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Surface plasmons in metamaterial cavities: Scattering by obstacles with negative wave speed

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Pith's one-line read For non-trapping metamaterial cavities with negative wave speed, every resonance sequence near the real axis consists of surface plasmons concentrated on the boundary, and their number satisfies an explicit boundary Weyl law.

desk verdict Near-real resonances in metamaterial cavities are exactly boundary-localized plasmons, with a Weyl law; the proof is careful and the main limitation (non-trapping exterior) is explicit. read the letter →

arxiv 2505.22253 v1 pith:F64WSTCF submitted 2025-05-28 math.SP math-phmath.APmath.MPphysics.optics

Yan-Long Fang , Jeffrey Galkowski This is my paper
classification math.SPmath-phmath.APmath.MPphysics.optics MSC 35P2535B3458J4058J50
keywords surfaceplasmonsmetamaterialsnegativeindexofrefractionscatteringresonancesnon-trappingdomainsDirichlet-to-NeumannmapWeyllawsemiclassicalanalysis
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 scattering of waves by a metamaterial obstacle in which the wave speed is negative, modeled by a sign-changing Laplace-type transmission problem. The central claim is a complete asymptotic description of the resonances that lie near the real axis: under a non-trapping assumption on the exterior, such resonances exist exactly when a certain boundary quantity is positive, they are all surface plasmons (resonant states concentrated on the interface and decaying faster than any polynomial away from it), and their number up to frequency $\lambda$ grows like $\lambda^{d-1}$ times the volume of an explicitly defined set in the boundary's phase space. When the boundary quantity has the opposite sign, the paper proves a resonance-free strip below the real axis. The result matters because it turns a physically important but mathematically slippery phenomenon, surface plasmon resonance, into a rigorously characterized spectral object with a precise counting law.

What carries the argument

The argument is carried by the difference of the boundary Dirichlet-to-Neumann maps, $\Lambda_O(z) - \tau\Lambda_I(z)$, viewed after the semiclassical rescaling $\lambda = h^{-1}(1+z)$ as a semiclassical pseudodifferential operator on $\partial\Omega$. Its principal symbol, $\rho_O\sqrt{|\xi'|^2_{g_O}-1} - \tau\rho_I\sqrt{|\xi'|^2_{g_I}+1}$, changes sign exactly on the set $V$ appearing in the counting formula, and the positivity or negativity of the associated quadratic form is precisely the dichotomy (1.6)/(1.7). The proof combines a semiclassical version of the Lee-Uhlmann factorization to compute the DtN symbols, defect-measure propagation with the non-trapping resolvent estimate to rule out interior mass, and a contour-integral Weyl-law argument with a complex absorbing potential to count zeros of the determinant $\det(I + iR_Q(z)Q)$.

What would settle it

Compute the resonances of the unit disk in $\mathbb{R}^2$ with constant index $n \equiv 3$ (the exterior is non-trapping). Theorem 1.11 predicts that $\#\{\lambda_j : 0 < \operatorname{Re}\lambda_j \le \lambda,\ \operatorname{Im}\lambda_j \ge -M\} = \frac{\lambda}{2\pi}\operatorname{vol}_{T^*\partial\Omega}(V) + o(\lambda)$; a numerical count that disagrees with this leading constant would falsify the counting claim. Alternatively, exhibit a resonance with $\operatorname{Im}\lambda \ge -M$ whose resonant state does not vanish super-polynomially away from the boundary, which would falsify Theorem 1.10.

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Extended reading notes

Core claim

The paper establishes that, for the transmission problem with negative wave speed and a non-trapping exterior, the dichotomy is governed by the boundary symbol inequality (1.7). If (1.7) holds, Theorem 1.9 shows resonances are either super-polynomially close to the real axis or have imaginary parts tending to $-\infty$; Theorem 1.10 shows that every sequence of resonances with bounded imaginary part is plasmonic in the sense of (1.2), with resonant states concentrating in a $|\lambda|^{-1}$ boundary layer and oscillating at frequency $\sim|\lambda|$; and Theorem 1.11 gives the asymptotic count $\#\{\lambda_j : 0 < \operatorname{Re}\lambda_j \le \lambda,\ \operatorname{Im}\lambda_j \ge -M\} = \frac{\lambda^{d-1}}{(2\pi)^{d-1}}\operatorname{vol}_{T^*\partial\Omega}(V) + o(\lambda^{d-1})$. In the opposite case (1.6), Theorem 1.8 gives a resonance-free region $\{\operatorname{Re}\lambda > C\} \cap \{\operatorname{Im}\lambda > -M\}$. In the constant-index scalar case $n|_{\partial\Omega} > 1$, this confirms that the heuristic quasimode sequence from the prior literature captures all near-real resonances.

Load-bearing premise

The exterior of the cavity is non-trapping: every ray outside the cavity must escape to infinity in finite time; if the exterior traps rays, propagating modes can create resonances near the real axis and the plasmon-only dichotomy fails.

Editorial extensions

If this is right

  • For a constant index of refraction with $n|_{\partial\Omega} > 1$, the sequence of boundary quasimodes predicted by earlier heuristic constructions is confirmed to exhaust the resonances near the real axis.
  • The number of near-real resonances grows like $\lambda^{d-1}$, the growth rate of a boundary phase-space volume; the interior volume plays no role in the leading term.
  • Under condition (1.6), a non-trapping negative-index cavity is resonance-free in any fixed strip $\{\operatorname{Im}\lambda > -M\}$ for large $|\operatorname{Re}\lambda|$.
  • Resonant states normalized on the boundary satisfy $\|\psi u_{\lambda_j}\|_{L^2} = O(|\lambda_j|^{-\infty})$ for any $\psi$ supported away from the boundary, giving a quantitative localization statement.

Reading between the lines

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

  • At the threshold where the boundary symbol vanishes, $\tau^2\rho_I^2|\xi'|^2_{g_I} = \rho_O^2|\xi'|^2_{g_O}$, the problem stops being uniformly elliptic at infinity; the paper excludes this by (1.4), so new, possibly non-plasmonic phenomena at exactly the threshold remain an open possibility.
  • If the non-trapping assumption fails, the exterior's trapped rays should produce additional resonances near the real axis carrying mass away from the boundary; a quantitative extension might express that excess count through a fractal Weyl law for the trapped set.
  • The same Dirichlet-to-Neumann difference machinery likely applies to other sign-changing transmission problems such as Maxwell or acoustic equations with negative index, yielding boundary phase-space Weyl laws for their surface modes.
  • The counting formula provides a concrete numerical test: in $d=2$, computing resonances of the unit disk with $n\equiv 3$ and comparing the count to $\frac{\lambda}{2\pi}\operatorname{vol}_{T^*\partial\Omega}(V)$ should reproduce the claimed leading term.
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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

1 major / 3 minor

Summary. The paper studies scattering resonances for a metamaterial cavity modeled by a transmission problem with a sign-changing coefficient (negative wave speed) inside a bounded obstacle. Under a non-trapping assumption on the exterior domain and a non-degeneracy condition on the boundary symbol, the authors prove a sharp dichotomy: under condition (1.6) all resonances have imaginary part tending to -infinity, while under the complementary condition (1.7) any sequence of resonances with bounded imaginary part is superpolynomially close to the real axis, the associated resonant states are boundary-localized (plasmons in the sense of (1.2)), and the counting function of such resonances satisfies an explicit Weyl law with a boundary phase-space volume. The proofs use semiclassical reductions, Dirichlet-to-Neumann maps, microlocal factorization, defect measures, and contour integration.

Significance. This is a substantial contribution to the spectral theory of transmission problems with sign-changing coefficients. It goes well beyond the earlier quasimode constructions in [CM23, DBCM24] by showing, under a natural non-trapping hypothesis, that all near-real resonances are necessarily boundary-localized and by giving an exact asymptotic count with a parameter-free constant. The paper contains detailed proofs of nontrivial microlocal statements, including a semiclassical Lee-Uhlmann factorization, and establishes the black-box framework in Appendix A. The main limitation, the non-trapping assumption on the exterior, is explicit and standard; the results are conditional on it, not circular.

major comments (1)
  1. [Section 6, proof of Theorem 1.11 (dyadic summation)] In the dyadic summation at the end of the proof of Theorem 1.11, the counting formula derived for V_epsilon(h_j) counts only resonances with |Im z| <= h_j, i.e., |Im lambda| <= 1, whereas the theorem claims a count for all Im lambda >= -M with arbitrary M. The proof does not explicitly justify that, for large Re lambda, every resonance with Im lambda >= -M satisfies |Im lambda| <= 1. This follows from Theorem 1.9: for fixed M and N, resonances with Re lambda > C lie either with Im lambda <= -M or with -|lambda|^{-N} < Im lambda < 0, so for large Re lambda they satisfy Im lambda > -1; hence the difference between the two counts is O(1). This short justification should be added to make the proof of Theorem 1.11 complete.
minor comments (3)
  1. [Introduction, after (1.2)] The sentence 'Throughout the text, we will assume that Omega_I is non-trapping' is inconsistent with all subsequent theorems, which assume the exterior domain (Omega_O, g_O) is non-trapping; the abstract and the theorem statements make clear that the exterior is meant, so this should be corrected.
  2. [Theorem 1.11 statement] The notation '0 < Re lambda_j <= lambda : Im lambda_j >= -M' uses a colon where a comma is intended; it should read '0 < Re lambda_j <= lambda, Im lambda_j >= -M'.
  3. [Section 5.4 heading] The heading contains a doubled period ('resonances. .'); this is a typographical issue that should be cleaned up.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: resonance-free regions, plasmonic localization, and the counting volume all derive from the PDE via proved semiclassical estimates; the cited prior results are technical tools, not the theorem being proved.

full rationale

The derivation chain is self-contained and non-circular. The set of resonances R(P) is defined through the meromorphic continuation of the resolvent of the transmission problem (1.5), not through the paper's own constants or conclusions. Theorems 1.8-1.11 are conditional on the exterior non-trapping hypothesis and on the boundary ellipticity/sign conditions (1.6)/(1.7). The dichotomy between resonance-free regions and plasmonic resonances follows from the principal symbols of the Dirichlet-to-Neumann maps computed in Propositions 4.17 and 4.20, whose symbols are derived from the operator via the factorization Lemma 4.2 and energy estimates. The counting volume V in Theorem 1.11 is the positive set of a symbol obtained from rho_O, rho_I, tau, g_O, and g_I, i.e. from the coefficients of the PDE itself; it is not fitted, and it reduces in the constant-index case to the known quasimode threshold (Remark 1.7 and Theorem 1.4), which is a consistency check rather than an input. The determinant identity in Lemma 6.2, relating resonances to zeros of F(z), is an operator-theoretic equivalence and is proved from the invertibility of Lambda_O - tau Lambda_I; the final Weyl law is obtained by a contour integral and stationary phase, not by assuming the answer. Self-citations such as [GMS21, GSW20, GL25, GLS24] are used for semiclassical microlocalization, defect-measure propagation, or resolvent estimates; these are imported tools with proofs elsewhere, not unique theorems that forbid alternatives, and the paper proves the specific lemmas it uses (e.g. Lemma 4.5 states the method was communicated from [GL25] but gives a complete proof). The only textual slip is the sentence after (1.2) saying 'Omega_I is non-trapping'; the abstract, all theorem statements, and estimate (3.1) consistently require (Omega_O,g_O) to be non-trapping, so this is a typo/limitation in presentation, not a circular step. No parameter is fitted to a subset of resonances and then renamed a prediction, and no known empirical pattern is merely renamed. Hence the central claims have independent content and no circularity is present.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim rests on geometric and dynamical assumptions and on standard microlocal tools, not on fitted parameters or new postulated entities. No free parameters appear in the statements.

assumptions (4)
  • domain assumption Exterior domain (Omega_O, g_O) is non-trapping.
    Invoked in Section 1 after (1.2) and used to obtain the high-frequency resolvent estimate (3.1) from [Bur02, Theorem 1.3] and for defect measure propagation in Lemma 5.5.
  • domain assumption Transmission boundary symbol has a strict sign, either (1.6) or (1.7).
    This excludes the threshold where tau^2 rho_I^2 |xi'|^2_gI = rho_O^2 |xi'|^2_gO; the sign selects whether surface plasmons are absent or present in Theorems 1.8 to 1.11.
  • domain assumption Smooth boundary and smooth positive coefficients n, rho_I, rho_O and metrics.
    Required for Fermi normal coordinates, semiclassical pseudodifferential calculus, and the Dirichlet-to-Neumann parametrix results in Sections 2 and 4.
  • standard math Standard black-box scattering and semiclassical analysis results.
    Used throughout via [DZ19, Chapter 4], [Bur02], [SZ99], and [GLS24] for resolvent estimates, defect measures, and meromorphic continuation.

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Pith. "Pith review of Surface plasmons in metamaterial cavities: Scattering by obstacles with negative wave speed." pith.science (2026). https://pith.science/paper/F64WSTCF

@misc{pith2026250522253,
  author       = {Pith},
  title        = {Pith review of: Surface plasmons in metamaterial cavities: Scattering by obstacles with negative wave speed},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/F64WSTCF}},
  note         = {Machine review of arXiv:2505.22253}
}
read the original abstract

We study scattering by metamaterials with negative indices of refraction, which are known to support \emph{surface plasmons} -- long-lived states that are highly localized at the boundary of the cavity. This type of states has found uses in a variety of modern technologies. In this article, we study surface plasmons in the setting of non-trapping cavities; i.e. when all billiard trajectories outside the cavity escape to infinity. We characterize the indices of refraction which support surface plasmons, show that the corresponding resonances lie super-polynomially close to the real axis, describe the localization properties of the corresponding resonant states, and give an asymptotic formula for their number.

Figures

Figures reproduced from arXiv: 2505.22253 by the authors.

Figure 1
Figure 1. Examples of trapping and non-trapping domains. We first determine conditions on the index of refraction, n, such that there are no resonances close to the real axis. Theorem 1.1. Suppose that ΩI is non-trapping and n ∈ C∞(ΩI ; (0,∞)) satisfies n|∂Ω < 1. Then for all M > 0 there is C > 0 such that R(n, ΩI ) ∩ {| Re λ| > C} ⊂ {Im λ < −M}. Next, in the complementary case, we describe the region in which resonances may … view at source ↗
Figure 2
Figure 2. The figure shows the resonances for (1.1) with n|∂Ω < 1 as x’s. The resonance free region is determined by Theorem 1.1 or 1.8. λL ≈ 8.4647 − 1.0396 × 10−2 i λR ≈ 13.145 − 8.5412 × 10−4 i [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Lemma 5.9 in fact shows that, modulo O(|λj | −∞), all surface plasmons are as pictured here (with ΩI = B(0, 1)). These plasmons concentrate in a |λj | −1 neighborhood of the boundary, ∂Ω and oscillate at frequency ∼ |λj | in ∂Ω. The functions plotted here are the real parts of the resonant state corresponding to n|B(0,1) ≡ 3 with resonance λL ≈ 8.4647 × 100 − 1.0396 × 10−2 i (on the left) and λR ≈ 13.145 − 8.5412 × … view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: The figure shows the resonances for (1.1) with n|∂Ω > 1. Non-plasmonic resonances are denoted with x’s and plasmonic resonances with o’s. The resonance free regions are those determined by Theorem 1.2 or 1.9. Theorems 1.4 and 1.11 determine the asymptotic number of pla…

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Works this paper leans on

38 extracted references · 36 canonical work pages

  1. [1]

    6, 1363--1387

    Anne-Sophie Bonnet-Ben Dhia, Lucas Chesnel, and Patrick Ciarlet, T-coercivity for scalar interface problems between dielectrics and metamaterials, ESAIM: Mathematical Modelling and Numerical Analysis 46 (2012), no. 6, 1363--1387

    work page 2012

  2. [2]

    Models Methods Appl

    Anne-Sophie Bonnet-Ben Dhia, Lucas Chesnel, and Xavier Claeys, Radiation condition for a non-smooth interface between a dielectric and a metamaterial, Math. Models Methods Appl. Sci. 23 (2013), no. 9, 1629--1662. 3062923

    work page 2013

  3. [3]

    6, 1007--1031

    Anne-Sophie Bonnet-Ben Dhia, Lucas Chesnel, and Patrick Ciarlet Jr, T-coercivity for the maxwell problem with sign-changing coefficients, Communications in Partial Differential Equations 39 (2014), no. 6, 1007--1031

    work page 2014

  4. [4]

    Anne-Sophie Bonnet-Ben Dhia , Christophe Hazard, and Florian Monteghetti, Complex-scaling method for the complex plasmonic resonances of planar subwavelength particles with corners, Journal of Computational Physics 440 (2021), 110433

    work page 2021

  5. [5]

    Spence, and Jared Wunsch, Sharp high-frequency estimates for the H elmholtz equation and applications to boundary integral equations , SIAM J

    Dean Baskin, Euan A. Spence, and Jared Wunsch, Sharp high-frequency estimates for the H elmholtz equation and applications to boundary integral equations , SIAM J. Math. Anal. 48 (2016), no. 1, 229--267. 3448345

    work page 2016

  6. [6]

    Nicolas Burq, Semi-classical estimates for the resolvent in nontrapping geometries, Int. Math. Res. Not. (2002), no. 5, 221--241. 1876933

    2002

  7. [7]

    Yves Colin de Verdi \`e re, On the spectral theory of sign-changing L aplace operators , arXiv:2502.03838 (2025)

    work page Pith review arXiv 2025

  8. [8]

    2, 215--257

    Camille Carvalho and Zo \" s Moitier, Scattering resonances in unbounded transmission problems with sign-changing coefficient, IMA Journal of Applied Mathematics 88 (2023), no. 2, 215--257

    work page 2023

Show all 38 references
  1. [9]

    II , Math

    Fernando Cardoso, Georgi Popov, and Georgi Vodev, Distribution of resonances and local energy decay in the transmission problem. II , Math. Res. Lett. 6 (1999), no. 3-4, 377--396. 1713138

    1999

  2. [10]

    Partial Differential Equations 26 (2001), no

    , Asymptotics of the number of resonances in the transmission problem, Comm. Partial Differential Equations 26 (2001), no. 9-10, 1811--1859. 1865946

    2001

  3. [11]

    2, 367--413

    Martin Costabel and Ernst Stephan, A direct boundary integral equation method for transmission problems, Journal of Mathematical Analysis and Applications 106 (1985), no. 2, 367--413

    1985

  4. [12]

    Monique Dauge, St \'e phane Balac, Gabriel Caloz, and Zo \" s Moitier, Whispering gallery modes and frequency combs: Two excursions in the world of photonic resonators, Book of Abstracts, The 16th International Conference on Mathematical and Numerical Aspects of Wave Propagati...

    2024

  5. [13]

    Kiril Datchev and Semyon Dyatlov, Fractal W eyl laws for asymptotically hyperbolic manifolds , Geom. Funct. Anal. 23 (2013), no. 4, 1145--1206. 3077910

    2013

  6. [14]

    12, 4301--4343

    Semyon Dyatlov and Jeffrey Galkowski, Fractal W eyl laws and wave decay for general trapping , Nonlinearity 30 (2017), no. 12, 4301--4343. 3734138

    2017

  7. [15]

    Semyon Dyatlov, Resonance projectors and asymptotics for r -normally hyperbolic trapped sets , J. Amer. Math. Soc. 28 (2015), no. 2, 311--381. 3300697

    2015

  8. [16]

    , Improved fractal W eyl bounds for hyperbolic manifolds (with an appendix by D avid B orthwick, S emyon D yatlov and T obias W eich) , J. Eur. Math. Soc. (JEMS) 21 (2019), no. 6, 1595--1639. 3945737

    2019

  9. [17]

    200, American Mathematical Society, Providence, RI, 2019

    Semyon Dyatlov and Maciej Zworski, Mathematical theory of scattering resonances, Graduate Studies in Mathematics, vol. 200, American Mathematical Society, Providence, RI, 2019. 3969938

    2019

  10. [18]

    Jeffrey Galkowski, Distribution of resonances in scattering by thin barriers, Mem. Amer. Math. Soc. 259 (2019), no. 1248, ix+152. 3948573

    2019

  11. [19]

    , The quantum S abine law for resonances in transmission problems , Pure Appl. Anal. 1 (2019), no. 1, 27--100. 3900029

    2019

  12. [20]

    Jeffrey Galkowski and Matthieu L\'eautaud, Semiclassical trace estimates in the glancing region, 2025

    2025

  13. [21]

    Spence, Local absorbing boundary conditions on fixed domains give order-one errors for high-frequency waves, IMA J

    Jeffrey Galkowski, David Lafontaine, and Euan A. Spence, Local absorbing boundary conditions on fixed domains give order-one errors for high-frequency waves, IMA J. Numer. Anal. 44 (2024), no. 4, 1946--2069. 4780829

    2024

  14. [22]

    Spence, Eigenvalues of the truncated H elmholtz solution operator under strong trapping , SIAM J

    Jeffrey Galkowski, Pierre Marchand, and Euan A. Spence, Eigenvalues of the truncated H elmholtz solution operator under strong trapping , SIAM J. Math. Anal. 53 (2021), no. 6, 6724--6770. 4344887

    2021

  15. [23]

    Spence, and Jared Wunsch, Optimal constants in nontrapping resolvent estimates and applications in numerical analysis, Pure Appl

    Jeffrey Galkowski, Euan A. Spence, and Jared Wunsch, Optimal constants in nontrapping resolvent estimates and applications in numerical analysis, Pure Appl. Anal. 2 (2020), no. 1, 157--202. 4041281

    2020

  16. [24]

    III , Classics in Mathematics, Springer, Berlin, 2007, Pseudo-differential operators, Reprint of the 1994 edition

    Lars H\"ormander, The analysis of linear partial differential operators. III , Classics in Mathematics, Springer, Berlin, 2007, Pseudo-differential operators, Reprint of the 1994 edition. 2304165

    2007

  17. [25]

    Lee and Gunther Uhlmann, Determining anisotropic real-analytic conductivities by boundary measurements, Comm

    John M. Lee and Gunther Uhlmann, Determining anisotropic real-analytic conductivities by boundary measurements, Comm. Pure Appl. Math. 42 (1989), no. 8, 1097--1112. 1029119

    1989

  18. [26]

    Maier, Plasmonics: Fundamentals and applications, Springer, New York, NY, 2007

    Stefan A. Maier, Plasmonics: Fundamentals and applications, Springer, New York, NY, 2007

    2007

  19. [27]

    William McLean, Strongly elliptic systems and boundary integral equations, Cambridge University Press, Cambridge, 2000. 1742312

    2000

  20. [28]

    Luc Miller, Refraction of high-frequency waves density by sharp interfaces and semiclassical measures at the boundary, J. Math. Pures Appl. (9) 79 (2000), no. 3, 227--269. 1750924

    2000

  21. [29]

    Spence, Acoustic transmission problems: wavenumber-explicit bounds and resonance-free regions, Math

    Andrea Moiola and Euan A. Spence, Acoustic transmission problems: wavenumber-explicit bounds and resonance-free regions, Math. Models Methods Appl. Sci. 29 (2019), no. 2, 317--354. 3917405

    2019

  22. [30]

    Georgi Popov and Georgi Vodev, Distribution of the resonances and local energy decay in the transmission problem, Asymptot. Anal. 19 (1999), no. 3-4, 253--265. 1696217

    1999

  23. [31]

    , Resonances near the real axis for transparent obstacles, Comm. Math. Phys. 207 (1999), no. 2, 411--438. 1724834

    1999

  24. [32]

    10, 3450--3455

    Takumi Sannomiya, Christian Hafner, and Janos Voros, In situ sensing of single binding events by localized surface plasmon resonance, Nano letters 8 (2008), no. 10, 3450--3455

    2008

  25. [33]

    Pure Appl

    John Sylvester and Gunther Uhlmann, Inverse boundary value problems at the boundary---continuous dependence, Comm. Pure Appl. Math. 41 (1988), no. 2, 197--219. 924684

    1988

  26. [34]

    183 (1999), no

    Johannes Sj\"ostrand and Maciej Zworski, Asymptotic distribution of resonances for convex obstacles, Acta Math. 183 (1999), no. 2, 191--253. 1738044

    1999

  27. [35]

    , Fractal upper bounds on the density of semiclassical resonances, Duke Math. J. 137 (2007), no. 3, 381--459. 2309150

    2007

  28. [36]

    E. C. Titchmarsh, The theory of the R iemann zeta-function , second ed., The Clarendon Press, Oxford University Press, New York, 1986, Edited and with a preface by D. R. Heath-Brown. 882550

    1986

  29. [37]

    Maciej Zworski, Distribution of poles for scattering on the real line, J. Funct. Anal. 73 (1987), no. 2, 277--296. 899652

    1987

  30. [38]

    138, American Mathematical Society, Providence, RI, 2012

    , Semiclassical analysis, Graduate Studies in Mathematics, vol. 138, American Mathematical Society, Providence, RI, 2012. 2952218

    2012

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