The bulk-edge correspondence for continuous dislocated systems
[Correspondance bulk-edge pour des dislocations continues]
Annales de l'Institut Fourier, Online first, 55 p.

Nous étudions les aspects topologiques de modes propres d’une famille d’opérateurs {𝒫(t)} t[0,2π] , perturbations d’un opérateur P 0 par une dislocation : un potentiel avec un défaut de phase t entre - and +. Quand t=π et la dislocation est adiabatique, Fefferman, Lee-Thorp and Weinstein ont montré que les dégénérescences coniques de P 0 bifurquent vers des états propres de 𝒫(t).

Nous montrons ici que ces états propres sont topologiquement protégés, même en dehors du régime adiabatique. Cela passe par une correspondance « bulk-edge » pour les systèmes de dislocations. Spécifiquement, nous démontrons que le nombre effectif d’états propres est égal à un nombre de Chern, calculé à partir du bulk. Nous montrons que ces nombres sont le degré d’une fonction définie à partir de la dégénérescence conique et de la dislocation. Finalement, nous illustrons la richesse du modèle à travers quelques exemples.

We study topological aspects of defect modes for a family of operators 𝒫(t), obtained as a Schrödinger operator P 0 perturbed by a phase defect t between - and +: a dislocation. When t=π and the dislocation is adiabatic, Fefferman, Lee-Thorp and Weinstein showed that conical degeneracies of P 0 bifurcate to defect modes of 𝒫(π).

We show that these modes are topologically protected even outside the adiabatic regime. This relies on a bulk-edge correspondence for dislocations. Specifically, we show that the signed number of defect modes equals a Chern number computed from the bulk. We next derive an explicit formula for these indices in terms of the dislocation and of the conical point Bloch modes. We illustrate the depth of our result through a few examples.

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DOI : https://doi.org/10.5802/aif.3420
Classification : 35P15,  35P25,  35Q40,  35Q41
Mots clés : matière quantinque, états de bord topologiques, correspondance bulk-edge
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Drouot, Alexis. The bulk-edge correspondence for continuous dislocated systems. Annales de l'Institut Fourier, Online first, 55 p.

[1] Asbóth, János K.; Oroszlány, László; Pályi, András A short course on topological insulators. Band structure and edge states in one and two dimensions, Lecture Notes in Physics, 919, Springer, 2016, xiii+168 pages | Zbl 1331.82002

[2] Atiyah, Michael F.; Patodi, Vijay K.; Singer, Isadore M. Spectral asymmetry and Riemannian geometry. I, Math. Proc. Camb. Philos. Soc., Volume 77 (1975), pp. 43-69 | Article | MR 397797 | Zbl 0297.58008

[3] Atiyah, Michael F.; Patodi, Vijay K.; Singer, Isadore M. Spectral asymmetry and Riemannian geometry. II, Math. Proc. Camb. Philos. Soc., Volume 78 (1975), pp. 405-432 | Article | MR 397798 | Zbl 0314.58016

[4] Atiyah, Michael F.; Patodi, Vijay K.; Singer, Isadore M. Spectral asymmetry and Riemannian geometry. III, Math. Proc. Camb. Philos. Soc., Volume 79 (1976), pp. 71-99 | Article | MR 397799 | Zbl 0325.58015

[5] Avila, Julio Cesar; Schulz-Baldes, Hermann; Villegas-Blas, Carlos Topological invariants of edge states for periodic two-dimensional models, Math. Phys. Anal. Geom., Volume 16 (2013) no. 2, pp. 137-170 | Article | MR 3063955 | Zbl 1271.81210

[6] Avron, Joseph E.; Seiler, Ruedi; Simon, Barry Charge deficiency, charge transport and comparison of dimensions, Commun. Math. Phys., Volume 159 (1994) no. 2, pp. 399-422 | Article | MR 1256994 | Zbl 0822.47056

[7] Bal, Guillaume Continuous bulk and interface description of topological insulators, J. Math. Phys., Volume 60 (2019) no. 8, 081506, 20 pages | MR 3993755 | Zbl 1435.82027

[8] Bal, Guillaume Topological protection of perturbed edge states, Commun. Math. Sci., Volume 17 (2019) no. 1, pp. 193-225 | MR 3960646 | Zbl 1421.35305

[9] Bär, Christian; Strohmaier, Alexander An index theorem for Lorentzian manifolds with compact spacelike Cauchy boundary, Am. J. Math., Volume 141 (2019) no. 5, pp. 1421-1455 | MR 4011805 | Zbl 1429.83004

[10] Barilari, Davide Trace heat kernel asymptotics in 3D contact sub-Riemannian geometry, J. Math. Sci., New York, Volume 195 (2013) no. 3, pp. 391-411 | Article | MR 3207127 | Zbl 1294.58004

[11] Berkolaiko, Gregory; Comech, Andrew Symmetry and Dirac points in graphene spectrum, J. Spectr. Theory, Volume 8 (2018) no. 3, pp. 1099-1147 | Article | MR 3831157 | Zbl 1411.35092

[12] Borisov, Denis I. Some singular perturbations of periodic operators, Theor. Math. Phys., Volume 151 (2007) no. 2, pp. 614-624 | Article | MR 2338076 | Zbl 1147.34060

[13] Borisov, Denis I. On the spectrum of a two-dimensional periodic operator with a small localized perturbation, Izv. Math., Volume 75 (2011) no. 3, pp. 471-505 | Article | MR 2847781 | Zbl 1233.35066

[14] Borisov, Denis I. On the band spectrum of a Schrödinger operator in a periodic system of domains coupled by small windows, Russ. J. Math. Phys., Volume 22 (2015) no. 2, pp. 153-160 | Article | Zbl 1319.35134

[15] Borisov, Denis I.; Gadyl’shin, Rustem R. On the spectrum of a periodic operator with small localized perturbation, Izv. Math., Volume 72 (2008), pp. 659-688 | Article | MR 2452233 | Zbl 1169.34056

[16] Bourne, Chris; Rennie, Adam Chern numbers, localisation and the bulk-edge correspondence for continuous models of topological phases, Math. Phys. Anal. Geom., Volume 21 (2018) no. 3, 16, 62 pages | MR 3819644 | Zbl 1398.82049

[17] Braverman, Maxim The spectral Flow of a family of Toeplitz operators, Lett. Math. Phys., Volume 109 (2019), pp. 2271-2289 | Article | MR 4001840 | Zbl 1426.58005

[18] Braverman, Maxim An index of strongly Callias operators on Lorentzian manifolds with non-compact boundary, Math. Z., Volume 294 (2020) no. 1-2, pp. 229-250 | Article | MR 4054812 | Zbl 1431.58010

[19] Carlsson, Ulf An infinite number of wells in the semi-classical limit, Asymptotic Anal., Volume 3 (1990) no. 3, pp. 189-214 | Article | MR 1076447 | Zbl 0727.35094

[20] Dang, Nguyen Viet; Guillarmou, Colin; Rivière, Gabriel; Shen, Shu Fried conjecture in small dimensions, Invent. Math., Volume 220 (2020) no. 2, pp. 525-579 | Article | MR 4081137 | Zbl 1439.37043

[21] Davies, Edward B.; Simon, Barry Scattering theory for systems with different spatial asymptotics on the left and right, Commun. Math. Phys., Volume 63 (1978), pp. 277-301 | Article | MR 513906 | Zbl 0393.34015

[22] Deift, Percy A.; Hempel, Rainer On the existence of eigenvalues of the Schrödinger operator H-λW in a gap of σ(H), Commun. Math. Phys., Volume 103 (1986), pp. 461-490 | Article | Zbl 0594.34022

[23] Delplace, Pierre; Marston, J. B.; Venaille, Antoine Topological origin of equatorial waves, Science, Volume 358 (2017) no. 6366, p. 1075 | Article | MR 3727800 | Zbl 1404.86016

[24] Dohnal, Tomáš; Pelinovsky, Dmitry Surface gap solitons at a nonlinearity interface, SIAM J. Appl. Dyn. Syst., Volume 7 (2008) no. 2, pp. 249-264 | Article | MR 2421105 | Zbl 1165.35443

[25] Dohnal, Tomáš; Plum, Michael; Reichel, Wolfgang Localized modes of the linear periodic Schrödinger operator with a nonlocal perturbation, SIAM J. Math. Anal., Volume 41 (2009) no. 5, pp. 1967-1993 | Article | Zbl 1204.34114

[26] Dohnal, Tomáš; Plum, Michael; Reichel, Wolfgang Surface gap soliton ground states for the nonlinear Schrödinger equation, Commun. Math. Phys., Volume 308 (2011) no. 2, pp. 511-542 | Article | Zbl 1230.35127

[27] Dombrowski, Nicolas; Hislop, Peter D.; Soccorsi, Eric Edge currents and eigenvalue estimates for magnetic barrier Schrödinger operators, Asymptotic Anal., Volume 89 (2014) no. 3-4, pp. 331-363 | Article | Zbl 1303.35080

[28] Drouot, Alexis Ubiquity of conical points in topological insulators (2020) (https://arxiv.org/abs/2004.07068)

[29] Drouot, Alexis Microlocal analysis of the bulk-edge correspondence., Commun. Math. Phys., Volume 383 (2021) no. 3, pp. 2069-2112 | Article | MR 4244266 | Zbl 07340020

[30] Drouot, Alexis; Fefferman, Charles L.; Weinstein, Michael I. Defect modes for dislocated periodic media, Commun. Math. Phys., Volume 377 (2020) no. 3, pp. 1637-1680 | Article | MR 4121609 | Zbl 07220093

[31] Dyatlov, Semyon; Zworski, Maciej Ruelle zeta function at zero for surfaces, Invent. Math., Volume 210 (2017), pp. 211-229 | Article | MR 3698342 | Zbl 1382.58022

[32] Elgart, Alexander; Graf, Gian M.; Schenker, Jeffrey H. Equality of the bulk and edge Hall conductances in a mobility gap, Commun. Math. Phys., Volume 259 (2005) no. 1, pp. 185-221 erratum in ibid. 261 (2006), no. 2, p. 545 | Article | MR 2169973 | Zbl 1086.81081

[33] Faure, F.; Zhilinskii, B. Topological Chern indices in molecular spectra, Phys. Rev. Lett., Volume 85 (2000) no. 3, pp. 960-963 | Article

[34] Fefferman, Charles L.; Lee-Thorp, James P.; Weinstein, Michael I. Topologically protected states in one-dimensional continuous systems and Dirac points, Proc. Natl. Acad. Sci. USA, Volume 111 (2014) no. 24, pp. 8759-8763 | Article | MR 3263410 | Zbl 1355.34124

[35] Fefferman, Charles L.; Lee-Thorp, James P.; Weinstein, Michael I. Edge states in honeycomb structures, Ann. PDE, Volume 2 (2016) no. 2, 12, 80 pages | MR 3595458 | Zbl 1404.35128

[36] Fefferman, Charles L.; Lee-Thorp, James P.; Weinstein, Michael I. Topologically protected states in one-dimensional systems, 1147, Springer, 2017, viii+118 pages

[37] Fefferman, Charles L.; Lee-Thorp, James P.; Weinstein, Michael I. Honeycomb Schrödinger operators in the strong binding regime, Commun. Pure Appl. Math., Volume 71 (2018) no. 6, pp. 1178-1270 | Article | Zbl 1414.35061

[38] Fefferman, Charles L.; Weinstein, Michael I. Edge States of continuum Schrödinger operators for sharply terminated honeycomb structures (to appear in Commun. Math. Phys.)

[39] Fefferman, Charles L.; Weinstein, Michael I. Honeycomb lattice potentials and Dirac points, J. Am. Math. Soc., Volume 25 (2012) no. 4, pp. 1169-1220 | Article | MR 2947949 | Zbl 1316.35214

[40] Fefferman, Charles L.; Weinstein, Michael I. Wave packets in honeycomb structures and two-dimensional Dirac equations, Commun. Math. Phys., Volume 326 (2014) no. 1, pp. 251-286 | Article | MR 3162492 | Zbl 1292.35195

[41] Figotin, Alexander; Klein, Abel Localized classical waves created by defects, J. Stat. Phys., Volume 86 (1997) no. 1-2, pp. 165-177 | Article | MR 1435195 | Zbl 0952.76543

[42] Fruchart, Michel; Carpentier, David An introduction to topological insulators, C. R. Physique, Volume 14 (2013), pp. 779-815 | Article

[43] Fruchart, Michel; Carpentier, David; Gawędzki, Krzysztof Parallel transport and band theory in crystals, Eur. Phys. Lett., Volume 106 (2014), 60002

[44] Fu, L.; Kane, C. L.; Mele, E. J. Topological insulators in three dimensions, Phys. Rev. Lett., Volume 98 (2007), p. 106803 | Article

[45] Fukui, Takahiro; Shiozaki, Ken; Fujiwara, Takanori; Fujimoto, Satoshi Bulk-edge correspondence for Chern topological phases: A viewpoint from a generalized index theorem, J. Phys. Soc. Japan, Volume 81 (2012) no. 11, 114602, 7 pages

[46] Gesztesy, Fritz; Latushkin, Yuri; Makarov, Konstantin A.; Sukochev, Fedor; Tomilov, Yuri The index formula and the spectral shift function for relatively trace class perturbations, Adv. Math., Volume 227 (2011) no. 1, pp. 319-420 | Article | MR 2782197 | Zbl 1220.47017

[47] Gontier, David Edge states in ordinary differential equations for dislocations, J. Math. Phys., Volume 61 (2020) no. 4, 043507, 21 pages | MR 4090721 | Zbl 1443.81030

[48] Graf, Gian M.; Porta, Marcello Bulk-edge correspondence for two-dimensional topological insulators, Commun. Math. Phys., Volume 324 (2013) no. 3, pp. 851-895 | Article | MR 3123539 | Zbl 1291.82120

[49] Graf, Gian M.; Shapiro, Jacob The bulk-edge correspondence for disordered chiral chains, Commun. Math. Phys., Volume 363 (2018) no. 3, pp. 829-846 | Article | MR 3858823 | Zbl 1401.82031

[50] Graf, Gian M.; Tauber, Clément Bulk-edge correspondence for two-dimensional Floquet topological insulators, Ann. Henri Poincaré, Volume 19 (2018) no. 3, pp. 709-741 | Article | MR 3769245 | Zbl 1392.82008

[51] Haldane, F.; Raghu, S. Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry, Phys. Rev. Lett., Volume 100 (2008), 013904 | Article

[52] Halperin, Bertrand I. Quantized Hall conductance, current-carrying edge states, and the existence of extended states in a two-dimensional disordered potential, Phys. Rev. B, Volume 25 (1982), p. 2185 | Article

[53] Harrell, Evans M. The band structure of a one dimensional periodic system in the scaling limit, Ann. Phys., Volume 119 (1974), pp. 351-369 | Article | MR 539149 | Zbl 0412.34013

[54] Hatsugai, Yasuhiro Chern number and edge states in the integer quantum Hall effect, Phys. Rev. Lett., Volume 71 (1993) no. 22, pp. 3697-3700 | Article | MR 1246070 | Zbl 0972.81712

[55] Helffer, Bernard; Sjostrand, Johannes Multiple wells in the semi-classical limit I, Commun. Partial Differ. Equations, Volume 9 (1984), pp. 337-408 | Article | Zbl 0546.35053

[56] Helffer, Bernard; Sjostrand, Johannes Puits multiples en limite semi-classique II Interaction moleculaire-symetries-perturbation, Ann. Inst. Henri Poincaré, Phys. Théor., Volume 42 (1985), pp. 127-212 | Numdam | MR 798695 | Zbl 0595.35031

[57] Hempel, Rainer; Kohlmann, Martin Spectral properties of grain boundaries at small angles of rotation, J. Spectr. Theory, Volume 1 (2011) no. 2, pp. 197-219 | MR 2822127 | Zbl 1222.35133

[58] Hempel, Rainer; Kohlmann, Martin A variational approach to dislocation problems for periodic Schrödinger operators, J. Math. Anal. Appl., Volume 381 (2011) no. 1, pp. 166-178 | Article | Zbl 1220.35028

[59] Hempel, Rainer; Kohlmann, Martin Dislocation problems for periodic Schrödinger operators and mathematical aspects of small angle grain boundaries, Spectral theory, mathematical system theory, evolution equations, differential and difference equations (Operator Theory: Advances and Applications), Volume 221, Birkhäuser, 2012, pp. 421-432 | Article | Zbl 1256.35013

[60] Hempel, Rainer; Kohlmann, Martin; Stautz, Marko; Voigt, Jürgen Bound states for nano-tubes with a dislocation, J. Math. Anal. Appl., Volume 431 (2015) no. 1, pp. 202-227 | Article | MR 3357583 | Zbl 1317.81102

[61] Hislop, Peter D.; Soccorsi, Eric Edge currents for quantum Hall systems. I. One-edge, unbounded geometries, Rev. Math. Phys., Volume 20 (2008) no. 1, pp. 71-115 | Article | MR 2379247 | Zbl 1145.81053

[62] Hislop, Peter D.; Soccorsi, Eric Edge currents for quantum Hall systems. II. Two-edge, bounded and unbounded geometries., Ann. Henri Poincaré, Volume 9 (2008) no. 6, pp. 1141-1175 | Article | MR 2453258 | Zbl 1172.81021

[63] Hoefer, Mark A.; Weinstein, Michael I. Defect modes and homogenization of periodic Schrödinger operators, SIAM J. Math. Anal., Volume 43 (2011) no. 2, pp. 971-996 | Article | Zbl 1242.35032

[64] Kane, C. L.; Mele, E. J. Quantum Spin Hall Effect in Graphene, Phys. Rev. Lett., Volume 95 (2005) no. 22, 226801

[65] Kane, C. L.; Mele, E. J. Z 2 Topological Order and the Quantum Spin Hall Effect, Phys. Rev. Lett., Volume 95 (2005) no. 14, 146802, 4 pages

[66] Kato, Tosio Perturbation theory for linear operators, Classics in Mathematics, Springer, 1995 | Article | Zbl 0836.47009

[67] Kellendonk, Johannes; Richter, Thomas; Schulz-Baldes, Hermann Edge current channels and Chern numbers in the integer quantum Hall effect, Rev. Math. Phys., Volume 14 (2002) no. 1, pp. 87-119 | Article | MR 1877916 | Zbl 1037.81106

[68] Kellendonk, Johannes; Schulz-Baldes, Hermann Boundary maps for C * -crossed products with with an application to the quantum Hall effect, Commun. Math. Phys., Volume 249 (2004) no. 3, pp. 611-637 | Article | MR 2084009 | Zbl 1084.46058

[69] Kellendonk, Johannes; Schulz-Baldes, Hermann Quantization of edge currents for continuous magnetic operators, J. Funct. Anal., Volume 209 (2004) no. 2, pp. 388-413 | Article | MR 2044228 | Zbl 1078.81024

[70] Keller, Rachael T.; Marzuola, Jeremy L.; Osting, Braxton; Weinstein, Michael I. Spectral band degeneracies of π/2-rotationally invariant periodic Schrödinger operators, Multiscale Model. Simul., Volume 16 (2018) no. 4, pp. 1684-1731 (erratum in ibid. 18, no. 3, p. 1371–1373) | Article | Zbl 1470.35295

[71] Kitaev, Alexei Periodic table for topological insulators and superconductors, Advances in theoretical physics (AIP Conference Proceedings), Volume 1134, American Institute of Physics, 2009, pp. 22-30 | Zbl 1180.82221

[72] Korotyaev, Evgeni Lattice dislocations in a 1-dimensional model, Commun. Math. Phys., Volume 213 (2000) no. 2, pp. 471-489 | Article | MR 1785464 | Zbl 0962.34064

[73] Korotyaev, Evgeni Schrödinger operator with a junction of two 1-dimensional periodic potentials, Asymptotic Anal., Volume 45 (2005) no. 1-2, pp. 73-97 | Zbl 1119.34067

[74] Korotyaev, Evgeni; Moller, Jacob Schach Schrödinger operators periodic in octants (2017) (https://arxiv.org/abs/1712.08893)

[75] Kuchment, Peter A. An overview of periodic elliptic operators, Bull. Am. Math. Soc., Volume 53 (2016) no. 3, pp. 343-414 | Article | MR 3501794 | Zbl 1346.35170

[76] Lee, Minjae Dirac cones for point scatterers on a honeycomb lattice, SIAM J. Math. Anal., Volume 48 (2016) no. 2, pp. 1459-1488 | MR 3490495 | Zbl 1342.35283

[77] Lee-Thorp, James P.; Weinstein, Michael I.; Zhu, Yi Elliptic operators with honeycomb symmetry: Dirac points, edge states and applications to photonic graphene, Arch. Ration. Mech. Anal., Volume 232 (2019) no. 1, pp. 1-63 | Article | MR 3916971 | Zbl 1410.78024

[78] Lu, Jianfeng; Watson, Alexander B.; Weinstein, Michael I. Dirac operators and domain walls, SIAM J. Math. Anal., Volume 52 (2020), pp. 1115-1145 | MR 4075334 | Zbl 1442.35275

[79] Martinez, André Estimations de l’effet tunnel pour le double puits, J. Math. Pures Appl., Volume 66 (1987), pp. 195-215 | MR 896187 | Zbl 0564.35028

[80] Martinez, André Estimations de l’effet tunnel pour le double puits. II. Etats hautement exités., Bull. Soc. Math. Fr., Volume 116 (1988) no. 2, pp. 199-229 | Article | Zbl 0666.35069

[81] Monaco, Domenico; Panati, Gianluca Symmetry and localization in periodic crystals: triviality of Bloch bundles with a fermionic time-reversal symmetry, Acta Appl. Math., Volume 137 (2015) no. 1, pp. 185-203 | Article | MR 3343378 | Zbl 1318.82045

[82] Moore, J. E.; Balents, L. Topological invariants of time-reversal-invariant band structures, Phys. Rev. B, Volume 75 (2007), 121306 | Article

[83] Nash, Lisa M.; Kleckner, Dustin; Read, Alismari; Vitelli, Vincenzo; Turner, Ari M.; Irvine, William T. M. Topological mechanics of gyroscopic materials, Proc. Natl. Acad. Sci. USA, Volume 24 (2015), pp. 14495-14500 | Article

[84] Outassourt, Abderrahim Comportement semi-classique pour l’operateur de Schrödinger à potentiel periodique, J. Funct. Anal., Volume 72 (1987), pp. 65-93 | Article | MR 883504

[85] Panati, Gianluca Triviality of Bloch and Bloch–Dirac bundles, Ann. Henri Poincaré, Volume 8 (2007) no. 5, pp. 995-1011 | Article | MR 2342883 | Zbl 1375.81102

[86] Post, Olaf Eigenvalues in spectral gaps of a perturbed periodic manifold, Math. Nachr., Volume 261 (2003), pp. 141-162 | Article | MR 2020392 | Zbl 1064.35123

[87] Prodan, Emil; Schulz-Baldes, Hermann Bulk and boundary invariants for complex topological insulators. From K-theory to physics, Mathematical Physics Studies, Springer, 2016 | Article

[88] Pushnitski, Alexander The spectral flow, the Fredholm index, and the spectral shift function, Spectral theory of differential operators (Advances in the Mathematical Sciences), Volume 62, American Mathematical Society, 2008, pp. 141-155 | MR 2509781 | Zbl 1168.47010

[89] Raghu, S.; Haldane, F. Analogs of quantum-Hall-effect edge states in photonic crystals, Phys. Rev. A, Volume 78 (2008) no. 3, 033834 | Article

[90] Reed, Michael; Simon, Barry Methods of modern mathematical physics. IV. Analysis of operators, Academic Press Inc., 1978

[91] Roy, Roy Topological phases and the quantum spin Hall effect in three dimensions, Phys. Rev. B, Volume 79 (2009) no. 19, 195322

[92] Sánchez-Morgado, Héctor R-torsion and zeta functions for analytic Anosov flows on 3-manifolds, Trans. Am. Math. Soc., Volume 348 (1996) no. 3, pp. 963-973 | Article | MR 1348868 | Zbl 0872.58044

[93] Simon, Barry Semiclassical analysis of low lying eigenvalues. III. Width of the ground state band in strongly coupled solids., Ann. Phys., Volume 158 (1984), pp. 415-420 | Article | MR 772619 | Zbl 0596.35028

[94] Taarabt, Amal Equality of bulk and edge Hall conductances for continuous magnetic random Schrödinger operators. (2014) (https://arxiv.org/abs/1403.7767)

[95] Thouless, D. J.; Kohmoto, M.; Nightingale, M. P.; den Nijs, M. Quantized Hall Conductance in a Two-Dimensional Periodic Potential, Phys. Rev. Lett., Volume 49 (1982) no. 6, p. 405 | Article

[96] Colin de Verdière, Yves Sur les singularités de van Hove génériques, Mém. Soc. Math. Fr., Nouv. Sér., Volume 46 (1991), pp. 99-110 | Numdam | MR 1125838 | Zbl 0773.47034

[97] Waterstraat, Nils Fredholm Operators and Spectral Flow. (2016) (https://arxiv.org/abs/1603.02009, lecture notes)

[98] Watson, Alexander B. Wave dynamics in locally periodic structures by multiscale analysis (2017) (Ph. D. Thesis) | MR 3732035

[99] Watson, Alexander B.; Weinstein, Michael I. Wavepackets in inhomogeneous periodic media: propagation through a one-dimensional band crossing, Commun. Math. Phys., Volume 363 (2018) no. 2, pp. 655-698 | Article | MR 3851826 | Zbl 1400.81103

[100] Yu, Zongfu; Veronis, Georgios; Wang, Zheng; Fan, Shanhui One-way electromagnetic waveguide formed at the interface between a plasmonic metal under a static magnetic field and a photonic crystal, Phys. Rev. B, Volume 100 (2008) no. 2, 023902

[101] Zelenko, Leonid Virtual bound levels in a gap of the essential spectrum of the weakly perturbed periodic Schrödinger operator, Integral Equations Oper. Theory, Volume 85 (2016) no. 3, pp. 307-345 | Article | MR 3523639 | Zbl 1422.47051

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