Pith. sign in

REVIEW 4 major objections 4 minor 67 references

On the eigenvalues of the Robin Laplacian with a complex parameter

T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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

Pith's one-line read For the Robin Laplacian with a complex boundary parameter, every analytic eigenvalue curve either converges to an eigenvalue of the Dirichlet Laplacian or diverges to infinity in the complex plane.

desk verdict A genuinely new paper on complex-α Robin eigenvalues with a solid DtN core, but the global eigenvalue-labelling theorem overreaches and should be reframed. read the letter →

arxiv 1908.06041 v2 pith:GL7VTVGG submitted 2019-08-16 math.SP math.AP

Sabine Bögli , James B. Kennedy , Robin Lang This is my paper · ORCID
classification math.SPmath.AP MSC 35J0535J2535P1035P1535S0547A1081Q12
keywords RobinLaplaciancomplexparameterDirichlet-to-Neumannoperatoreigenvalueasymptoticsnon-self-adjointspectraltheoryanalyticcurvesnumericalrangeAbelbasis
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 studies the Robin Laplacian on a bounded Lipschitz domain when the boundary parameter α is complex, and it establishes that the large-α behaviour of every eigenvalue is fully dichotomous: each analytic eigenvalue curve λ(α) either converges to an eigenvalue of the Dirichlet Laplacian or diverges to infinity in C. The proof replaces the variational min-max techniques that work for real α with a duality between the Robin eigenvalue problem and the Dirichlet-to-Neumann operator, whose eigenvalues as functions of λ are exactly the Robin parameters. This duality gives a short proof of the dichotomy and, together with new numerical-range bounds, shows that when α escapes to infinity away from the negative real semi-axis all accumulation points are Dirichlet eigenvalues, whereas arbitrary complex accumulation points become possible if α is allowed to approach the negative real semi-axis. Explicit interval, hyperrectangle and ball examples yield divergent eigenvalue curves with leading behaviour -$α^{2}$, and the paper formulates a conjecture that on smooth domains in dimension at least two, all eigenvalues converge to the Dirichlet spectrum whenever Re α remains bounded below.

What carries the argument

The load-bearing object is the Dirichlet-to-Neumann operator M(λ), which maps given Dirichlet data g for solutions of -Δu=λu to the negative outer normal derivative of that solution. The duality is that λ is a Robin eigenvalue with parameter α if and only if α is an eigenvalue of M(λ), so the roles of α and λ are exchanged: instead of following λ(α), one follows the eigenvalue branches α(λ) of a meromorphic family whose poles are exactly the Dirichlet eigenvalues. That meromorphy, together with the fact that an eigenvalue branch of a holomorphic family cannot appear or disappear without crossing a singularity, is what forces the dichotomy of Theorem 1.5. A secondary set of tools supports this: sectoriality of the associated form (its numerical range lies in a fixed sector), a derivative formula λ'(α)=∫∂Ωψ²dσ/∫Ωψ²dx for simple eigenvalues, the numerical-range estimate that confines eigenvalues to a parabolic region, and the basis theorem for forms with a self-adjoint principal part showing the eigenfunctions form an Abel basis but never an orthonormal basis for non-real α.

What would settle it

On a bounded Lipschitz domain, numerically follow a simple Robin eigenvalue curve λ(α) along a sequence α_k→∞ with Re α_k≥0; if a finite accumulation point outside the Dirichlet spectrum of -Δ_D is found, then Theorem 1.5 (and the accumulation-point theorem) would be false, and the Dirichlet-to-Neumann pole argument would have to break down.

Watch

Extended reading notes

Core claim

The central discovery is that the spectrum of the Robin Laplacian with complex parameter is globally organised into analytic eigenvalue curves, and as α→∞ in C each such curve either has all its finite accumulation points in the Dirichlet spectrum or leaves every compact set. This is proved by studying the Dirichlet-to-Neumann operator M(λ), which is meromorphic in λ with poles exactly at the Dirichlet eigenvalues; boundedness of an eigenvalue branch forces λ to approach a pole, and divergence of λ forces the corresponding parameter α to diverge. The same machinery yields a complementary classification of accumulation points: if α diverges in a sector away from the negative real semi-axis, all Robin eigenvalues have only Dirichlet eigenvalues as accumulation points, regardless of curve choice, while for arbitrary α_k→∞ near the negative real semi-axis every λ∈C can be realised as a Robin eigenvalue. For the interval, hyperrectangle and ball, the dichotomy is made quantitative, giving asymptotic expansions whose leading divergent term is -$α^{2}$ in all three cases.

Load-bearing premise

The whole framework rests on the global labelling of eigenvalues into analytic curves λ_k(α) on all of C, which the paper derives from self-adjoint-holomorphic family theory; the stated identity A(α)*=A(α) is false for non-real α, so the correct identity A(α)*=A(ar α) must be used for that derivation to go through.

Editorial extensions

If this is right

  • For large complex impedance on a bounded Lipschitz domain, spectral accumulation cannot occur outside the Dirichlet spectrum as long as Re α is bounded below or |Re α/Im α| is controlled.
  • The dichotomy applies to real α as well, closing a previously listed open problem about whether Robin eigenvalues can converge to non-Dirichlet limits as α→∞.
  • On an interval, α→∞ in a left half-plane sector produces exactly two divergent eigenvalues with λ(α)=-α²+O(α²e^{2a Re α}), while all other eigenvalues converge to Dirichlet eigenvalues.
  • On hyperrectangles there are infinitely many divergent eigenvalue curves at levels -jα² for j=1,...,d-1 and exactly 2^d curves at level -dα², and on the ball the divergent curves satisfy λ(α)=-α²+(d-1)α+O(1).
  • The numerical-range bound Re λ≥-(C₁²/4)|Re α|²-C₂|Re α| is new even for real negative α on general Lipschitz domains and gives a two-sided bound on the principal eigenvalue as α→-∞.

Reading between the lines

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

  • If the global analytic eigencurve labelling is repaired by using the correct self-adjoint-holomorphic identity A(α)*=A(ar α) in place of the paper's stated A(α)*=A(α), the dichotomy and all subsequent results should survive unchanged, since the misstatement does not affect the meromorphy argument.
  • The Dirichlet-to-Neumann method should transfer directly to other non-self-adjoint boundary-value problems, including quantum graphs with complex δ-couplings, where the same duality can turn a spectral asymptotics problem into a pole analysis of a meromorphic matrix family.
  • The conjecture that all eigenvalues converge when Re α is bounded below, if true, implies that large imaginary parts of Robin eigenvalues can only be generated by α approaching the negative real semi-axis; this gives a concrete spectral-instability criterion for impedance-type boundary conditions.
  • For variable boundary coefficients α∈L∞(∂Ω), the numerical-range estimates persist but the analytic-curve dichotomy is likely to fail, and one could test this by constructing a boundary function that creates eigenvalue crossings or non-meromorphic parameter dependence.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. This paper studies the Robin Laplacian on bounded Lipschitz domains with a complex Robin parameter α. It establishes basic operator-theoretic properties (m-sectoriality, holomorphic dependence of eigenvalues and eigenprojections, Abel/Riesz basis results and failure of orthonormal bases for non-real α), gives new numerical-range and trace bounds with explicit constants, and uses a Dirichlet-to-Neumann duality to prove a dichotomy for eigenvalue branches as |α|→∞: each branch either converges to a point in the Dirichlet spectrum or diverges to infinity. The final sections work out detailed asymptotic expansions for intervals, hyperrectangles, and balls, and the paper closes with a conjecture for general smooth domains.

Significance. If the central claims can be made fully rigorous, this would be a substantial contribution to the spectral theory of non-self-adjoint Robin Laplacians. The Dirichlet-to-Neumann duality is developed carefully and is genuinely useful; the numerical-range bounds appear new even for real negative α; and the explicit interval, rectangle, and ball computations provide concrete, testable asymptotics. The paper is essentially self-contained, and I see no circularity: the DtN duality is independently proved, and the numerical-range bounds rest on trace inequalities rather than on the conclusions being drawn. The main reservations concern the global labelling of eigenvalue curves and a few statements whose proofs are too terse or one-sided.

major comments (4)
  1. [Theorem 4.1(1), §4.1] The asserted identity A(α)* = A(α) is false for non-real α. The adjoint of the Robin Laplacian with parameter α is the Robin Laplacian with parameter Ā, so the correct self-adjoint-holomorphic condition is A(α)* = A(Ā). This matters because the subsequent proof of Theorem 4.1(2) relies on Kato's theory of self-adjoint holomorphic families; the reader must supply the corrected identity to even invoke that theory. Please correct the statement and make explicit that self-adjointness holds only for real α.
  2. [Theorem 4.1(2) and proof, §4.1] The claim that each λ_k(α) extends to a meromorphic function on all of C, with only algebraic singularities at non-real crossings, is not justified by the cited Kato theory and is likely false in general. Kato's theorem gives local analytic branches away from crossing points, but at crossings the branches can have square-root-type branch points. For example, the family A(κ)=[[κ,1],[1,-κ]] satisfies A(κ)*=A(Ā) and has eigenvalues ±√(κ²+1), with branch points at κ=±i; no single-valued meromorphic extension to C exists. The compact-exhaustion argument in the proof does not address this monodromy obstruction. Since Theorem 1.5 and Remark 4.3 quantify over 'each analytic eigenvalue curve', the global-labelling issue is load-bearing: the dichotomy must either be proved for locally defined analytic branches along paths, or an additional no-branching property must be established for the Robin Laplacian.
  3. [Theorem 1.6(2) and §8] The statement that for every λ∈C there exist α_k with |α_k|→∞ such that λ is itself a Robin eigenvalue for every k is stronger than the preceding accumulation statement, and the proof for λ in the Dirichlet spectrum is not supplied. The one-sentence appeal to a 'multivalued operator' in Remark 7.3(2) does not define the eigenvalue equation for that multivalued operator or show that it has solutions α_k with |α_k|→∞. If only an accumulation statement is intended, the 'more precisely' sentence should be weakened; if the exact statement is intended, it needs a real proof, especially at points of σ(−Δ_D).
  4. [Theorem 5.1, §5] The proof assumes the inclusion σ(A(α)) ⊂ {Im z ≥ 0} and then that all eigenvalues have Im λ_k ≥ 0. This is valid only for Im α > 0. For Im α < 0 the numerical range lies in the opposite half-plane and the contradiction as written does not apply. Since the theorem claims the result for all α∈C\R, the proof must cover both half-planes, for example by arguing with the adjoint or by replacing α by Ā after justifying that the assumed orthonormal basis property transfers.
minor comments (4)
  1. [Theorem 4.1(2)] The phrase 'meromorphic function with at most algebraic singularities' is internally contradictory: algebraic branch points are not poles of a single-valued meromorphic function. Please rephrase the assertion in terms of analytic continuation and Puiseux series.
  2. [Definition 2.2 and §2.1] The sector notation is unnecessarily confusing, and the sentence beginning 'if π/2<θ′<π, we set define T+...' is incomplete. Also, the statement that λ=0 is a Dirichlet eigenvalue of the interval is inconsistent with the standard Dirichlet spectrum (π²j²/(4a²), j≥1); please clarify whether 0 is being treated only as a singular point of the DtN representation.
  3. [Theorem 9.8] The index of the Bessel function in the statement and proof appears inconsistent: formula (9.19) and the definition of m_l in the proof point to m = d/2 + l − 1, while the displayed result uses d/2 − l + 1. Please check and correct.
  4. [§9.1] There are numerous typos in this section, including 'tideous' and 'ourselved'. The phrase 'α→∞ in the left half-plane away from the imaginary axis' should presumably be 'Re α→−∞'. These do not affect the mathematics but should be cleaned up.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main eigenvalue dichotomy is derived from independently proved Dirichlet-to-Neumann duality and trace estimates, with self-citations only contextual.

full rationale

The paper's central claims are derived from first-principles arguments rather than from fitted inputs or self-citations. Theorem 1.5 is proved in Section 7 using the Dirichlet-to-Neumann duality stated and proved in Theorem 7.6: the weak forms of the Robin and DtN eigenvalue equations are compared directly, and the meromorphic structure of M(λ) with poles exactly at the Dirichlet spectrum is obtained from the explicit perturbation formula (7.9) and the resolvent of the Dirichlet Laplacian. No eigenvalue asymptotics are assumed to prove the dichotomy; the conclusion that a bounded analytic branch must approach a Dirichlet eigenvalue follows from the pole structure of M(λ). Likewise, the numerical-range bounds in Theorem 6.1 and the new real-α bound in Corollary 1.4 rest on the trace inequality Lemma 6.5, which is proved by geometric arguments (distance-function and local graph estimates) with constants chosen from the geometry, not fitted to spectral data. The analytic-eigencurve framework in Theorem 4.1 cites Kato's standard theory, an external source, not a self-citation. Even if the global labelling statement in Theorem 4.1(2) is open to a mathematical correctness objection concerning branch points for non-real crossings, that is not a circularity: the statement does not presuppose the conclusion of Theorem 1.5. The authors' self-citation [18] (Bucur-Freitas-Kennedy) is used for background, open-problem context, and comparisons; it is not load-bearing in any proof. Other cited results such as [19], [37], and [52] are independent external works. The explicit interval, hyperrectangle, and ball examples are solved by direct calculation of transcendental equations and Bessel-function asymptotics. There is no fitted parameter renamed as a prediction and no output quantity used as an input. The paper is therefore self-contained with respect to its main claimed derivation chain.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

All inputs are standard theorems from the literature; no free parameters are fitted. The constants C1,C2 in Theorem 6.1 are proven to exist via trace inequalities and are not tuned to data. The only ad hoc element is the sector decomposition in Definition 2.2, which is a proof tool, not an assumption.

assumptions (6)
  • standard math Kato's perturbation theory for holomorphic families of operators, including self-adjoint holomorphic families
    Used in Section 4 to obtain analytic eigenvalue curves and eigenprojections; cited [40].
  • standard math Trace theorem and compact embeddings H1(Ω) into L2(Ω) and L2(∂Ω)
    Basis for the form (3.4), sectoriality, and basis properties; used throughout.
  • standard math Weyl asymptotics for Robin Laplacians
    Used in proof of Theorem 5.7(ii) to determine the Abel basis order; cited [39,65].
  • standard math Agranovich's theorem on Abel, Riesz and Bari bases for forms with self-adjoint principal part
    Quoted as Theorem 5.6 and applied in Theorem 5.7.
  • standard math Standard asymptotics of Bessel and Hankel functions
    Used in Section 9.3 for ball asymptotics; cited [1].
  • standard math Rouché's theorem
    Used in interval and ball inversion of Dirichlet-to-Neumann asymptotics.

how reviews work

0 comments
Cite this review

Pith. "Pith review of On the eigenvalues of the Robin Laplacian with a complex parameter." pith.science (2026). https://pith.science/paper/GL7VTVGG

@misc{pith2026190806041,
  author       = {Pith},
  title        = {Pith review of: On the eigenvalues of the Robin Laplacian with a complex parameter},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GL7VTVGG}},
  note         = {Machine review of arXiv:1908.06041}
}
abstract

We study the spectrum of the Robin Laplacian with a complex Robin parameter $\alpha$ on a bounded Lipschitz domain $\Omega$. We start by establishing a number of properties of the corresponding operator, such as generation properties, local analytic dependence of the eigenvalues and eigenspaces on $\alpha \in \mathbb C$, and basis properties of the eigenfunctions. Our focus, however, is on bounds and asymptotics for the eigenvalues as functions of $\alpha$: we start by providing estimates on the numerical range of the associated operator, which lead to new eigenvalue bounds even in the case $\alpha \in \mathbb R$. For the asymptotics of the eigenvalues as $\alpha \to \infty$ in $\mathbb C$, in place of the min-max characterisation of the eigenvalues and Dirichlet-Neumann bracketing techniques commonly used in the real case, we exploit the duality between the eigenvalues of the Robin Laplacian and the eigenvalues of the Dirichlet-to-Neumann map. We use this to show that every Robin eigenvalue either diverges to $\infty$ in $\mathbb C$ or converges to a point in the spectrum of the Dirichlet Laplacian, and also to give a comprehensive treatment of the special cases where $\Omega$ is an interval, a hyperrectangle or a ball. This leads to the conjecture that on a general smooth domain in dimension $d\geq 2$ all eigenvalues converge to the Dirichlet spectrum if ${\rm Re}\, \alpha$ remains bounded from below as $\alpha \to \infty$, while if ${\rm Re}\, \alpha \to -\infty$, then there is a family of divergent eigenvalue curves, each of which behaves asymptotically like $-\alpha^2$.

Figures

Figures reproduced from arXiv: 1908.06041 by the authors.

Figure 2.1
Figure 2.1. (2) If θ = π/2, the sectors S ± θ vanish and T ± θ are defined as in (2.7). Furthermore, if π/2 < θ0 < π, we set define T + θ 0 by (2.7), that is, a partition of the complex plane in two sectors T + θ 0 and T − π−θ 0 [PITH_FULL_IMAGE:figures/full_fig_p007_2_1.png] view at source ↗
Figure 6.1
Figure 6.1. The set ΛΩ,α, which contains the numerical range W(aα), for a represen￾tative choice of Re α > 0 and Im α > 0, corresponding to the region between the curve ∂ΛΩ,α and the real axis. The region is composed of the union of segments of the form {t + α · s ∈ C : s ∈ [0, C1 √ t + C2]}, each of slope Im α/Re α, for different values of t ≥ 0; the dotted lines show these segments for selected values of t1, . . . , t4 > 0. T… view at source ↗
Figure 6.2
Figure 6.2. The set ΛΩ,α for Re α < 0 and two different choices of Im α > 0 (whose upper boundaries correspond to the solid and dashed curves, respectively). As Im α → 0, the region collapses to the part of the real axis from − C 2 1 4 |Re α| 2 − C2|Re α| to +∞. Remark 6.2. (1) We recall the bound λ1(α) < −|α| 2 (6.2) on the principal Robin eigenvalue λ1(α) = λ1(−∆α Ω) of any bounded Lipschitz domain Ω ⊂ R d for α < 0, which ma… view at source ↗
Figures from the paper (1 more)
Figure 9.1
Figure 9.1. Figure 9.1: On the left a path in the λ-plane from one pole λ1 of the Dirichlet-to￾Neumann operator to the next one λ2 while passing a zero (the second Neumann eigen￾value µ2). On the right-hand side the real curve λ(α) ∈ R increasing from λ1 to λ2 as α ∈ R tends from −∞ to +∞ […

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

67 extracted references · 66 canonical work pages

  1. [18]

    The Robin problem

    Bucur, D., Freitas, P., and Kennedy, J. The Robin problem. In Shape optimization and spectral theory. De Gruyter Open, Warsaw, 2017, pp. 78–119

    work page 2017

  2. [1]

    Abramowitz, M., and Stegun, I. A. Handbook of mathematical functions with formulas, graphs, and mathemat- ical tables , vol. 55 of National Bureau of Standards Applied Mathematics Series . U.S. Government Printing Office, Washington, D.C., 1964

    work page 1964

  3. [2]

    A., and Brown, K

    Afrouzi, G. A., and Brown, K. J. On principal eigenvalues for boundary value problems with indefinite weight and Robin boundary conditions. Proc. Amer. Math. Soc. 127 , 1 (1999), 125–130

    work page 1999

  4. [3]

    S.On series in root vectors of operators defined by forms with a selfadjoint principal part.Funktsional

    Agranovich, M. S.On series in root vectors of operators defined by forms with a selfadjoint principal part.Funktsional. Anal. i Prilozhen. 28 , 3 (1994), 1–21, 95

    work page 1994

  5. [4]

    Alber, H.-D., and Ramm, A. G. Asymptotics of the solution to Robin problem. J. Math. Anal. Appl. 349 , 1 (2009), 156–164

    work page 2009

  6. [5]

    Antunes, P. R. S., Freitas, P., and Krejˇciˇr´ık, D. Bounds and extremal domains for Robin eigenvalues with negative boundary parameter. Adv. Calc. Var. 10 , 4 (2017), 357–379. ON THE EIGENVALUES OF THE ROBIN LAPLACIAN 37

    work page 2017

  7. [6]

    Arendt, W., Batty, C. J. K., Hieber, M., and Neubrander, F. Vector-valued Laplace transforms and Cauchy problems, second ed., vol. 96 of Monographs in Mathematics. Birkh¨ auser/Springer Basel AG, Basel, 2011

    work page 2011

  8. [7]

    Friedlander’s eigenvalue inequalities and the Dirichlet-to-Neumann semigroup

    Arendt, W., and Mazzeo, R. Friedlander’s eigenvalue inequalities and the Dirichlet-to-Neumann semigroup. Com- mun. Pure Appl. Anal. 11 , 6 (2012), 2201–2212

    work page 2012

Show all 67 references
  1. [8]

    Arendt, W., and ter Elst, A. F. M.The Dirichlet-to-Neumann operator on rough domains.J. Differential Equations 251, 8 (2011), 2100–2124

    2011

  2. [9]

    Arendt, W., and ter Elst, A. F. M. Sectorial forms and degenerate differential operators. J. Operator Theory 67 , 1 (2012), 33–72

    2012

  3. [10]

    A., Evans, W

    Balinsky, A. A., Evans, W. D., and Lewis, R. T. The analysis and geometry of Hardy’s inequality . Universitext. Springer, Cham, 2015

    2015

  4. [11]

    An inverse problem of Calder´ on type with partial data.Comm

    Behrndt, J., and Rohleder, J. An inverse problem of Calder´ on type with partial data.Comm. Partial Differential Equations 37, 6 (2012), 1141–1159

    2012

  5. [12]

    Behrndt, J., and ter Elst, A. F. M. Dirichlet-to-Neumann maps on bounded Lipschitz domains. J. Differential Equations 259, 11 (2015), 5903–5926

    2015

  6. [13]

    Behrndt, J., and ter Elst, A. F. M. Jordan chains of elliptic partial differential operators and Dirichlet-to-Neumann maps. arXiv preprints (2019), arXiv:1905.12041

    2019 arXiv

  7. [14]

    PT -symmetric waveguides

    Borisov, D., and Krej ˇciˇr´ık, D. PT -symmetric waveguides. Integral Equations Operator Theory 62 , 4 (2008), 489– 515

    2008

  8. [15]

    The effective Hamiltonian for thin layers with non-Hermitian Robin-type boundary conditions

    Borisov, D., and Krej ˇciˇr´ık, D. The effective Hamiltonian for thin layers with non-Hermitian Robin-type boundary conditions. Asymptot. Anal. 76 , 1 (2012), 49–59

    2012

  9. [16]

    Boundary triplets and M-functions for non-selfadjoint operators, with applications to elliptic PDEs and block operator matrices

    Brown, M., Marletta, M., Naboko, S., and Wood, I. Boundary triplets and M-functions for non-selfadjoint operators, with applications to elliptic PDEs and block operator matrices. J. Lond. Math. Soc. (2) 77 , 3 (2008), 700–718

    2008

  10. [17]

    On the negative spectrum of the Robin Laplacian in corner domains

    Bruneau, V., and Popoff, N. On the negative spectrum of the Robin Laplacian in corner domains. Anal. PDE 9 , 5 (2016), 1259–1283

    2016

  11. [19]

    On the asymptotics of a Robin eigenvalue problem

    Cakoni, F., Chaulet, N., and Haddar, H. On the asymptotics of a Robin eigenvalue problem. C. R. Math. Acad. Sci. Paris 351 , 13-14 (2013), 517–521

    2013

  12. [20]

    The direct and inverse scattering problems for partially coated obstacles

    Cakoni, F., Colton, D., and Monk, P. The direct and inverse scattering problems for partially coated obstacles. Inverse Problems 17 , 6 (2001), 1997–2015

    2001

  13. [21]

    Chandler-Wilde, S. N. The impedance boundary value problem for the Helmholtz equation in a half-plane. Math. Methods Appl. Sci. 20 , 10 (1997), 813–840

    1997

  14. [22]

    Absence of eigenvalues of non-self-adjoint Robin Laplacians on the half-space

    Cossetti, L., and Krejˇciˇr´ık, D. Absence of eigenvalues of non-self-adjoint Robin Laplacians on the half-space. arXiv preprints (2018), arXiv:1812.05348

    2018 arXiv

  15. [23]

    Non-positivity of the semigroup generated by the Dirichlet-to-Neumann operator

    Daners, D. Non-positivity of the semigroup generated by the Dirichlet-to-Neumann operator. Positivity 18, 2 (2014), 235–256

    2014

  16. [24]

    Daners, D., and Kennedy, J. B. On the asymptotic behaviour of the eigenvalues of a Robin problem. Differential Integral Equations 23, 7-8 (2010), 659–669

    2010

  17. [25]

    F., and Xu, Y

    Dunkl, C. F., and Xu, Y. Orthogonal polynomials of several variables , vol. 81 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 2001

    2001

  18. [26]

    C., and Gariepy, R

    Evans, L. C., and Gariepy, R. F. Measure theory and fine properties of functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992

    1992

  19. [27]

    Asymptotic eigenvalue estimates for a Robin problem with a large parameter

    Exner, P., Minakov, A., and Parnovski, L. Asymptotic eigenvalue estimates for a Robin problem with a large parameter. Port. Math. 71 , 2 (2014), 141–156

    2014

  20. [28]

    Curvature measures

    Federer, H. Curvature measures. Trans. Amer. Math. Soc. 93 (1959), 418–491

    1959

  21. [29]

    On the eigenvalues of a Robin problem with a large parameter

    Filinovskiy, A. On the eigenvalues of a Robin problem with a large parameter. Math. Bohem. 139, 2 (2014), 341–352

    2014

  22. [30]

    Filinovskiy, A. V. On the asymptotic behavior of eigenvalues and eigenfunctions of the Robin problem with large parameter. Math. Model. Anal. 22 , 1 (2017), 37–51

    2017

  23. [31]

    The first Robin eigenvalue with negative boundary parameter.Adv

    Freitas, P., and Krejcirik, D. The first Robin eigenvalue with negative boundary parameter.Adv. Math. 280 (2015), 322–339

    2015

  24. [32]

    Generalized Robin boundary conditions, Robin-to-Dirichlet maps, and Krein-type resolvent formulas for Schr¨ odinger operators on bounded Lipschitz domains

    Gesztesy, F., and Mitrea, M. Generalized Robin boundary conditions, Robin-to-Dirichlet maps, and Krein-type resolvent formulas for Schr¨ odinger operators on bounded Lipschitz domains. In Perspectives in partial differential equations, harmonic analysis and applications , vol. ...

    2008

  25. [33]

    Gilbarg, D., and Trudinger, N. S. Elliptic partial differential equations of second order. Springer-Verlag, Berlin-New York, 1977. Grundlehren der Mathematischen Wissenschaften, Vol. 224

    1977

  26. [34]

    Eigenvalue estimates and critical temperature in zero fields for enhanced surface supercon- ductivity

    Giorgi, T., and Smits, R. Eigenvalue estimates and critical temperature in zero fields for enhanced surface supercon- ductivity. Z. Angew. Math. Phys. 58 , 2 (2007), 224–245

    2007

  27. [35]

    Bounds and monotonicity for the generalized Robin problem

    Giorgi, T., and Smits, R. Bounds and monotonicity for the generalized Robin problem. Z. Angew. Math. Phys. 59 , 4 (2008), 600–618

    2008

  28. [36]

    Eigenvalues for the Robin Laplacian in domains with variable curvature.Trans

    Helffer, B., and Kachmar, A. Eigenvalues for the Robin Laplacian in domains with variable curvature.Trans. Amer. Math. Soc. 369 , 5 (2017), 3253–3287

    2017

  29. [37]

    Non-self-adjoint graphs

    Hussein, A., Krejˇciˇr´ık, D., and Siegl, P. Non-self-adjoint graphs. Trans. Amer. Math. Soc. 367, 4 (2015), 2921–2957

    2015

  30. [38]

    I.Inner product structures , vol

    Istr˘at ¸escu, V. I.Inner product structures , vol. 25 of Mathematics and its Applications . D. Reidel Publishing Co., Dordrecht, 1987. Theory and applications

    1987

  31. [39]

    Ivri˘ı, V. J. The second term of the spectral asymptotics for a Laplace-Beltrami operator on manifolds with boundary. Funktsional. Anal. i Prilozhen. 14 , 2 (1980), 25–34. 38 SABINE B ¨OGLI, JAMES B. KENNEDY, AND ROBIN LANG

    1980

  32. [40]

    Perturbation theory for linear operators , second ed

    Kato, T. Perturbation theory for linear operators , second ed. Springer-Verlag, Berlin-New York, 1976. Grundlehren der Mathematischen Wissenschaften, Band 132

    1976

  33. [41]

    Spectral asymptotics for Robin Laplacians on polygonal domains

    Khalile, M. Spectral asymptotics for Robin Laplacians on polygonal domains. J. Math. Anal. Appl. 461 , 2 (2018), 1498–1543

    2018

  34. [42]

    Effective operators for robin eigenvalues in domains with corners

    Khalile, M., Ourmi `eres-Bonafos, T., and Pankrashkin, K. Effective operators for robin eigenvalues in domains with corners. arXiv preprints (2018), arXiv:1809.04998

    2018 arXiv

  35. [43]

    Eigenvalues of Robin Laplacians in infinite sectors

    Khalile, M., and Pankrashkin, K. Eigenvalues of Robin Laplacians in infinite sectors. Math. Nachr. 291, 5-6 (2018), 928–965

    2018

  36. [44]

    Kovaˇr´ık, H., and Pankrashkin, K.On thep-Laplacian with Robin boundary conditions and boundary trace theorems. Calc. Var. Partial Differential Equations 56 , 2 (2017), Art. 49, 29

    2017

  37. [45]

    Closed formula for the metric in the Hilbert space of a PT -symmetric model

    Krejˇciˇr´ık, D., B ´ıla, H., and Znojil, M. Closed formula for the metric in the Hilbert space of a PT -symmetric model. J. Phys. A 39 , 32 (2006), 10143–10153

    2006

  38. [46]

    Reduction of dimension as a consequence of norm-resolvent convergence and applications

    Krejˇciˇr´ık, D., Raymond, N., Royer, J., and Siegl, P. Reduction of dimension as a consequence of norm-resolvent convergence and applications. Mathematika 64, 2 (2018), 406–429

    2018

  39. [47]

    A., Ockendon, J

    Lacey, A. A., Ockendon, J. R., and Sabina, J. Multidimensional reaction diffusion equations with nonlinear bound- ary conditions. SIAM J. Appl. Math. 58 , 5 (1998), 1622–1647

    1998

  40. [48]

    Resonance regimes of scattering by small bodies with impedance boundary conditions

    Lakshtanov, E., and Vainberg, B. Resonance regimes of scattering by small bodies with impedance boundary conditions. J. Phys. A 43 , 41 (2010), 415205, 16

    2010

  41. [49]

    Langdon, S., and Chandler-Wilde, S. N. A wavennumber independent boundary element method for an acoustic scattering problem. SIAM J. Numer. Anal. 43 , 6 (2006), 2450–2477

    2006

  42. [50]

    On the principal eigenvalue of a Robin problem with a large parameter

    Levitin, M., and Parnovski, L. On the principal eigenvalue of a Robin problem with a large parameter. Math. Nachr. 281, 2 (2008), 272–281

    2008

  43. [51]

    Lidski˘ı, V. B. Summability of series in terms of the principal vectors of non-selfadjoint operators. Trudy Moskov. Mat. Obˇ sˇ c. 11(1962), 3–35

    1962

  44. [52]

    A singularly perturbed linear eigenvalue problem inC1 domains

    Lou, Y., and Zhu, M. A singularly perturbed linear eigenvalue problem inC1 domains. Pacific J. Math. 214 , 2 (2004), 323–334

    2004

  45. [53]

    Eigenvalue problems on exterior domains and Dirichlet to Neumann maps

    Marletta, M. Eigenvalue problems on exterior domains and Dirichlet to Neumann maps. J. Comput. Appl. Math. 171, 1-2 (2004), 367–391

    2004

  46. [54]

    McCartin, B. J. Laplacian eigenstructure of the equilateral triangle . Hikari Ltd., Ruse, 2011

    2011

  47. [55]

    Strongly elliptic systems and boundary integral equations

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

    2000

  48. [56]

    Les m´ ethodes directes en th´ eorie des ´ equations elliptiques

    Neˇcas, J. Les m´ ethodes directes en th´ eorie des ´ equations elliptiques. Masson et Cie, Paris, 1967

    1967

  49. [57]

    Bound states in waveguides with complex Robin boundary conditions

    Nov´ak, R. Bound states in waveguides with complex Robin boundary conditions. Asymptot. Anal. 96 , 3-4 (2016), 251–281

    2016

  50. [58]

    Resonant alteration of propagation in guiding structures with complex Robin parameter and its magnetic- field-induced restoration

    Olendski, O. Resonant alteration of propagation in guiding structures with complex Robin parameter and its magnetic- field-induced restoration. Ann. Physics 326 (2011), 1479–1500

    2011

  51. [59]

    Guiding structures with multiply connected cross sections: Evolution of propagation in external fields at complex R obin parameters

    Olendski, O. Guiding structures with multiply connected cross sections: Evolution of propagation in external fields at complex R obin parameters. Ann. Physics 327 (2012), 1365–1390

    2012

  52. [60]

    The inverse Robin boundary value problem in a half-space

    P¨aiv¨arinta, L., and Zubeldia, M. The inverse Robin boundary value problem in a half-space. Appl. Anal. 94 , 12 (2015), 2565–2587

    2015

  53. [61]

    Mean curvature bounds and eigenvalues of Robin Laplacians

    Pankrashkin, K., and Popoff, N. Mean curvature bounds and eigenvalues of Robin Laplacians. Calc. Var. Partial Differential Equations 54 , 2 (2015), 1947–1961

    2015

  54. [62]

    Titchmarsh-Weyl Theory and Inverse Problems for Elliptic Differential Operators , 1 ed., vol

    Rohleder, J. Titchmarsh-Weyl Theory and Inverse Problems for Elliptic Differential Operators , 1 ed., vol. 19 of Monographic Series TU Graz: Computation in Engineering and Science . Verlag der Technischen Universit¨ at Graz, 2013

    2013

  55. [63]

    Real and complex analysis , third ed

    Rudin, W. Real and complex analysis , third ed. McGraw-Hill Book Co., New York, 1987

    1987

  56. [64]

    Shanin, A. V. Excitation of a wave field in a triangular domain with impedance boundary conditions. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 250 , Mat. Vopr. Teor. Rasprostr. Voln. 27 (1998), 300–318, 340

    1998

  57. [65]

    G., and Safarov, Y

    Vasil′ev, D. G., and Safarov, Y. G. The asymptotic distribution of eigenvalues of differential operators. In Spectral theory of operators (Novgorod, 1989), vol. 150 of Amer. Math. Soc. Transl. Ser. 2 . Amer. Math. Soc., Providence, RI, 1992, pp. 55–110

    1989

  58. [66]

    T., and Watson, G

    Whittaker, E. T., and Watson, G. N. A course of modern analysis . Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1996. Reprint of the fourth (1927) edition

    1927

  59. [67]

    Differential-operator equations, vol

    Yakubov, S., and Yakubov, Y. Differential-operator equations, vol. 103 of Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics . Chapman & Hall/CRC, Boca Raton, FL, 2000. Ordinary and partial differential equations. Sabine B¨ogli, Department of Mathematical ...

    2000

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.