An overview of periodic elliptic operators

The article surveys the main techniques and results of the spectral theory of periodic operators arising in mathematical physics and other areas. Close attention is paid to studying analytic properties of Bloch and Fermi varieties, which influence significantly most properties of such operators. The approaches described are applicable not only to the standard model example of Schr\"odinger operator with periodic electric potential $-\Delta+V(x)$, but to a wide variety of elliptic periodic equations and systems, equations on graphs, $\overline{\partial}$-operator, and other operators on abelian coverings of compact bases. Many important applications are mentioned. However, due to the size restrictions, they are not dealt with in details.

[1]  Yoshimi Saito,et al.  Eigenfunction Expansions Associated with Second-order Differential Equations for Hilbert Space-valued Functions , 1971 .

[2]  About isospectral deformations of discrete laplacians , 1979 .

[3]  C. Gérard.  Resonance theory for periodic Schrödinger operators , 1990 .

[4]  Peter Li.  Geometric Analysis: Index , 2012 .

[5]  R. Pinsky.  Second Order Elliptic Operators with Periodic Coefficients: Criticality Theory, Perturbations, and Positive Harmonic Functions , 1995 .

[6]  J. S. Blakemore.  Solid State Physics: ELECTRONS IN METALS , 1985 .

[7]  E. Bierstone,et al.  Semianalytic and subanalytic sets , 1988 .

[8]  Absolute continuity of the spectrum of a Schrodinger operator with a potential which is periodic in some directions and decays in others , 2004, math-ph/0402013.

[9]  R. Hempel,et al.  Spectral properties of periodic media in the large coupling limit , 1999 .

[10]  S. Levendorskii,et al.  On the structure of spectra of periodic elliptic operators , 2001 .

[11]  B. Simon.  On the genericity of nonvanishing instability intervals in Hills equation , 1976 .

[12]  Large gaps in point-coupled periodic systems of manifolds , 2002, math-ph/0212052.

[13]  Y. Kordyukov.  Semiclassical Asymptotics and Spectral Gaps for Periodic Magnetic Schrödinger Operators on Covering Manifolds , 2006 .

[14]  Carlos Conca,et al.  Bloch approximation in homogenization on bounded domains , 2005 .

[15]  G. Folland.  A course in abstract harmonic analysis , 1995 .

[16]  R. Shterenberg,et al.  Complete asymptotic expansion of the integrated density of states of multidimensional almost-periodic Schrödinger operators , 2010, 1004.2939.

[17]  Konstantin Pankrashkin,et al.  Quantum waveguides with small periodic perturbations: gaps and edges of Brillouin zones , 2012, 1211.5786.

[18]  Toshiaki Adachi,et al.  Density of states in spectral geometry , 1993 .

[19]  J. Ziman.  Principles of the Theory of Solids , 1965 .

[20]  Bethe–Sommerfeld Conjecture for Pseudodifferential Perturbation , 2008, 0804.3488.

[21]  Peter Li.  CURVATURE AND FUNCTION THEORY ON RIEMANNIAN MANIFOLDS , 2002 .

[22]  Maciej Dunajski,et al.  Integrable Systems , 2012 .

[23]  V. Meshkov.  ON THE POSSIBLE RATE OF DECAY AT INFINITY OF SOLUTIONS OF SECOND ORDER PARTIAL DIFFERENTIAL EQUATIONS , 1992 .

[24]  Mariya Vorobets.  On the Bethe–Sommerfeld conjecture for certain periodic Maxwell operators , 2011 .

[25]  Kaoru Ono,et al.  Periodic Schrödinger Operators on a Manifold , 1989 .

[26]  N. Filonov,et al.  Absolute continuity of the Schrödinger operator spectrum in a multidimensional cylinder , 2009 .

[27]  C. Gérard.  A proof of the abstract limiting absorption principle by energy estimates , 2008 .

[28]  P. Kuchment,et al.  On resonant spectral gap opening in quantum graph networks , 2016, 1601.04774.

[29]  A. Sobolev,et al.  Absence of the singular continuous component in spectra of analytic direct integrals , 2006 .

[30]  Spectral Gaps for Periodic Schrödinger Operators with Strong Magnetic Fields , 2003, math/0311200.

[31]  M. Shubin.  THE SPECTRAL THEORY AND THE INDEX OF ELLIPTIC OPERATORS WITH ALMOST PERIODIC COEFFICIENTS , 1979 .

[32]  Carlos Conca,et al.  The Bloch transform and applications , 1998 .

[33]  Green's function asymptotics near the internal edges of spectra of periodic elliptic operators. Spectral edge case , 2012 .

[34]  G. Panati,et al.  Bloch Bundles, Marzari-Vanderbilt Functional and Maximally Localized Wannier Functions , 2011, 1112.6197.

[35]  刘楠.  Graphene , 2012 .

[36]  I. Kunin.  Inverse Scattering Method , 1982 .

[37]  Measures of Fermi Surfaces and Absence of Singular Continuous Spectrum for Magnetic Schrödinger Operators , 1999, math-ph/9908026.

[38]  SPHERICAL REPRESENTATION OF SOLUTIONS OF INVARIANT DIFFERENTIAL EQUATIONS ON A RIEMANNIAN SYMMETRIC SPACE OF NONCOMPACT TYPE , 1986 .

[39]  U. Feige,et al.  Spectral Graph Theory , 2015 .

[40]  N. Filonov,et al.  Absolute continuity of the “even" periodic Schrödinger operator with nonsmooth coefficients , 2005 .

[41]  Joe Calloway,et al.  Energy Band Theory , 1964 .

[42]  B. Mityagin,et al.  Spectral gaps of Schr\"odinger operators with periodic singular potentials , 2009, 0903.5210.

[43]  C. Conca,et al.  Fourier approach to homogenization problems , 2002 .

[44]  On the quantum graph spectra of graphyne nanotubes , 2013, 1311.4162.

[45]  Absolute continuity of the spectrum of a Landau Hamiltonian perturbed by a generic periodic potential , 2009, 0904.2891.

[46]  Mark S. C. Reed,et al.  Method of Modern Mathematical Physics , 1972 .

[47]  S. Agmon.  On Positive Solutions of Elliptic Equations with Periodic Coefficients in N, Spectral Results and Extensions to Elliptic Operators on Riemannian Manifolds , 1984 .

[48]  The Creation of Spectral Gaps by Graph Decoration , 2000, math-ph/0008013.

[49]  H. McKean,et al.  The spectrum of Hill's equation , 1975 .

[50]  R. Young.  The Fermi Surfaces of Metals , 1972 .

[51]  Leonid Parnovski,et al.  Bethe–Sommerfeld Conjecture , 2008, 0801.3096.

[52]  R. Tolimieri,et al.  Abelian harmonic analysis, theta functions and function algebras on a nilmanifold , 1975 .

[53]  Liouville theorems and spectral edge behavior on abelian coverings of compact manifolds , 2005, math-ph/0503010.

[54]  J. Neumann,et al.  Über merkwürdige diskrete Eigenwerte , 1993 .

[55]  G. Floquet,et al.  Sur les équations différentielles linéaires à coefficients périodiques , 1883 .

[56]  T. Sunada,et al.  On the spectrum of gauge-periodic elliptic operators , 1992 .

[57]  H. Knörrer,et al.  There is no two dimensional analogue of Lamé's equation , 1992 .

[58]  Peter Li.  Harmonic Functions on Complete Riemannian Manifolds , 2008 .

[59]  Charles L Fefferman,et al.  Topologically protected states in one-dimensional continuous systems and Dirac points , 2014, Proceedings of the National Academy of Sciences.

[60]  F. M. Arscott,et al.  Periodic Differential Equations , 1963 .

[61]  Vladimir Igorevich Arnold,et al.  Geometrical Methods in the Theory of Ordinary Differential Equations , 1983 .

[62]  On absence of embedded eigenvalues for schrÖdinger operators with perturbed periodic potentials , 1999, math-ph/9904016.

[63]  J. M. Harrison,et al.  On occurrence of spectral edges for periodic operators inside the Brillouin zone , 2007, math-ph/0702035.

[64]  Joseph B. Keller,et al.  Partial Differential Equations with Periodic Coefficients and Bloch Waves in Crystals , 1964 .

[65]  Zhongwei Shen.  On the Bethe-Sommerfeld conjecture for higher-order elliptic operators , 2003 .

[66]  I. M. Gelfand,et al.  Eigenfunction expansions for equations with periodic coefficients , 1987 .

[67]  S. Novikov.  Two-dimensional Schrödinger operators in periodic fields , 1985 .

[68]  Zhongwei Shen,et al.  Uniform Sobolev inequalities and absolute continuity of periodic operators , 2007 .

[69]  H. Knörrer,et al.  A directional compactification of the complex Bloch variety , 1990 .

[70]  L. I. Danilov.  On absolute continuity of the spectrum of three- and four-dimensional periodic Schrödinger operators , 2010 .

[71]  Spectral asymptotics of periodic elliptic operators , 1997, funct-an/9707002.

[72]  E. Green.  Spectral Theory of Laplace–Beltrami Operators with Periodic Metrics , 1997 .

[73]  Kristian Kirsch,et al.  Theory Of Ordinary Differential Equations , 2016 .

[74]  Claudia Baier,et al.  Elektronentheorie der Metalle , 1937, Nature.

[75]  Toshikazu Sunada,et al.  Fundamental groups and Laplacians , 1988 .

[76]  L. Brillouin.  Wave propagation in periodic structures : electric filters and crystal lattices , 1953 .

[77]  Michael S. Maier.  Michael S , 1966 .

[78]  Av Sobolev.  Recent results on the Bethe-Sommerfeld conjecture , 2007 .

[79]  Vassilis G. Papanicolaou,et al.  Some Results on Ordinary Differential Operators with Periodic Coefficients , 2014, 1412.5423.

[80]  S. Fujita,et al.  The Fermi Surface , 2007 .

[81]  C. Fefferman,et al.  Bifurcations of edge states—topologically protected and non-protected—in continuous 2D honeycomb structures , 2015, 1509.08957.

[82]  Kristian Kirsch,et al.  Methods Of Modern Mathematical Physics , 2016 .

[83]  A. Scherer,et al.  Coupled-resonator optical waveguide: a proposal and analysis. , 1999, Optics letters.

[84]  EXISTENCE OF SPECTRAL GAPS, COVERING MANIFOLDS AND RESIDUALLY FINITE GROUPS , 2005, math-ph/0503005.

[85]  Toshikazu Sunada,et al.  Riemannian coverings and isospectral manifolds , 1985 .

[86]  B. Simon,et al.  Scattering theory for systems with different spatial asymptotics on the left and right , 1978 .

[87]  A. Sobolev.  Absolute continuity of the periodic magnetic Schrödinger operator , 1999 .

[88]  K. Yajima.  Scattering theory for Schrödinger equations with potentials periodic in time , 1977 .

[89]  M. Kreĭn,et al.  Introduction to the theory of linear nonselfadjoint operators , 1969 .

[90]  Zhongwei Shen.  On Absolute Continuity of the Periodic Schrödinger Operators , 2001 .

[91]  J. Sjöstrand.  Microlocal analysis for the periodic magnetic schrodinger equation and related questions , 1991 .

[92]  On isospectral potentials on flat tori II , 1995 .

[93]  Semiclassical Asymptotics and Gaps in the Spectra of Magnetic Schrödinger Operators , 2001, math/0102021.

[94]  Absolute Continuity of a Two-Dimensional Magnetic Periodic Schrödinger Operator ith Potentials of the Type of Measure Derivative , 2003 .

[95]  I. M. Glazman.  Direct methods of qualitative spectral analysis of singular differential operators , 1965 .

[96]  Rachel J. Steiner,et al.  The spectral theory of periodic differential equations , 1973 .

[97]  P. Kuchment,et al.  Green's function asymptotics near the internal edges of spectra of periodic elliptic operators. Spectral gap interior , 2015, 1508.06703.

[98]  Alexander Figotin,et al.  Spectral Properties of Classical Waves in High-Contrast Periodic Media , 1998, SIAM J. Appl. Math..

[99]  K. Schmidt,et al.  Periodic Differential Operators , 2012 .

[100]  M. G. Kreïn.  Stability of solutions of differential equations in Banach space , 2007 .

[101]  Inverse spectral problem for the Schrödinger equation with periodic vector potential , 1989 .

[102]  P. Kuchment,et al.  Integral representations and Liouville theorems for solutions of periodic elliptic equations , 2000, math/0007051.

[103]  Resolvent estimates and spectrum of the Dirac operator with periodic potential , 1995 .

[104]  E. Wohlfarth.  Electrons in Metals , 1954, Nature.

[105]  F. Nier,et al.  Scattering theory for the perturbations of periodic Schrödinger operators , 1998 .

[106]  Boris Mityagin,et al.  Smoothness of Schrödinger Operator Potential in the Case of Gevrey Type Asymptotics of the Gaps , 2002 .

[107]  Y. Pinchover.  On positive solutions of second-order elliptic equations, stability results, and classification , 1988 .

[108]  David L. Webb,et al.  Inverse spectral results on even dimensional tori , 2008 .

[109]  W. Woess.  Random walks on infinite graphs and groups, by Wolfgang Woess, Cambridge Tracts , 2001 .

[110]  Yulia E. Karpeshina,et al.  Perturbation Theory for the Schrödinger Operator with a Periodic Potential , 1997 .

[111]  V. Zhikov,et al.  Homogenization of Differential Operators and Integral Functionals , 1994 .

[112]  M. Wegener,et al.  Periodic nanostructures for photonics , 2007 .

[113]  A. Khrabustovskyi,et al.  Gaps in the spectrum of the Neumann Laplacian generated by a system of periodically distributed traps , 2013, 1301.2926.

[114]  M. Kreĭn,et al.  Stability of Solutions of Differential Equations in Banach Spaces , 1974 .

[115]  Yulia Karpeshina.  Spectral Properties of the Periodic Magnetic Schrödinger Operator in the High-Energy Region. Two-Dimensional Case , 2004 .

[116]  M. M. Skriganov,et al.  Geometric and arithmetic methods in the spectral theory of multidimensional periodic operators , 1987 .

[117]  M. Eastham,et al.  Schrödinger-type operators with continuous spectra , 1982 .

[118]  T. O’Neil.  Geometric Measure Theory , 2002 .

[119]  Mark J. Ablowitz,et al.  Multidimensional nonlinear evolution equations and inverse scattering , 1986 .

[120]  F. Klopp,et al.  Endpoints of the spectrum of periodic operators are generically simple , 2000 .

[121]  H. Grauert.  Analytische Faserungen über holomorph-vollständigen Räumen , 1958 .

[122]  Yulia Karpeshina.  On the density of states for the periodic Schrödinger operator , 2000 .

[123]  S. Chu.  Nonperturbative Approaches to Atomic and Molecular Multiphoton (Half-Collision) Processes in Intense Laser Fields , 1997 .

[124]  P. Kuchment,et al.  On Embedded Eigenvalues of Perturbed Periodic Schrödinger Operators , 1998 .

[125]  F. Lund.  Solitons and Geometry , 1978 .

[126]  B. Simon.  Szegő's Theorem and Its Descendants: Spectral Theory for L 2 Perturbations of Orthogonal Polynomials , 2010 .

[127]  N. Filonov,et al.  Absolute continuity of the spectrum of the periodic Schrödinger operator in a layer and in a smooth cylinder , 2010, 1011.1394.

[128]  L. Friedlander,et al.  Institute for Mathematical Physics on the Spectrum of a Class of Second Order Periodic Elliptic Differential Operators on the Spectrum of a Class of Second Order Periodic Elliptic Differential Operators , 2022 .

[129]  M. Berry,et al.  Diabolical points in the spectra of triangles , 1984, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[130]  L. Ehrenpreis.  Fourier analysis in several complex variables , 1970 .

[131]  Y. Pinchover,et al.  Manifolds With Group Actions and Elliptic Operators , 1995 .

[132]  H. Knörrer,et al.  Perturbatively unstable eigenvalues of a periodic Schrödinger operator , 1991 .

[133]  Hans L. Cycon,et al.  Schrodinger Operators: With Application to Quantum Mechanics and Global Geometry , 1987 .

[134]  Yves Colin de Verdière.  Sur les singularités de van Hove génériques , 1991 .

[135]  Barry Simon,et al.  Analytic properties of band functions , 1978 .

[136]  Spectrum of the periodic Dirac operator , 2000, 0905.4622.

[137]  V. A. I︠A︡kubovich,et al.  Linear differential equations with periodic coefficients , 1975 .

[138]  The Schrödinger operator in a periodic waveguide on a plane and quasiconformal mappings , 2005 .

[139]  Kurt Busch,et al.  The Wannier function approach to photonic crystal circuits , 2003 .

[140]  V. Guillemin.  Spectral theory on S2: Some open questions , 1981 .

[141]  P. Kuchment,et al.  On the location of spectral edges in -periodic media , 2010, 1006.3001.

[142]  O. Veliev.  Spectrum of multidimensional periodic operators , 1990 .

[143]  On the scattering theory of the Laplacian with a periodic boundary condition. I. Existence of wave operators , 2003 .

[144]  E. Jaynes,et al.  The Fermi Surface , 1962 .

[145]  R. Shterenberg,et al.  Asymptotic expansion of the integrated density of states of a two-dimensional periodic Schrödinger operator , 2008, 0804.3561.

[146]  F. Smithies.  Linear Operators , 2019, Nature.

[147]  P. Kuchment.  Graph models for waves in thin structures , 2002 .

[148]  P. Kuchment,et al.  GREEN ’ S FUNCTION ASYMPTOTICS NEAR THE INTERNAL EDGES OF SPECTRA OF PERIODIC ELLIPTIC OPERATORS , 2015 .

[149]  B. Harris.  On the spectrum of the periodic Dirac operator , 1981, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[150]  M. Gavrilă.  Atomic Structure and Decay in High-Frequency Fields , 1992 .

[151]  J. Combes,et al.  Localization Near Band Edges For Random Schr Odinger Operators , 1997 .

[152]  Steven G. Johnson,et al.  Photonic Crystals: Molding the Flow of Light , 1995 .

[153]  Homogenization with corrector for a multidimensional periodic elliptic operator near an edge of an inner gap , 2011 .

[154]  J. Moser,et al.  On a Liouville-type theorem for linear and nonlinear elliptic differential equations on a torus , 1992 .

[155]  Homogenization with corrector for a periodic elliptic operator near an edge of inner gap , 2009 .

[156]  V. Palamodov,et al.  Harmonic synthesis of solutions of elliptic equation with periodic coefficients , 1993 .

[157]  H. Knörrer,et al.  The geometry of algebraic Fermi curves , 1992 .

[158]  Y. Zel’dovich.  METHODOLOGICAL NOTES: Scattering and emission of a quantum system in a strong electromagnetic wave , 1973 .

[159]  P. Wallace.  The Band Theory of Graphite , 1947 .

[160]  An example of a periodic magnetic Schrödinger operator with degenerate lower edge of the spectrum , 2005 .

[161]  Ole A. Nielsen.  Direct Integral Theory , 1980 .

[162]  P. Hartman.  Ordinary Differential Equations , 1965 .

[163]  T. Colding,et al.  HARMONIC FUNCTIONS ON MANIFOLDS , 1997 .

[164]  A. Pliś.  Non-Uniqueness in Cauchy's Problem for Differential Equations of Elliptic Type , 1960 .

[165]  B. Simon.  Trace ideals and their applications , 1979 .

[166]  B. Helffer,et al.  Asymptotic of the density of states for the Schrödinger operator with periodic electric potential , 1998 .

[167]  Absolutely Continuous Spectrum for the Isotropic Maxwell Operator with Coefficients that are Periodic in Some Directions and Decay in Others , 2004, math-ph/0406045.

[168]  R. Bolstein,et al.  Expansions in eigenfunctions of selfadjoint operators , 1968 .

[169]  R. Brooks.  The fundamental group and the spectrum of the laplacian , 1981 .

[170]  N. Zimbovskaya.  Local Geometry of the Fermi Surface: And High-Frequency Phenomena in Metals , 2001 .

[171]  V. Grushin,et al.  Multiparameter perturbation theory of Fredholm operators applied to bloch functions , 2009 .

[172]  Threshold Effects near the Lower Edge of the Spectrum for Periodic Differential Operators of Mathematical Physics , 2001 .

[173]  Edward Bierstone,et al.  A simple constructive proof of Canonical Resolution of Singularities , 1991 .

[174]  Walter Kohn,et al.  Analytic Properties of Bloch Waves and Wannier Functions , 1959 .

[175]  G. Nenciu.  Existence of the exponentially localised Wannier functions , 1983 .

[176]  J. Zak,et al.  MAGNETIC TRANSLATION GROUP , 1964 .

[177]  Gianluca Panati.  Triviality of Bloch and Bloch–Dirac Bundles , 2006 .

[178]  M. Birman,et al.  Homogenization of a multidimensional periodic elliptic operator in a neighborhood of the edge of an internal gap , 2006 .

[179]  M. Shubin,et al.  The spectral asymptotics of elliptic operators of Schrödinger type on a hyperbolic space , 1992 .

[180]  J. S. Howland.  Floquet operators with singular spectrum. II , 1989 .

[181]  P. Kuchment.  Floquet Theory for Partial Differential Equations , 1993 .

[182]  M. Babillot.  Asymptotics of Green Functions on a Class of Solvable Lie Groups , 1998 .

[183]  C. Conca,et al.  Fluids And Periodic Structures , 1995 .

[184]  C. Fefferman,et al.  Edge States in Honeycomb Structures , 2015, 1506.06111.

[185]  Konstantin Pankrashkin,et al.  On the extrema of band functions in periodic waveguides , 2013 .

[186]  Eberhard Freitag,et al.  Analytic Functions of Several Complex Variables , 2011 .

[187]  A. Soshnikov,et al.  Splitting of the Low Landau Levels into a Set of Positive Lebesgue Measure under Small Periodic Perturbations , 1997 .

[188]  Complete Asymptotic Expansion of the Integrated Density of States of Multidimensional Almost-Periodic Pseudo-Differential Operators , 2012, 1204.1076.

[189]  P. Kuchment.  The mathematics of photonic crystals , 2001 .

[190]  Periodic manifolds with spectral gaps , 2002, math-ph/0207017.

[191]  P. Kuchment.  Tight frames of exponentially decaying Wannier functions , 2008, 0807.1342.

[192]  P. McClintock,et al.  Graphene: Carbon in Two Dimensions , 2012 .

[193]  V. I. Arnold,et al.  On teaching mathematics , 1998 .

[194]  B. Mityagin.  The Zero Set of a Real Analytic Function , 2015, Mathematical Notes.

[195]  V. Palamodov,et al.  Linear Differential Operators with Constant Coefficients , 1970 .

[196]  The periodic Dirac operator is absolutely continuous , 1999 .

[197]  A R Plummer,et al.  Introduction to Solid State Physics , 1967 .

[198]  Peter D. Lax,et al.  Periodic solutions of the KdV equation , 2010 .

[199]  John Mallet-Paret,et al.  Floquet bundles for scalar parabolic equations , 1995 .

[200]  Asymptotic of the density of states for the Schrödinger operator with periodic electromagnetic potential , 1997 .

[201]  A. Khrabustovskyi.  Opening up and control of spectral gaps of the Laplacian in periodic domains , 2013, 1308.4091.

[202]  Mo-Lin Ge,et al.  Theory of Solitons , 2016 .

[203]  P. Kuchment,et al.  BANACH BUNDLES AND LINEAR OPERATORS , 1975 .

[204]  R. Remmert,et al.  Theory Of Stein Spaces , 1979 .

[205]  M. Shubin,et al.  ASYMPTOTIC EXPANSION OF THE STATE DENSITY AND THE SPECTRAL FUNCTION OF A HILL OPERATOR , 1987 .

[206]  I. Kachkovskiy,et al.  On the structure of band edges of 2d periodic elliptic operators , 2015, 1510.04367.

[207]  T. Sunada.  Group C*-Algebras and the Spectrum of a Periodic Schrödinger Operator on a Manifold , 1992, Canadian Journal of Mathematics.

[208]  Spectral Theory for Periodic Schrödinger Operators with Reflection Symmetries , 2003 .

[209]  Positive measure spectrum for Schrödinger operators with periodic magnetic fields , 2002, math-ph/0209039.

[210]  L. I. Danilov.  On Absolute Continuity of the Spectrum of a 3D Periodic Magnetic Dirac Operator , 2011 .

[211]  Shui-Nee Chow,et al.  Floquet Theory for Parabolic Differential Equations , 1994 .

[212]  G. Weiss,et al.  EIGENFUNCTION EXPANSIONS. Associated with Second-order Differential Equations. Part I. , 1962 .

[213]  Francis Nier,et al.  The Mourre Theory for Analytically Fibered Operators , 1998 .

[214]  V. Zhikov,et al.  On spectrum gaps of some divergent elliptic operators with periodic coefficients , 2005 .

[215]  Integrated Density of States for the Periodic Schrödinger Operator in Dimension Two , 2005 .

[216]  Anatolii A. Logunov,et al.  Analytic functions of several complex variables , 1965 .

[217]  Göran Borg.  Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe , 1946 .

[218]  N. Marzari,et al.  Maximally localized generalized Wannier functions for composite energy bands , 1997, cond-mat/9707145.

[219]  Minjae Lee.  Conic dispersion surfaces for point scatterers on a honeycomb lattice , 2014 .

[220]  Wilhelm Schlag,et al.  Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon's 60th Birthday , 2007 .

[221]  Nate Orlow,et al.  Hill ’ s Equation , 2010 .

[222]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[223]  David W. Zingg,et al.  for Linear Ordinary Differential Equations , 1997 .

[224]  H. Knörrer,et al.  The perturbatively stable spectrum of a periodic Schrödinger operator , 1990 .

[225]  Asymptotic bounds for spectral bands of periodic Schrödinger operators , 2006 .

[226]  A. Bensoussan,et al.  Asymptotic analysis for periodic structures , 1979 .

[227]  P. Kuchment.  Quantum graphs , 2004 .

[228]  S. Łojasiewicz.  Introduction to Complex Analytic Geometry , 1991 .

[229]  On the Structure of Eigenfunctions Corresponding to Embedded Eigenvalues of Locally Perturbed Periodic Graph Operators , 2005, math-ph/0511084.

[230]  A. Sobolev,et al.  Absolute Continuity in Periodic Waveguides , 2002 .

[231]  E. Korotyaev,et al.  Schrödinger Operators on Zigzag Nanotubes , 2007 .

[232]  L. Parnovski,et al.  Critical Dimensions for counting Lattice Points in Euclidean Annuli , 2010 .

[233]  Motoko Kotani,et al.  Discrete Geometric Analysis , 2004 .

[234]  Floquet-Bloch Theory for Elliptic Problems with Discontinuous Coefficients , 2011 .

[235]  M. Kha.  Green's function asymptotics of periodic elliptic operators on abelian coverings of compact manifolds , 2015, 1511.00276.

[236]  T. Sunada,et al.  On the spectrum of periodic elliptic operators , 1992, Nagoya Mathematical Journal.

[237]  Lower bound on the density of states for periodic Schr , 2009, 0907.4465.

[238]  M. Birman,et al.  Second order periodic differential operators. Threshold properties and homogenization , 2004 .

[239]  Noncommutative Bloch theory , 2000, math-ph/0006021.

[240]  V. Rich.  Personal communication , 1989, Nature.

[241]  Asymptotics of Green functions and Martin boundaries for elliptic operators with periodic coefficients (Spectral and Scattering Theory and Related Topics) , 2002 .

[242]  Carlos Conca,et al.  Bloch Approximation in Homogenization and Applications , 2002, SIAM J. Math. Anal..

[243]  Alden Waters.  Isospectral periodic Torii in dimension 2 , 2015 .

[244]  P. Kuchment,et al.  Introduction to Quantum Graphs , 2012 .

[245]  F. Greenleaf,et al.  Representations of nilpotent Lie groups and their applications , 1989 .

[246]  Integral representations of solutions of periodic elliptic equations , 2006, math/0604139.

[247]  Absolute Continuity of the Spectrum for Periodically Modulated Leaky Wires in $${\mathbb{R}^{3}}$$ , 2005, math/0508525.

[248]  M. Born,et al.  Wave Propagation in Periodic Structures , 1946, Nature.

[249]  On an abstract integrodifferential equation with periodic coefficient. I , 1992 .

[250]  M. Seifert,et al.  Generalized Eigenfunctions for Waves in Inhomogeneous Media , 2002 .

[251]  R. Shterenberg,et al.  On the scattering theory of the Laplacian with a periodic boundary condition. II. Additional channels of scattering , 2004, Documenta Mathematica.

[252]  ABSOLUTE CONTINUITY OF THE SPECTRUM FOR PERIODICALLY MODULATED LEAKY WIRES IN R3 , 2005 .

[253]  N. Filonov.  Second-Order Elliptic Equation of Divergence Form Having a Compactly Supported Solution , 2001 .

[254]  K. Yajima.  Large Time Behaviors of Time-Periodic Quantum Systems , 1984 .

[255]  Michael I. Weinstein,et al.  Wave Packets in Honeycomb Structures and Two-Dimensional Dirac Equations , 2012, 1212.6072.

[256]  Stephen M. Stigler,et al.  STIGLER'S LAW OF EPONYMY† , 1980 .

[257]  On the structure of the lower edge of the spectrum of the periodic magnetic Schrödinger operator with small magnetic potential , 2006 .

[258]  J. Howland.  Scattering theory for Hamiltonians periodic in time , 2012, 1212.2931.

[259]  O. Veliev.  Perturbation Theory for the Periodic Multidimensional Schrodinger Operator and the Bethe-Sommerfeld Conjecture , 2006, math-ph/0610057.

[260]  S. Yau,et al.  GEOMETRIC ANALYSIS , 2005 .

[261]  Carlos Conca,et al.  Homogenization of Periodic Structures via Bloch Decomposition , 1997, SIAM J. Appl. Math..

[262]  Michael I. Weinstein,et al.  Honeycomb Lattice Potentials and Dirac Points , 2012, 1202.3839.

[263]  N. Filonov.  Gaps in the Spectrum of the Maxwell Operator with Periodic Coefficients , 2003 .

[264]  Absence of eigenvalues for the generalized two-dimensional periodic Dirac operator , 2007, math-ph/0703029.

[265]  L. Saloff‐Coste.  RANDOM WALKS ON INFINITE GRAPHS AND GROUPS (Cambridge Tracts in Mathematics 138) , 2001 .

[266]  V. Hoang,et al.  Absence of bound states for waveguides in 2D periodic structures , 2011, 1111.4578.

[267]  Harold R. Parks,et al.  A Primer of Real Analytic Functions , 1992 .

[268]  M. Avellaneda,et al.  Un theoreme de Liouville pour des equation elliptiques a coefficient periodiques (French): [A Liouville theorem for elliptic equations with peridic coefficients] , 1989 .

[269]  P. Kuchment,et al.  Quantum graph spectra of a graphyne structure , 2013, 1302.5176.

[270]  Gregory Berkolaiko,et al.  Symmetry and Dirac points in graphene spectrum , 2014, Journal of Spectral Theory.

[271]  Grégoire Allaire,et al.  BLOCH WAVE HOMOGENIZATION AND SPECTRAL ASYMPTOTIC ANALYSIS , 1998 .

[272]  G. Johnson.  The Schrödinger equation , 1998 .

[273]  Barry Simon,et al.  Comparison theorems for the gap of Schrödinger òperators , 1987 .

[274]  Robert W. Boyd,et al.  SCISSOR solitons and other novel propagation effects in microresonator-modified waveguides , 2002 .

[275]  Peter Kuchment,et al.  Spectral Properties of High Contrast Band-Gap Materials and Operators on Graphs , 1999, Exp. Math..

[276]  M. Skriganov.  The spectrum band structure of the three-dimensional Schrödinger operator with periodic potential , 1985 .

[277]  Absolute Continuity in Periodic Thin Tubes and Strongly Coupled Leaky Wires , 2003, math/0307188.

[278]  L. I. Danilov.  On absolute continuity of the spectrum of a periodic magnetic Schrödinger operator , 2009, 0902.3371.

[279]  A. Wigderson,et al.  ENTROPY WAVES, THE ZIG-ZAG GRAPH PRODUCT, AND NEW CONSTANT-DEGREE , 2004, math/0406038.

[280]  C. Wilcox.  Theory of Bloch waves , 1978 .

[281]  J. Cooper,et al.  Les Algebres d'Operateurs dans l'Espace Hilbertien , 1958 .

[282]  S. Shipman.  Eigenfunctions of Unbounded Support for Embedded Eigenvalues of Locally Perturbed Periodic Graph Operators , 2013, 1307.6838.

[283]  Tetsuo Tsuchida,et al.  Asymptotics of Green functions and the limiting absorption principle for elliptic operators with periodic coefficients , 2006 .

[284]  Absolute Continuity of the Spectrum of the Periodic Operator of Elasticity Theory for Constant Shear Modulus , 2002 .

[285]  Peter Kuchment,et al.  Waves in Periodic and Random Media , 2003 .

[286]  D. Thouless.  Wannier functions for magnetic sub-bands , 1984 .

[287]  Absolute Continuity of the Spectrum of a Periodic Schrödinger Operator , 2003 .

[288]  Sébastien Guenneau,et al.  Bloch dispersion and high frequency homogenization for separable doubly-periodic structures , 2012 .

[289]  L. Friedlander.  ON THE DENSITY OF STATES OF PERIODIC MEDIA IN THE LARGE COUPLING LIMIT , 2002 .

[290]  Andrii Khrabustovskyi,et al.  Periodic elliptic operators with asymptotically preassigned spectrum , 2012, Asymptot. Anal..

[291]  V. I. I︠U︡dovich.  The linearization method in hydrodynamical stability theory , 1989 .

[292]  On the Spectra of Carbon Nano-Structures , 2006, math-ph/0612021.

[293]  B. Dahlberg,et al.  A remark on two dimensional periodic potentials , 1982 .

[294]  On the spectrum of the two-dimensional periodic Dirac operator , 1999 .

[295]  F. Greenleaf,et al.  Basic theory and examples , 1990 .

[296]  Peter Kuchment.  Quantum graphs: II. Some spectral properties of quantum and combinatorial graphs , 2005 .

[297]  M. Babillot,et al.  Théorie du renouvellement pour des chaînes semi-markoviennes transientes , 1988 .

[298]  M. Birman,et al.  The limit absorption principle and homogenization procedure for periodic elliptic operators , 2008 .

[299]  Inverse spectral results on two-dimensional tori , 1990 .

[300]  B. Mityagin,et al.  Instability zones of periodic 1-dimensional Schrödinger and Dirac operators , 2006 .

[301]  T. Kappeler,et al.  Asymptotics of spectral quantities of Schrodinger operators , 2011, 1107.4542.

[302]  O. Post,et al.  Spectral Gaps for Periodic Elliptic Operators with High Contrast: an Overview , 2002, math-ph/0207020.

[303]  On discreteness of the spectrum of some operator sheaves associated with a periodic Schrödinger equation , 1988 .

[304]  M. Saboormaleki.  On the spectrum of non-selfadjoint differential operators. , 2008 .

[305]  V. A. Kondrat'ev,et al.  On Positive Solutions of Elliptic Equations , 1971 .

[306]  J. Lions,et al.  Un théorème de Liouville pour des équations elliptiques à coefficients périodiques , 1989 .

[307]  J. Garnett,et al.  Gaps and bands of one dimensional periodic Schrödinger operators , 1984 .

[308]  Bands and Gaps for Periodic Magnetic Hamiltonians , 1995 .

[309]  L. I. Danilov.  Absolute continuity of the spectrum of a periodic dirac operator , 2000 .

[310]  A. Sobolev,et al.  Bethe-Sommerfeld conjecture for periodic operators with strong perturbations , 2009, 0907.0887.

[311]  The absolute continuity of the spectrum of Maxwell operator in a periodic media , 2000 .

[312]  T. Hoffmann-Ostenhof,et al.  L2-lower bounds to solutions of one-body Schrödinger equations , 1983, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[313]  L. Thomas,et al.  Time dependent approach to scattering from impurities in a crystal , 1973 .

[314]  Absolutely Continuous Spectrum of a Polyharmonic Operator with a Limit Periodic Potential in Dimension Two , 2007, 0711.4404.

[315]  B. Bernevig.  Topological Insulators and Topological Superconductors , 2013 .

[316]  G. V. Chester,et al.  Solid State Physics , 2000 .

[317]  The Bethe-Sommerfeld conjecture for the 3-Dimensional periodic Landau operator. , 2004 .

[318]  K. Yajima.  Quantum dynamics of time periodic systems , 1984 .

[319]  S. Agmon.  Lectures on Elliptic Boundary Value Problems , 1965 .

[320]  E. Trubowitz,et al.  On isospectral periodic potentials in Rn. II , 1984 .

[321]  I. S. Lapin.  One Version of the Bethe–Sommerfeld Conjecture , 2003 .

[322]  Alexander Figotin,et al.  Band-Gap Structure of Spectra of Periodic Dielectric and Acoustic Media. I. Scalar Model , 1996, SIAM J. Appl. Math..

[323]  M. Birman.  On homogenization procedure for periodic operators near the edge of an internal gap , 2004 .

[324]  L. Thomas,et al.  Asymptotic behaviour of eigenfunctions for multiparticle Schrödinger operators , 1973 .

[325]  Izidor Gertner,et al.  The Finite Zak Transform and the Finite Fourier Transform , 1992 .

[326]  E. Wigner,et al.  Theory of Brillouin Zones and Symmetry Properties of Wave Functions in Crystals , 1936 .

[327]  L. Brillouin,et al.  Propagation des ondes dans les milieux périodiques , 1957 .

[328]  Willy Dörfler,et al.  Photonic Crystals: Mathematical Analysis and Numerical Approximation , 2011 .

[329]  H. Knörrer,et al.  A directional compactification of the complex Fermi surface and isospectrality , 1990 .

[330]  Henry P. McKean,et al.  Hill’s Operator and Hyperelliptic Function Theory in the Presence of Infinitely Many Branch Points , 1976 .