitted total-loss computation would change the constant or the index shift between σ_m and λ_m. Please provide the full computation for Proposition 5.32, or at least an explicit derivation of the total-loss formula and its comparison with Definition 5.30.
  2. [5.7, Propositions 5.33 and 5.34] The gluing propositions for exceptional components are introduced with 'a straightforward adaptation of the proof of Proposition 5.13', but no proof or even a precise statement of the required adaptation is given. These propositions are load-bearing for Theorem 5.31: they propagate the natural enumeration from the basic two-sided case to arbitrary exceptional zigzags, and the half-integer counting functions in Definition 5.30 make the correctness of this propagation non-obvious. Please either give the detailed proofs of Propositions 5.33 and 5.34 or state them as lemmas with the relevant analogue of Lemma 5.13 and its proof.
minor comments (3)
  1. [2.7, Proposition 2.41] The proof of Proposition 2.41 is omitted with the note that it is 'almost identical' to that of Proposition 2.27. Since this proposition is used later in the proof of Theorem 4.30 for zigzag domains, a brief indication of the differences (e.g., the treatment of the endpoint conditions) would help the reader verify the result without reconstructing the whole argument.
  2. [9, numerical examples] The numerical tests in Section 9 mostly concern symmetric cases such as the right isosceles triangle or regular polygons, where high multiplicities make index shifts less sensitive. Adding a numerical example with mixed-parity exceptional angles and incommensurable side lengths, where the half-integer shifts in Definition 5.30 are individually observable, would provide additional confidence in the exceptional-case enumeration.
  3. [2.6, Theorem 2.31] The notation in Theorem 2.31 uses both ε0 (from Theorem 1.4) and a parameter written as '~ε'; the relation between the two is stated but could be made clearer by explicitly defining the admissible range for the exponent in the error term.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: quasi-eigenvalues are geometric definitions, and the comparison to Steklov eigenvalues uses quasimode construction and independent sloshing results.

full rationale

The paper's central objects, the quasi-eigenvalues σ_m, are defined purely from the side lengths and angles of the polygon through the vertex and side transfer matrices (Definitions 2.3, 2.10); no Steklov eigenvalue or eigenfunction is used in their definition. Theorem 1.4 is then obtained by an explicit quasimode construction (Section 4) followed by a separate enumeration argument (Section 5) that compares zigzag problems to the sloshing problem asymptotics of [LPPS17]. The reliance on [LPPS17] is substantial but not circular: it is a prior published theorem about a different mixed boundary value problem, not a restatement of the Steklov result being proved, and it is not fitted to the present paper's data. The quantum graph reformulation is presented as an equivalent characterization, not as the source of the quasimode approximation. The main gap noted by a skeptical reader is that the exceptional-angle enumeration (Theorem 5.31, Proposition 5.32) is outlined rather than fully proved; this is a completeness or correctness concern, not a circularity. No equation reduces to its own input by definition, no fitted parameter is renamed as a prediction, and no load-bearing uniqueness assertion is imported solely from the authors' prior work. The derivation chain is therefore self-contained in the sense relevant here, so the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 3 invented entities

The paper introduces no free fitted parameters: the quasi-eigenvalues are determined by the geometry (angles and side lengths) with no constants fitted to the Steklov spectrum. It does rely on the prior sloshing paper LPPS17 for the sector solutions, and on standard results for quantum graphs and Lipschitz boundary value problems, all of which are independent of the final claim.

assumptions (5)
  • domain assumption The boundary of the domain is piecewise C^5 (or at least C^M with M>=3) with internal angles alpha_j in (0,pi), and the domain is simply connected.
    Used throughout; Theorem 4.14 assumes M>=3 and Theorem 4.19 piecewise C^5; cf. Sections 4.5-4.6.
  • domain assumption Peters solutions for sloping beach problems exist with the decay rates stated in [LPPS17, Theorem 2.1]; in particular the remainder in the sector satisfies |R| + rho |grad R| <= C rho^{-r} with r = mu.
    This is the key building block for quasimodes; invoked in Section 3, Lemma 3.5 and Theorem 3.1.
  • standard math The quantum graph Laplacian Weyl law N(sigma) = (|boundary|/pi) sigma + O(1) applies to the quasi-eigenvalue counting function.
    Used in Section 2.6, equation (2.31), from [BeKu13, Lemma 3.7.4].
  • domain assumption Layer potential operators on curvilinear polygons have kernel estimates sufficient to compare the Dirichlet-to-Neumann map of a curved side with a straight side (following Costabel [Cos83]).
    Used in Section 8 to handle full curvilinearity; relies on the boundary being piecewise smooth and the angles less than pi.
  • standard math The Riemann mapping theorem provides conformal maps Theta_j near vertices in the Hoelder class C^{1,gamma} for every gamma < 1.
    Used in Section 4.6 to straighten the corner for the quasimode construction; cited from [PePu14].
invented entities (3)
  • Quasi-eigenvalues sigma_m independent evidence
    purpose: Explicit sequence approximating the Steklov eigenvalues; defined via transfer matrices and trigonometric polynomials.
    They are computed from geometry alone and match the true eigenvalues to error m^{-epsilon}; numerics in Section 9 confirm the match.
  • Vertex transfer matrix A(alpha) and side transfer matrix B(ell,sigma) independent evidence
    purpose: Encoding the scattering at corners and propagation along sides; these matrices generate the quasi-eigenvalue condition.
    A(alpha) is derived directly from the sector scattering solutions (Theorem 3.1), not postulated; B(ell,sigma) is a diagonal phase rotation.
  • Scattering Peters solutions Phi(h_in,h_out)_alpha independent evidence
    purpose: Building blocks for the quasimodes; they are exact harmonic functions in a sector that behave like a sum of two plane waves plus a decaying remainder.
    Their existence is proved in Section 3 via explicit formulas (Lemma 3.5 and Proposition 3.6), and they reproduce the transfer matrices.

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Pith. "Pith review of Sloshing, Steklov and corners: Asymptotics of Steklov eigenvalues for curvilinear polygons." pith.science (2026). https://pith.science/paper/5ZY2QWZR

@misc{pith2026190806455,
  author       = {Pith},
  title        = {Pith review of: Sloshing, Steklov and corners: Asymptotics of Steklov eigenvalues for curvilinear polygons},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5ZY2QWZR}},
  note         = {Machine review of arXiv:1908.06455}
}
read the original abstract

We obtain asymptotic formulae for the Steklov eigenvalues and eigenfunctions of curvilinear polygons in terms of their side lengths and angles. These formulae are quite precise: the errors tend to zero as the spectral parameter tends to infinity. The Steklov problem on planar domains with corners is closely linked to the classical sloshing and sloping beach problems in hydrodynamics; as we show it is also related to quantum graphs. Somewhat surprisingly, the arithmetic properties of the angles of a curvilinear polygon have a significant effect on the boundary behaviour of the Steklov eigenfunctions. Our proofs are based on an explicit construction of quasimodes. We use a variety of methods, including ideas from spectral geometry, layer potential analysis, and some new techniques tailored to our problem.

Figures

Figures reproduced from arXiv: 1908.06455 by the authors.

Figure 1
Figure 1. A curvilinear polygon ` = (`1, . . . , `n) ∈ R n +, and we will use cyclic subscript identification n + 1 ≡ 1. Our choice of orientation ensures that an internal angle αj is measured from Ij to Ij+1 in the counter-clockwise direction, as in Figure1. The perimeter of P is |∂P| = `1 + · · · + `n. In what follows we will have to distinguish the cases when some of the polygon’s angles belong to the sets of exceptional a… view at source ↗
Figure 2
Figure 2. Boundary traces of u18 and u19 for the equilateral triangle [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Boundary traces of u18 and u19 for the right-angled isosceles triangle 1.5 Plan of the paper and further directions We begin in section 2 by defining and studying the sequence {σm} of quasi-eigenvalues which appears in The￾orem 1.4. The quasi-eigenvalues and, importantly, their multiplicitiesare originally defined in terms of a combi￾nation of vertex transfer matrices A(αj ) and side transfer matrices B(`j ), which … view at source ↗
Figures from the paper (20 more)
Figure 4
Figure 4. Figure 4: An example of re-labelling for a pentagon with two exceptional angles and therefore two exceptional [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]
Figure 5
Figure 5. Figure 5: A quantum graph. Remark 2.22. For αj ∈ E / , we can re-write the matching conditions as f|Vj+0 = cot  π 2 4αj  f|Vj−0, f 0 |Vj+0 = tan  π 2 4αj  f 0 |Vj−0, (2.25) For αj ∈ E, the matching conditions are given by ( f|Vj−0 = f 0 |Vj+0 = 0 if O(αj ) = 1, f|Vj+0 = f 0 …
Figure 6
Figure 6. Figure 6: A zigzag Z and a zigzag domain Ω We will call a zigzag straight if its arcs I1, . . . , In, are straight-line intervals, and partially curvilinear if the arcs are straight in a neighbourhood of each vertex. We will call a zigzag non-exceptional orexceptional if α ∈ (Π\…
Figure 7
Figure 7. Figure 7: A path L Consider the Dirac operator (2.27) on L acting on vector functions f(s) with the matching conditions (2.28) at internal vertices V1, . . . , Vn−1 and with the boundary conditions f|V0+ · ℵ ⊥ = f|Vn− · i ⊥ = 0. (2.41) We have the following Proposition 2.41. The…
Figure 8
Figure 8. Figure 8: Sectors Sα and Sα/2 . Let, for t ∈ R, e(t) := e −it e it ! . (3.1) For any fixed vector h = [PITH_FULL_IMAGE:figures/full_fig_p024_8.png]
Figure 9
Figure 9. Figure 9: A straight polygon with local coordinates [PITH_FULL_IMAGE:figures/full_fig_p030_9.png]
Figure 10
Figure 10. Figure 10: Boundary orthogonal coordinates for a curvilinear boundary. [PITH_FULL_IMAGE:figures/full_fig_p036_10.png]
Figure 11
Figure 11. Figure 11: Partition of unity and boundary coordinates in a neighbourhood of [PITH_FULL_IMAGE:figures/full_fig_p039_11.png]
Figure 12
Figure 12. Figure 12: Decomposition of a three-arc partially curvilinear zigzag into the union of two two-piece straight [PITH_FULL_IMAGE:figures/full_fig_p057_12.png]
Figure 13
Figure 13. Figure 13: Symmetric decomposition of Steklov–Neumann symmetric zigzag domain. Solid lines denote [PITH_FULL_IMAGE:figures/full_fig_p059_13.png]
Figure 14
Figure 14. Figure 14: Symmetric decomposition of Steklov–Dirichlet symmetric zigzag domain. Solid lines denote Steklov [PITH_FULL_IMAGE:figures/full_fig_p059_14.png]
Figure 15
Figure 15. Figure 15: Two isospectral Steklov–Neumann–Dirichlet problems. Solid lines denote Steklov conditions, [PITH_FULL_IMAGE:figures/full_fig_p060_15.png]
Figure 16
Figure 16. Figure 16: A polygon with a cut Denote also by N q P (σ), N q Z(DD) (σ)and N q Z(NN) (σ)the corresponding quasi-eigenvalue counting functions. By the Dirichlet–Neumann bracketing we have for all positive σ NZ(DD) (σ) ≤ NP(σ) ≤ NZ(NN) (σ). (5.44) In view of Remark 5.16, we need t…
Figure 17
Figure 17. Figure 17: Relative FEM errors for the disk and the square, and asymptotics error for the square [PITH_FULL_IMAGE:figures/full_fig_p099_17.png]
Figure 18
Figure 18. Figure 18: Family T (α) of curvilinear triangles The asymptotic accuracy for T1, T2, T3, and T4 is plotted in [PITH_FULL_IMAGE:figures/full_fig_p100_18.png]
Figure 19
Figure 19. Figure 19: Asymptotic accuracy for T1, T2, T3, and T4 Solving (9.1) for non-negative σ gives the expressions for quasi-eigenvalues in the statement of Proposition 1.13. To prove the statement on multiplicities, we remark that if c 6= ±1, the matrix C(α, `, σ) has two linearly in…
Figure 20
Figure 20. Figure 20: Asymptotic accuracy for P3, P5, P6, and R Additionally, we consider a family of (non-symmetric) one-angled droplets Dα = P1(α, `α) shown in [PITH_FULL_IMAGE:figures/full_fig_p101_20.png]
Figure 21
Figure 21. Figure 21: A family of one-gons (droplets) Dα 20, sharply bend upwards around m ≈ 20. This means that for higher eigenvalues the errors of numerical computations exceed asymptotic errors (with the asymptotics in these cases converging rather rapidly), and the results become unre…
Figure 22
Figure 22. Figure 22: Asymptotic accuracy for droplets Dα, α ∈ π 5 , π 3 , 3π 7 , 5π 8 [PITH_FULL_IMAGE:figures/full_fig_p103_22.png]
Figure 23
Figure 23. Figure 23: Asymptotic accuracy for sectors Sα, α ∈ 3π 4 , π, 3π 2 , 11π 6 [PITH_FULL_IMAGE:figures/full_fig_p103_23.png]

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Works this paper leans on

50 extracted references · 50 canonical work pages

  1. [1]

    M. S. Agranovich, On a mixed Poincare-Steklov type spectral problem in a Lipschitz domain, Russ. J. Math. Phys. 13(3), 281--290 (2006)

    work page 2006

  2. [2]

    M. S. Agranovich, Mixed problems in a Lipschitz domain for strongly elliptic second-order systems, Func. Anal. Appl. 45(2), 81--98 (2011)

    work page 2011

  3. [3]

    Wave Enhancement through Optimization of Boundary Conditions

    H. Ammari, O. Bruno, K. Imeri, and N. Nigam, Wave enhancement through optimization of boundary conditions, arXiv:1907.11170

    work page Pith review arXiv 1907

  4. [4]

    Optimization of Steklov-Neumann eigenvalues

    H. Ammari, K. Imeri, and N. Nigam, Optimization of Steklov-Neumann eigenvalues, arXiv:1907.11147

    work page Pith review arXiv 1907

  5. [5]

    Berkolaiko, An elementary introduction to quantum graphs, in Geometric and Computational Spectral Theory, eds

    G. Berkolaiko, An elementary introduction to quantum graphs, in Geometric and Computational Spectral Theory, eds. A. Girouard, D. Jakobson, M. Levitin, N. Nigam, I. Polterovich, and F. Rochon, Contemporary Mathematics 700 , American Mathematical Society, Providence, Rhode Island and Centre de Recherches Math\' e matique, Montr\' e al, Qu\' e bec, 2017, 41--72

    work page 2017

  6. [6]

    Berkolaiko and P

    G. Berkolaiko and P. Kuchment, Introduction to quantum graphs, Mathematical Surveys and Monographs 186 , American Mathematical Society, Providence, Rhode Island, 2013

    work page 2013

  7. [7]

    Bolte and S

    J. Bolte and S. Endres, The trace formula for quantum graphs with general self adjoint boundary conditions , Ann. Henri Poincar\'e 10, 189--223 (2009)

    work page 2009

  8. [8]

    Brown, The mixed problem for Laplace's equation in a class of Lipschitz domains, Comm

    R. Brown, The mixed problem for Laplace's equation in a class of Lipschitz domains, Comm. PDEs. 19(7--8), 1217--1233 (1994)

    work page 1994

Show all 50 references
  1. [9]

    Costabel, Boundary integral operators on curved polygons, Ann

    M. Costabel, Boundary integral operators on curved polygons, Ann. Mat. Pur. Appl. 133(1), 305--326 (1983)

    1983

  2. [10]

    Davis, Short surface waves in a canal: dependence of frequency on the curvatures and their derivatives , Q

    A.M.J. Davis, Short surface waves in a canal: dependence of frequency on the curvatures and their derivatives , Q. J. Mech. Appl. Math. 27 , 523--535 (1974)

    1974

  3. [11]

    Duclos, P

    P. Duclos, P. Exner, and P. S t ov\' c ek, Curvature-induced resonances in a two-dimensional Dirichlet tube, Annales de l'I.H.P. Section A 62(1), 81--101 (1995)

    1995

  4. [12]

    Edward, An inverse spectral result for the Neumann operator on planar domains, J

    J. Edward, An inverse spectral result for the Neumann operator on planar domains, J. Func. Anal. 111, 312--322 (1993)

    1993

  5. [13]

    S. A. Fulling, P. Kuchment, and J. H. Wilson, Index theorems for quantum graphs, J. Phys. A 40 14165--14180 (2007)

    2007

  6. [14]

    E. M. Garau and P. Morin, Convergence and quasi-optimality of adaptive FEM for Steklov eigenvalue problems, IMA J. Num. Anal. 31(3), 914--946 (2011)

    2011

  7. [15]

    Gilkey, The spectral geometry of the higher order Laplacian , Duke Math

    P. Gilkey, The spectral geometry of the higher order Laplacian , Duke Math. J. 47(3), 511--528 (1980)

    1980

  8. [16]

    Girouard, J

    A. Girouard, J. Lagac\'e, I. Polterovich and A. Savo, The Steklov spectrum of cuboids , Mathematika 65(2), 272--310 (2019)

    2019

  9. [17]

    Girouard, L

    A. Girouard, L. Parnovski, I. Polterovich, and D. Sher, The Steklov spectrum of surfaces: asymptotics and invariants , Math. Proc. Cambridge Philos. Soc. 157(3), 379--389 (2014)

    2014

  10. [18]

    Girouard and I

    A. Girouard and I. Polterovich, Spectral geometry of the Steklov problem, J. Spectral Theory 7(2), 321--359 (2017)

    2017

  11. [19]

    Gordon, P

    C. Gordon, P. Herbrich, and D. Webb, Robin and Steklov isospectral manifolds, arXiv:1808.10741

    arXiv

  12. [20]

    Grubb and R

    G. Grubb and R. Seeley, Zeta and eta functions for Atiyah-Patodi-Singer operators , J. Geom. Anal. 6(1), 31--77 (1996)

    1996

  13. [21]

    Hecht, New development in FreeFem++ , J

    F. Hecht, New development in FreeFem++ , J. Numer. Math. 20(3-4), 251--266 (2012)

    2012

  14. [22]

    Ivrii, Spectral asymptotics for Dirichlet to Neumann operator , arXiv:1802.07524

    V. Ivrii, Spectral asymptotics for Dirichlet to Neumann operator , arXiv:1802.07524

    arXiv

  15. [23]

    Khalile, Spectral asymptotics for Robin Laplacians on polygonal domains , J

    M. Khalile, Spectral asymptotics for Robin Laplacians on polygonal domains , J. Math. Anal. Appl. 461, 1498--1543 (2018)

    2018

  16. [24]

    Khalile, T

    M. Khalile, T. Ourmi\`eres-Bonafos and K. Pankrashkin, Effective operator for Robin eigenvalues in domains with corners , arXiv:1809.04998

    arXiv

  17. [25]

    Kottos and U

    T. Kottos and U. Smilansky, Periodic orbit theory and spectral statistics for quantum graphs , Ann. Physics 274, 76--124 (1999)

    1999

  18. [26]

    Kuchment and L

    P. Kuchment and L. A. Kunyansky, Spectral properties of high contrast band-gap materials and operators on graphs , Exp. Math. 8(1), 1--28 (1999)

    1999

  19. [27]

    Kuchment and L

    P. Kuchment and L. Kunyansky, Differential operators on graphs and photonic crystals , Adv. in Comput. Math. 16, 263--290 (2002)

    2002

  20. [28]

    Kurasov, Private communication , (2019)

    P. Kurasov, Private communication , (2019)

    2019

  21. [29]

    Kurasov and M

    P. Kurasov and M. Nowaczyk, Geometric properties of quantum graphs and vertex scattering matrices , Opus. Math. 30(3), 295--309 (2010)

    2010

  22. [30]

    Laptev and A

    A. Laptev and A. Hassannezhad, Eigenvalue bounds of mixed Steklov problems , arXiv:1712.00753

    arXiv

  23. [31]

    Levitin, A very brief introduction to eigenvalue computations with FreeFem++ , electronic notes available at http://michaellevitin.net/FreeFem++.html (2018)

    M. Levitin, A very brief introduction to eigenvalue computations with FreeFem++ , electronic notes available at http://michaellevitin.net/FreeFem++.html (2018)

    2018

  24. [32]

    Levitin, L

    M. Levitin, L. Parnovski, and I. Polterovich, Isospectral domains with mixed boundary conditions, J. Phys. A: Math. Gen. 39, 2073--2082 (2006)

    2006

  25. [33]

    Levitin, L

    M. Levitin, L. Parnovski, I. Polterovich, and D. Sher, Sloshing, Steklov and corners: Asymptotics of sloshing eigenvalues , arXiv:1709.01891

    arXiv

  26. [34]

    Matou s ek, Lectures on discrete geometry , Graduate Texts in Mathematics 212, Springer-Verlag, New York, 2002

    J. Matou s ek, Lectures on discrete geometry , Graduate Texts in Mathematics 212, Springer-Verlag, New York, 2002

    2002

  27. [35]

    Mazzeo and J

    R. Mazzeo and J. Rowlett, A heat trace anomaly on polygons , Math. Proc. Cambridge Philos. Soc. 159(2), 303--319 (2015)

    2015

  28. [36]

    L. G. Molinari, Determinants of block tridiagonal matrices , Lin. Algebra Applic. 429, 2221--2226 (2008)

    2008

  29. [37]

    Nursultanov, J

    M. Nursultanov, J. Rowlett and D. Sher, The heat kernel on curvilinear polygonal domains in surfaces , arXiv:1905.00259

    1905 arXiv

  30. [38]

    McLean, Strongly elliptic systems and boundary integral equations , Cambridge University Press, 2000

    W. McLean, Strongly elliptic systems and boundary integral equations , Cambridge University Press, 2000

    2000

  31. [39]

    Perfekt and M

    K.-M. Perfekt and M. Putinar, Spectral bounds for the Neumann-Poincar\'e operator on planar domains with corners , J. d'Anal. Math. 124(1) , 39--57 (2014)

    2014

  32. [40]

    A. S. Peters, The effect of a floating mat on water waves, Comm. Pure and Appl. Math. 3(4), 319--354 (1950)

    1950

  33. [41]

    Polterovich and D

    I. Polterovich and D. Sher, Heat invariants of the Steklov problem , J. Geom. Anal. 25(2), 924--950 (2015)

    2015

  34. [42]

    Polterovich, D

    I. Polterovich, D. Sher, and J. Toth, Nodal length of Steklov eigenfunctions on real-analytic Riemannian surfaces , J. Reine Angew. Math. 754, 17--47 (2019)

    2019

  35. [43]

    Rellich, Perturbation theory of eigenvalue problems , Courant Inst

    F. Rellich, Perturbation theory of eigenvalue problems , Courant Inst. Math. Sci., New York (1954)

    1954

  36. [44]

    Rondi, Continuity properties of Neumann-to-Dirichlet maps with respect to the H -convergence of the coefficient matrices , Inverse Problems 31, 045002 (2015)

    L. Rondi, Continuity properties of Neumann-to-Dirichlet maps with respect to the H -convergence of the coefficient matrices , Inverse Problems 31, 045002 (2015)

    2015

  37. [45]

    G. V. Rozenblyum, On the asymptotics of the eigenvalues of certain two-dimensional spectral problems , Sel. Math. Sov. 5, 233--244 (1986)

    1986

  38. [46]

    Rueckriemen, Heat trace asymptotics for quantum graphs , arXiv:1212.2840

    R. Rueckriemen, Heat trace asymptotics for quantum graphs , arXiv:1212.2840

    arXiv

  39. [47]

    Safarov, Riesz means of a function of the distribution of the eigenvalues of an elliptic operator , Zap

    Yu. Safarov, Riesz means of a function of the distribution of the eigenvalues of an elliptic operator , Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 163, 143--145 (1987) (Russian); translation in J. Soviet Math. 49(5), 1210--1212 (1990)

    1987

  40. [48]

    Taylor, Partial Differential Equations II: Qualitative Studies of Linear Equations , Springer, New York, 1996

    M. Taylor, Partial Differential Equations II: Qualitative Studies of Linear Equations , Springer, New York, 1996

    1996

  41. [49]

    Ursell, Short surface waves in a canal: dependence of frequency on curvature , J

    F. Ursell, Short surface waves in a canal: dependence of frequency on curvature , J. Fluid Mech. 63(1), 177--181 (1974)

    1974

  42. [50]

    Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains , J

    G. Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains , J. Func. Anal. 59, 572-611 (1984)

    1984

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