Dispersion relations and spectra of periodically perforated structures
DDISPERSION RELATIONS AND SPECTRA OFPERIODICALLY PERFORATED STRUCTURES
PETER KUCHMENT AND JARI TASKINEN
Dedicated to Professor Shmuel Agmon, a great mathematician and inspiration
Abstract.
We establish absolute continuity of the spectrum ofa periodic Schr¨odiner operator in R n with periodic perforations.We also prove analytic dependece of the dispersion relation on theshape of the perforation. Introduction
The theory of periodic partial differential equations is at least a cen-tury old (see, e.g., [5, 24, 25, 35, 38]), but is still widely active, mostlydue to its importance for various areas of mathematical physics, suchas solid state physics, photonic crystal theory, topological insulatorstheory, and nano-science, to name just a few (see [1,3,5,6,10–12,20,23,24, 26, 28, 30]. Periodicity is usually introduced by crystalline structureof materials, or its optical analogs. Among the topics being discussedone can mention as examples analytic structure of the correspondingdispersion relations and spectral structure of periodic operators. An-other option is to study media with periodically modulated shapes, ormulti-periodic perforations, see e.g. [7,13,31,33] and references therein.The latter is the topic we address here. We will be concentrating on thefollowing two questions: dependence of dispersion relations (and thusspectra) on the shape of perforation and absolute continuity of thespectrum. The former has been considered for instance for the case ofcircular perforations with respect to a varying radius in R [31]. Thelatter is a well known, much studied, but still not completely finishedtopic (see [24] for references and discussion), when periodicity arisesdue to periodic coefficients, rather periodic perforations. Date : July 27, 2020.P.K. acknowledges support of the NSF DMS grants a r X i v : . [ m a t h - ph ] J u l PETER KUCHMENT AND JARI TASKINEN
The goal of this article is to show that a combination of several knownfor other situations approaches and results enable one to establish withease some very general properties of interest for periodically perforateddomains in any dimension.The text is structured as follows: Section 1 briefly refers to somepowerful (and not always that well known) techniques of domain per-turbations [9,15,18,19], following [18]. An important for the further dis-cussions Theorem 2 on analytic dependence on domain perturbationsis established. The next Section 2 describes the perforated geometry ofinterest. Theorem 5 of Section 3 establishes a very general analyticitywith respect to shape variations result. Section 4 contains the proof ofTheorem 7 on absolute continuity of spectra of such structures (i.e., im-possibility of creating a bound state by periodic perforations), allowingalso for presence of periodic electric potential. Section 5 contains someadditional remarks. It is followed by the Acknowledgments section.1.
Domain perturbations
Let Ω ⊂ R n be a smooth bounded domain and l ( x, z, D ) be a linearelliptic partial differential expression of order m with “sufficiently nice”coefficients defined in a neighborhood of the closure of Ω, where we usethe standard PDE notation D for 1 i ∂∂x . The coefficients are allowedto depend analytically on a parameter z in a domain C ⊂ C l for someinteger l ≥
1, or a complex analytic space, or even domain in a complexBanach space E . We assume that boundary conditions Bu | Γ = 0 areimposed on Γ := ∂ Ω that lead to an elliptic boundary value problemfor the operator L ( z ) acting as l ( x, z, D ) in Ω. The coefficients of theboundary operators are also allowed to depend analytically on z .Let us denote by H mz the closed subspace of H m (Ω) consisting of allfunctions satisfying the boundary conditions B ( z ) u | Γ = 0.The following claims are standard: Proposition 1. (1)
There exists, locally in z , a projector P ( z ) : H (Ω) (cid:55)→ H z an-alytically dependent on z , and thus its range forms an analytic “Sufficiently nice” means that the only property that we need is that the bound-ary value problem produces a Fredholm operator from the Sobolev space H m in Ωwith the corresponding boundary conditions to L (Ω). ISPERSION RELATIONS AND SPECTRA OF PERIODICALLY PERFORATED STRUCTURES3 subbundle (1) F := (cid:71) z ∈C H mz in the trivial bundle C × H m (Ω) over C . (2) The operator L ( z ) produces a Fredholm morphism between bun-dles F and C × L (Ω) . (3) The “dispersion relation” (2) D := { ( λ, z ) ∈ C × C | L ( z ) u = λu has a non-zero solution } is an analytic subset in C × C . It is principal (i.e., is definedas the set of zeros of a single analytic function f ( λ, z ) ) if theFredholm index of the operator is equal to zero. Indeed, local existence of an analytic projector is a simple exercise(see, e.g. [37, 42]. The main notions and results concerning analyticBanach bundles and Fredholm morphisms that explain the rest of thefirst two claims can be found in [42]. The last statement of the theoremfollows from [42, Theorem 4.11 and its Corollary].We now show how domain variations fit into this scheme. Since ana-lyticity is a local property, we will be looking at small shape variationsonly. A convenient (although over-determined and often neglected)way to parameterize domain variations is by varying its natural em-bedding into the ambient space, as opposed to parameterizations bynormal perturbations of the boundary. Thus, let Ω ∈ R n be a boundeddomain with the smooth boundary Γ. We denote by I Ω its natural Figure 1.
A smooth domain Ω ⊂ R n and its boundary Γ.embedding into R n and consider the Banach space C m (Ω , R n ) of m times continuously (and uniformly) differentiable mappings from Ω If the domain C is holomorphically convex (or is an abstracts complex Steinspace), then an analytic projector exists globally in z (see [37, 42]), but this is notneeded for our results. PETER KUCHMENT AND JARI TASKINEN to R n . Then I Ω ∈ C m (Ω , R n ). Consider h ∈ C m (Ω , R n ) such that (cid:107) ( h − I Ω ) (cid:107) is sufficiently small and denote the corresponding small ballby R ⊂ C m (Ω , R n ). Then h is still a diffeomorphic embedding of Ω into R n . The domains Ω h := h (Ω) are “small perturbations” of domain Ω,and their boundaries Γ h := h (Γ) are small perturbations of Γ. Let usnow define the operator L ( z, h ) acting as l ( x, z, D ) on the domain Ω h with elliptic boundary conditions B ( z, h ) u | Γ h = 0, where the boundaryoperators B may depend analytically on h .The diffeomorphisms h enable us to pull-back the BVPs from the do-mains Ω h back to Ω, ending up with a new family of elliptic operators,which we will call M ( h, z ). A simple calculation (see, e.g., [18, bottomof page 20]) shows that the coefficients of the operator M depend an-alytically on h ∈ R . Thus, the following result is just a corollary ofthe proposition 1: Theorem 2.
The “dispersion relation” (3) D := { ( λ, z, h ) ∈ C × C × R | L ( z, h ) u = λu has a non-zero solution } is an analytic subset in C × C × R . It is principal (i.e., is defined asthe set of zeros of a single analytic function f ( λ, z, h ) ) if the Fredholmindex of the operator is equal to zero. One simple consequence of this result is:
Corollary 3. If λ is a simple eigenvalue of the operator L ( z ) , then itextends analytically to a simple eigenvalue λ ( z, h ) of L ( z, h ) for suffi-ciently small | z − z | and (cid:107) h − I Ω (cid:107) . Remark 4.
In particular, the results of Theorem 2 and Corollary 3apply to the simplest case of a spherical domain of changing radius [31],or in fact to homothetic perturbation of a star-shaped domain. Perforated geometry
Our main goal here is to consider a periodically perforated mediumof the following kind: The domain of consideration is the space R n with a Z n -periodic arrangement of non-overlapping smooth contractiblebounded domains (“holes”) removed (see the shaded domain W in Fig.2). Consider in W a periodic elliptic operator L ( x, D ) of order m withelliptic conditions on the boundary of W (i.e., on the boundaries of theholes) and with sufficiently nice coefficients for the Fredholm propertybetween the appropriate Sobolev spaces to hold. We are interested This analyticity does not have anything to do with smoothness of the coefficientsof the operator or the surface Γ.
ISPERSION RELATIONS AND SPECTRA OF PERIODICALLY PERFORATED STRUCTURES5
Figure 2.
The space R n being periodically perforatedby contractible smooth holes.in its spectrum, and thus first of all in its dispersion relation (see,e.g. [24, 25] and references therein)(4) D := { ( k, λ ) ∈ C n × C | ∃ u (cid:54) = 0 Z n -periodic, L ( x, D + k ) u = λu } . It is well known (see [24] and references therein) that the spectrum ofthe operator coincides with the projection of D on the λ -plane. Also,according to Proposition 1, it is an analytic set.We are interested now in dependence of this picture on variations ofthe shape of the holes.3. Analyticity of the extended dispersion relation
Since in (4) only Z n -periodic functions are of interest, we can con-centrate on a single fundamental domain Ω folded into a torus, see Fig.3 for such a fundamental domain).We consider a smooth surface Γ (cid:15) near to the boundary Γ and inside Ω.We split the domain between Γ and Γ (cid:15) into two nested annular domains V ⊂ U with another smooth surface approximating Γ as a boundaryof V . We can make variations of the hole’s boundary Γ by considering C m -embeddings h from the annular domain U into R n that are close to I U and coincide with I V on the sub-annulus V . These extend as identityto the whole domain Ω. Using these diffeomorphisms, analogously towhat was done in Section 1, one can rewrite the eigenvalue problem(4) on the h -modified domain as a problem on the original domain, butwith a modified operator L ( k, D, h ). As it was explained in Section 1,this elliptic operator depends analytically on h and thus Proposition 1is applicable with the quasi-momenta k playing the role of parameters z . Thus, one obtains the following result: Theorem 5.
The “dispersion relation” (5) D := { ( λ, k, h ) ∈ C × C × R | ∃ u (cid:54) = 0 Z n -periodic, L ( k, h ) u = λu } PETER KUCHMENT AND JARI TASKINEN
Figure 3.
The unit cell of the structure. is an analytic subset in C × C n × R . It is principal (i.e., is defined asthe set of zeros of a single analytic function f ( λ, k, h ) ) if the Fredholmindex of the operator is equal to zero.In particular, if λ is a simple eigenvalue of the operator L ( k ) , thenit extends analytically to a simple eigenvalue λ ( k, h ) of L ( k, h ) for suf-ficiently small (cid:107) k − k (cid:107) and (cid:107) h − I U (cid:107) . Absolute continuity of the spectrum
We are now interested in the structure of the spectrum of the periodicelliptic BVP in periodically perforated R n (see the previous sections forthe exact description). The standard proof (see [25, 35]) applies thatleads to the absence of singular continuous spectrum: Theorem 6.
The singular continuous part of the spectrum of the pe-riodic operator described above is empty.
Thus, in proving the absolute continuity of the spectrum the onlyhurdle is to exclude the possibility of existence of the pure point partof the spectrum. In the generality considered above (an arbitrary or-der periodic elliptic operator) the statement that the spectrum is ab-solutely continuous would be incorrect, as the well known examplesof elliptic operators of higher order with compactly supported eigen-functions [34](see also usage of this result in the periodic situation
ISPERSION RELATIONS AND SPECTRA OF PERIODICALLY PERFORATED STRUCTURES7 in [24, 25]). So, the only realistic option is to restrict ourselves to op-erators of second order. So, we consider now the Schr¨odinger operator(6) − ∆ + V ( x ) , where V ( x ) is a real Z n -periodic bounded measurable potential. Wealso impose zero Dirichlet conditions on the boundaries of the per-forations.
Theorem 7.
The spectrum of the periodic operator (6) with Dirichletconditions in the perforated domain is absolutely continuous.Proof.
Let us write the operator as(7) (cid:18) i ∂∂x (cid:19) + V ( x )acting in the natural way in L ( W ) with the domain H ( W ). Thespectral parameter λ can be absorbed by the potential, and thus it issufficient to consider the case when λ = 0.The standard Floquet theory (see [24, 25, 38] and references therein)reduces this operator to the direct integral over the Brillouin zone ofthe quasimomenta of the operators(8) L ( k ) := (cid:18) i ∂∂x + k (cid:19) + V ( x )acting on Z n -periodic functions on W , i.e. on functions on the sub-domain (cid:102) W := W/ Z n of the torus T n = R n / Z n . Here k ∈ R n is thequasi-momentum. As it is standard, the family of the Fredholm oper-ators L ( k ) : H ( (cid:102) W ) (cid:55)→ L ( (cid:102) W ) extends analytically to the whole C n .Again, as the standard L. Thomas’ argument shows (see [24, 35, 41]),existence of the point λ = 0 in the pure point spectrum of L is equiv-alent to the following: Pure point spectrum condition : For any k ∈ C n there exists anon-trivial function u ∈ H ( (cid:102) W ) such that (9) L ( k ) u = 0 . Like in most of the known proofs of absolute continuity, we will tryto find a complex value of the quasimomentum k = a + ib, a, b ∈ R n for which the equation (9) has no non-trivial (periodic) solutions.We rewrite (9) as follows:(10) (cid:18) i ∂∂x + k (cid:19) u = − V ( x ) u. PETER KUCHMENT AND JARI TASKINEN
The L -norm of the right hand side, due to boundedness of the potentialcan be estimated for any u ∈ L ( (cid:102) W ) as follows: (cid:107) V u (cid:107) L ≤ C (cid:107) u (cid:107) L . If now we find a value of k such that for any non-zero u one has(11) (cid:107) (cid:18) i ∂∂x + k (cid:19) u (cid:107) L > C (cid:107) u (cid:107) L , the theorem will be proven.So far, everything went along the standard L. Thomas’ proof. Find-ing a complex value of k such that (11) holds also resembles the case ofnon-perforated domain, with a little additional caveat, due to (cid:102) W beingonly a part of the torus, rather than the whole torus.So, let k = a + ib and rewrite the differential expression in the lefthand side of (11) as follows:(12) (cid:18) − ∆ + | a | − | b | − ia · ∂∂x (cid:19) + 2 ib · (cid:18) a + i ∂∂x (cid:19) . Let us denote the expression in the first parentheses by A and in thesecond as B , so in the left hand side of (11) is Au + Bu . A directcomputation of inner products in L ( (cid:102) W ) shows that(13) (cid:107) Au + Bu (cid:107) L = (cid:107) Au (cid:107) L + (cid:107) Bu (cid:107) L . The formal reason is that A and B commute, A is symmetric, and B is skew-symmetric. However, the direct calculation works nicely for u ∈ H ( (cid:102) W ), avoiding composition of A and B .So, it is sufficient for a given constant C > a, b ∈ R n insuch a way that(14) (cid:107) Bu (cid:107) L > C (cid:107) u (cid:107) L for any non-zero u ∈ H ( (cid:102) W )).Now let us extend u as a function ˜ u on the whole torus T n , settingit equal to zero outside of (cid:102) W . The resulting function is not in H onthe torus anymore, but is still in H , due to the Dirichlet conditionimposed. Let us expand ˜ u into Fourier series:(15) ˜ u ( x ) = (cid:88) m ∈ Z n u m e πix · m . Let us choose a = ( α, , . . . , , b = ( β, , . . . , Bu = B ˜ u | (cid:102) W = (cid:88) m ∈ Z n iβ ( α − πm ) u m e ix · m . ISPERSION RELATIONS AND SPECTRA OF PERIODICALLY PERFORATED STRUCTURES9
Figure 4.
Choosing now α = π and β > C/
6, one sees that the absolute values ofall multipliers in the Fourier series (16) exceed C , which implies (14),and hence (11). This finishes the proof of the theorem. (cid:3) Remarks and conclusions (1) The geometry considered in this article seems to be restrictedby two secretly made assumptions: the group of periods being Z n (changing variables would lead the Laplacian to look some-what different), as well as by the perforations being holes in thecells of a cubic tiling of the space. This seems to be excludingperforations like the one in Fig. 4. In fact, the proof carries outwithout difficulty to arbitrary Brave lattice and any “strange”shape of the fundamental domain instead of a cube. In partic-ular, in the main consideration, any fundamental domain, notnecessarily a cube, would fold onto the torus and thus the proofcarries through.(2) It was mentioned in [24] that the “extended” dispersion re-lation, i.e. including for instance potentials, is also analytic.This, however did not include the domain shape variations. Weshow that these can also be included, which in particular im-plies analyticity of an isolated eigenvalue. However, the globalanalyticity of the dispersion relation carries more information,including also its structure near non-smooth points as well. It isalso known to be important for other issues of spectral theory,see e.g. [8, 16, 17, 21, 22, 24, 25, 29, 30].(3) One expects that the statement of Theorem 7 about absolutecontinuity of the spectrum holds for any second order periodicelliptic operator with sufficiently “nice” coefficients, instead ofjust the Schr¨odinger operator (6). This is known to be not asimple problem even without perforations (see [4, 14, 24, 25, 27,
32, 36, 39–41]). The famous work by Friedlander [14] proves ab-solute continuity under an additional symmetry condition thatthe operator should be even with respect to one of the coordi-nates corresponding to the periodicity axes. This proof applieswithout any modification to the perforated domain case, if thesymmetry condition is imposed on the shape of the perforationas well:
Theorem 8.
Let ( x , ..., x n ) be the coordinates in R n , such thatthe natural generators of the group of periods Z n act by shiftingthe appropriate coordinates.Suppose that the periodic 2nd order elliptic operator L in theperforate space, equipped with Dirichlet boundary conditions, aswell as the shape of the perforation are symmetric w.r.t. thetransformation ( x , . . . , x n ) (cid:55)→ ( − x , . . . , x n ) . Then the spec-trum is absolutely continuous. Acknowledgments
P.K. acknowledges support of the NSF DMS grants
References [1] N. W. Ashcroft and N. D. Mermin,
Solid state physic s, Holt, Rinehart andWinston, New York-London, 1976.[2] G. Berkolaiko and P. Kuchment,
Introduction to quantum graphs , AmericanMathematical Society, Providence, RI, 2013.[3] B. A. Bernevig,
Topological insulators and topological superconductors , Prince-ton University Press, Princeton, NJ, 2013.[4] M. Sh. Birman and T. A. Suslina, A periodic magnetic Hamiltonian with avariable metric. The problem of absolute continuity (Russian, with Russiansummary), Algebra i Analiz 11 (1999), no. 2, 1–40; English transl., St. Peters-burg Math. J. 11 (2000), no. 2, 203–232.[5] L. Brillouin,
Wave propagation in periodic structures. Electric filters and crys-tal lattices , Dover Publications, Inc., New York, N. Y. 1953.[6] K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili,and M. Wegener, Periodic nanostructures for photonics, Physics Reports 444(2007), no. 3–6, p. 101–202.[7] G. Cardone, S. Nazarov, J. Taskinen, Spectra of open waveguides in periodicmedia, J. Functional Anal. (2015), 2328–2364.[8] Y. Colin de Verdi‘ere, Sur les singularit´es de van Hove g´en´eriques (French),Mem. Soc. Math. France (N.S.) 46 (1991), 99–110.
ISPERSION RELATIONS AND SPECTRA OF PERIODICALLY PERFORATED STRUCTURES11 [9] R. Courant, D. Hilbert,
Methods of mathematical physics.
Vol. I. IntersciencePublishers, Inc., New York, N.Y., 1953.[10] N. Do, On the quantum graph spectra of graphyne nanotubes, Anal. Math.Phys. 5 (2015), no. 1, 39–65[11] Ngoc T. Do, P. Kuchment, Quantum graph spectra of a graphyne struc-ture, Nanoscale Systems: Mathematical Modeling, Theory and Applications (2013), 107–123.[12] C. L. Fefferman, J. P. Lee-Thorp, and M. I. Weinstein, Topologically protectedstates in onedimensional continuous systems and Dirac points, Proc. Natl.Acad. Sci. USA 111 (2014), no. 24, 8759–8763.[13] F. Ferraresso, J. Taskinen, Singular perturbation Dirichlet problem in a dou-bleperiodic perforated plane, Ann. Univ. Ferrara (2015), 277–290.[14] L. Friedlander, On the Spectrum of a Class of Second Order Periodic EllipticDifferential Operators, Comm. Math. Phys. 229 (2002), 49–55.[15] P. R. Garabedian, Partial differential equations . Reprint of the 1964 original.AMS Chelsea Publishing, Providence, RI, 1998.[16] C. G´erard, Resonance theory for periodic Schr¨odinger operators, Bull. Soc.Math. France 118 (1990), no. 1, 27–54.[17] D. Gieseker, H. Knorrer, and E. Trubowitz,
The geometry of algebraic Fermicurves , Academic Press Inc., Boston, MA, 1993.[18] D. Henry,
Perturbation of the boundary in boundary-value problems of partialdifferential equations , London Mathematical Society Lecture Note Series, v.318. Cambridge University Press, Cambridge, 2005.[19] L. Ivanov, L. Kotko, S. Kreˇin,
Boundary value problems in variable domains ,(Russian. Lithuanian, English summary), Differencialnye Uravnenija i Prime-nen.Trudy Sem. Processy (1977), 161 pp.[20] J. D. Joannopoulos, S. Johnson, R. D. Meade, and J. N. Winn, Photonic crys-tals: Molding the flow of light , 2nd ed., Princeton University Press, Princeton,N.J., 2008.[21] Y. E. Karpeshina,
Perturbation theory for the Schrodinger operator with aperiodic potential , Lecture Notes in Mathematics, vol. 1663, Springer–Verlag,Berlin, 1997.[22] H. Kn¨orrer and E. Trubowitz, A directional compactification of the complexBloch variety, Comment. Math. Helv. 65 (1990), no. 1, 114–149.[23] E. Korotyaev and I. Lobanov, Schrodinger operators on zigzag nanotubes, Ann.Henri Poincar´e 8 (2007), no. 6, 1151–1176.[24] P. Kuchment, An overview of periodic elliptic operators, Bull. Amer. Math.Soc., 53 (2016), 343–414.[25] P. Kuchment,
Floquet theory for partial differential equations , Birkh¨auser,Basel, 1993.[26] P. Kuchment, The Mathematics of Photonic Crystals, Ch. 7 in
MathematicalModeling in Optical Science , Gang Bao, Lawrence Cowsar, and Wen Masters(Editors), Frontiers in Applied Mathematics v. 22, SIAM, 2001, 207–272.[27] P. Kuchment, S. Levendorskii, On the structure of spectra of periodic ellipticoperators, Trans. AMS 354 (2002), 537-569.[28] P. Kuchment, O. Post, On the spectra of carbon nano-structures , Comm.Math. Phys. 275 (2007), no. 3, 805–826. [29] P. Kuchment, B. Vainberg, On absence of embedded eigenvalues forSchr¨odinger operators with perturbed periodic potentials, Commun. Part. Diff.Equat. 25(2000), no. 9-10, 1809 - 1826.[30] P. Kuchment, B. Vainberg, On the structure of eigenfunctions correspondingto embedded eigenvalues of locally perturbed periodic graph operators, Comm.Math. Phys. 268 (2006), 673–686.[31] M. Lanza de Cristoforis, P. Musolino, J. Taskinen. Band-gap spectrum andreal analytic dependence on a geometric parameter for the Dirichlet Laplacian,unpublished, 2019.[32] A. Morame, The absolute continuity of the spectrum of Maxwell operator ina periodic media, J. Math. Phys. 41 (2000), no. 10, 7099–7108,[33] S.A. Nazarov, K. Ruotsalainen, J. Taskinen, Spectral gaps in the Dirichletand Neumann problems on the plane perforated by a double-periodic familyof circular holes, J. Math. Sci. (N.Y.), 181 (2012), 164–222.[34] A. Plis, Non-uniqueness in Cauchys problem for differential equations of elliptictype, J. Math. Mech. 9 (1960), 557–562.[35] M. Reed and B. Simon, Methods of modern mathematical physics. v. IV,Academic Press, New York, 1980.[36] Z. Shen, On absolute continuity of the periodic Schrodinger operators, Inter-nat. Math. Res. Notices (2001), no. 1, 1–31.[37] M. A. Shubin, Holomorphic families of subspaces of a Banach space. (Russian)Mat. Issled. (1970), no. 4 (18) 153–165.[38] M. M. Skriganov, Geometric and arithmetic methods in the spectral theory ofmultidimensional periodic operators (Russian), Trudy Mat. Inst. Steklov. 171(1985), 122 pp.[39] A. V. Sobolev, Absolute continuity of the periodic magnetic Schrodinger op-erator, Invent. Math. 137 (1999), no. 1, 85–112.[40] A. V. Sobolev and J. Walthoe, Absolute continuity in periodic waveguides,Proc. London Math. Soc. (3) 85 (2002), no. 3, 717-741.[41] L. E. Thomas, Time dependent approach to scattering from impurities in acrystal, Comm. Math. Phys. 33 (1973), 335–343.[42] M.G. Zaidenberg, S.G. Krein, P.A. Kuchment, A.A. Pankov, Banach bundlesand linear operators, Russian Math Surveys 30,5 (1975), 115–175.
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