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The article surveys the main techniques and results of the spectral theory of periodic operators arising in mathematical physics and other areas.
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by same author, A perturbation of a Hill’s operator, that has a first moment and a non-zero integral, introduces a single discrete level into each of the distant spectral lacunae , Mathematical physics and functional analysis, No. 4 (Russian), Fiz.-Tehn. Inst. Nizkih Temperatur, Akad. Nauk Ukrain. SSR, Kharkov, 1973, pp. 158–159, 163. MR MR0477257 (57 #16798)
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by same author, Floquet operators with singular spectrum. I, II , Ann. Inst. H. Poincaré Phys. Théor. 50
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by same author, The linearization method in hydrodynamical stability theory , Translations of Mathematical Monographs, vol. 74, American Mathematical Society, Providence, RI, 1989, Translated from the Russian by J. R. Schulenberger. MR 1003607 (90h:76001)
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by same author, On the homogenization for periodic operators in a neighborhood of an edge of an internal gap , Algebra i Analiz 15
2003
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by same author, Periodic second-order differential operators. Threshold properties and averaging , Algebra i Analiz 15
2003
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2003
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K. Busch, S. Mingaleev, A. Garcia-Martin, M. Schillinger, and D. Hermann, The Wannier function approach to photonic crystal circuits , Journal of Physics: Condensed Matter 15
2003
Later among the works it cites.
by same author, On the absolute continuity of the spectrum of a periodic Schrödinger operator , Mat. Zametki 73
2003
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1990
Cited alongside, same era.
L. J. Corwin and F. P. Greenleaf, Representations of nilpotent Lie groups and their applications. Part I , Cambridge Studies in Advanced Mathematics, vol. 18, Cambridge University Press, Cambridge, 1990, Basic theory and examples. MR 1070979 (92b:22007)
1990
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L. Danilov, On the spectrum of the Dirac operator in 𝐑 n {\bf R}^{n} with periodic potential , Teoret. Mat. Fiz. 85
1990
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J. Feldman, H. Knörrer, and E. Trubowitz, The perturbatively stable spectrum of a periodic Schrödinger operator , Invent. Math. 100
1990
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L. Friedlander, On the spectrum of the periodic problem for the Schrödinger operator , Comm. Partial Differential Equations 15
1990
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C. Gérard, Resonance theory for periodic Schrödinger operators , Bull. Soc. Math. France 118
1990
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by same author, Inverse spectral results on two-dimensional tori , Journal of the American Mathematical Society 3
1990
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H. Knörrer and E. Trubowitz, A directional compactification of the complex Bloch variety , Comment. Math. Helv. 65
1990
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by same author, On the spectrum of the two-dimensional periodic Schrödinger operator , Teoret. Mat. Fiz. 134
2003
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by same author, Gaps in the spectrum of the Maxwell operator with periodic coefficients , Comm. Math. Phys. 240
2003
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R. Frank, On the scattering theory of the Laplacian with a periodic boundary condition. I. Existence of wave operators , Doc. Math. 8
2003
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by same author, Absolute continuity of the spectra of periodic waveguides , Waves in periodic and random media (South Hadley, MA, 2002), Contemp. Math., vol. 339, Amer. Math. Soc., Providence, RI, 2003, pp. 37–42. MR MR2042530 (2005a:35211)
2003
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by same author, Positive measure spectrum for Schrödinger operators with periodic magnetic fields , J. Math. Phys. 44
2003
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B. Helffer and T. Hoffmann-Ostenhof, Spectral theory for periodic Schrödinger operators with reflection symmetries , Comm. Math. Phys. 242
2003
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R. Hempel and O. Post, Spectral gaps for periodic elliptic operators with high contrast: an overview , Progress in analysis, Vol. I, II (Berlin, 2001), World Sci. Publ., River Edge, NJ, 2003, pp. 577–587. MR 2032728 (2004k:35281)
2003
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E. Korotyaev and A. Pushnitski, On the high-energy asymptotics of the integrated density of states , Bull. London Math. Soc. 35
2003
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I. S. Lapin, One version of the Bethe-Sommerfeld conjecture , J. Math. Sci. (N. Y.) 117
2003
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N. Marzari, I Souza, I., and D. Vanderbilt, An introduction to maximally-localized wannier functions , Highlight of the Month, Psi-K Newsletter 57
2003
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M. Murata and T. Tsuchida, Asymptotics of Green functions and Martin boundaries for elliptic operators with periodic coefficients , J. Differential Equations 195
2003
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O. Post, Periodic manifolds with spectral gaps , J. Differential Equations 187
2003
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by same author, On the Bethe-Sommerfeld conjecture for higher-order elliptic operators , Math. Ann. 326
2003
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by same author, The Schrödinger operator in a periodic waveguide on a plane and quasiconformal mappings , Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 295
2003
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by same author, High energy asymptotics of the density of states for certain periodic pseudo-differential operators in dimension one , Waves in periodic and random media (South Hadley, MA, 2002), Contemp. Math., vol. 339, Amer. Math. Soc., Providence, RI, 2003, pp. 171–184. MR 2042537 (2005a:35216)
2003
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by same author, Homogenization of a multidimensional periodic elliptic operator in a neighborhood of an edge of an inner gap , Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 318
2004
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by same author, Topological and algebraic geometry methods in contemporary mathematical physics , Classic Reviews in Mathematics and Mathematical Physics, vol. 2, Cambridge Scientific Publishers, Cambridge, 2004. MR 2187099 (2006f:37002)
2004
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D. Elton, The Bethe-Sommerfield conjecture for the 3-dimensional periodic Landau operator , Rev. Math. Phys. 16
2004
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N. Filonov and F. Klopp, Absolute continuity of the spectrum of a Schrödinger operator with a potential which is periodic in some directions and decays in others , Doc. Math. 9
2004
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by same author, Erratum to: “Absolute continuity of the spectrum of a Schrödinger operator with a potential which is periodic in some directions and decays in others” [Doc. Math. 9
2004
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N. Filonov and A. V. Sobolev, Absence of the singular continuous component in the spectrum of analytic direct integrals , Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 318
2004
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R. Frank and R. Shterenberg, On the scattering theory of the Laplacian with a periodic boundary condition. II. Additional channels of scattering , Doc. Math. 9
2004
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Y. Higuchi and T. Shirai, Some spectral and geometric properties for infinite graphs , Discrete geometric analysis, Contemp. Math., vol. 347, Amer. Math. Soc., Providence, RI, 2004, pp. 29–56
2004
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by same author, Spectral properties of the periodic magnetic Schrödinger operator in the high-energy region. Two-dimensional case , Comm. Math. Phys. 251
2004
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by same author, An example of a periodic magnetic Schrödinger operator with a degenerate lower edge of the spectrum , Algebra i Analiz 16
2004
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M. Tikhomirov and N. Filonov, Absolute continuity of an “even” periodic Schrödinger operator with nonsmooth coefficients , Algebra i Analiz 16
2004
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V. V. Zhikov, Gaps in the spectrum of some elliptic operators in divergent form with periodic coefficients , Algebra i Analiz 16
2004
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by same author, Bloch approximation in homogenization on bounded domains , Asymptot. Anal. 41
2005
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by same author, On the nonexistence of eigenvalues in the spectrum of a generalized two-dimensional periodic Dirac operator , Algebra i Analiz 17
2005
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by same author, Absolutely continuous spectrum for the isotropic Maxwell operator and coefficients that are periodic in some directions and decay in others , Comm. Math. Phys. 258
2005
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Y. Kordyukov, Spectral gaps for periodic Schrödinger operators with strong magnetic fields , Comm. Math. Phys. 253
2005
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by same author, Quantum graphs. II. Some spectral properties of quantum and combinatorial graphs , J. Phys. A 38
2005
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by same author, On the structure of the lower edge of the spectrum of a periodic magnetic Schrödinger operator with small magnetic potential , Algebra i Analiz 17
2005
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by same author, Trace ideals and their applications , second ed., Mathematical Surveys and Monographs, vol. 120, American Mathematical Society, Providence, RI, 2005. MR 2154153 (2006f:47086)
2005
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M. M. Skriganov and A. V. Sobolev, Asymptotic estimates for spectral bands of periodic Schrödinger operators , Algebra i Analiz 17
2005
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by same author, Integrated density of states for the periodic Schrödinger operator in dimension two , Ann. Henri Poincaré 6
2005
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by same author, Instability zones of one-dimensional periodic Schrödinger and Dirac operators , Uspekhi Mat. Nauk 61
2006
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by same author, Semiclassical asymptotics and spectral gaps for periodic magnetic Schrödinger operators on covering manifolds , C ∗ C^{\ast} -algebras and elliptic theory, Trends Math., Birkhäuser, Basel, 2006, pp. 129–150. MR 2276917 (2008f:58034)
2006
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by same author, On the structure of eigenfunctions corresponding to embedded eigenvalues of locally perturbed periodic graph operators , Comm. Math. Phys. 268
2006
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by same author, Asymptotics of Green functions and the limiting absorption principle for elliptic operators with periodic coefficients , J. Math. Kyoto Univ. 46
2006
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Beng-Seong Ong, Spectral problems of optical waveguides and quantum graphs , Ph.D. thesis, Texas A&M University, 2006
2006
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by same author, Asymptotics of the integrated density of states for periodic elliptic pseudo-differential operators in dimension one , Rev. Mat. Iberoam. 22
2006
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K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, Periodic nanostructures for photonics , Physics Reports 444
2007
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P. Exner and R. Frank, Absolute continuity of the spectrum for periodically modulated leaky wires in ℝ 3 \mathbb{R}^{3} , Ann. Henri Poincaré 8
2007
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J. M. Harrison, P. Kuchment, A. Sobolev, and B. Winn, On occurrence of spectral edges for periodic operators inside the Brillouin zone , J. Phys. A 40
2007
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E. Korotyaev and I. Lobanov, Schrödinger operators on zigzag nanotubes , Ann. Henri Poincaré 8
2007
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by same author, Integral representations of solutions of periodic elliptic equations , Probability and mathematical physics, CRM Proc. Lecture Notes, vol. 42, Amer. Math. Soc., Providence, RI, 2007, pp. 323–339. MR 2352277 (2008k:35077)
2007
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by same author, On Sunada’s no gap conjecture , unpublished (2007)
2007
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by same author, Liouville theorems and spectral edge behavior on abelian coverings of compact manifolds , Trans. Amer. Math. Soc. 359
2007
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P. Kuchment and O. Post, On the spectra of carbon nano-structures , Comm. Math. Phys. 275
2007
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G. Panati, Triviality of Bloch and Bloch-Dirac bundles , Ann. Henri Poincaré 8
2007
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V. P. Smyshlyaev and P. Kuchment, Slowing down and transmission of waves in high contrast periodic media via “non-classical” homogenization , unpublished (2007)
2007
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by same author, Recent results on the Bethe-Sommerfeld conjecture , Spectral theory and mathematical physics: a Festschrift in honor of Barry Simon’s 60th birthday, Proc. Sympos. Pure Math., vol. 76, Amer. Math. Soc., Providence, RI, 2007, pp. 383–398. MR 2310211 (2009d:35245)
2007
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by same author, Perturbation theory for the periodic multidimensional Schrödinger operator and the Bethe-Sommerfeld conjecture , Int. J. Contemp. Math. Sci. 2
2007
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by same author, The limiting absorption principle and the homogenization procedure for periodic elliptic operators , Funktsional. Anal. i Prilozhen. 42
2008
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by same author, A proof of the abstract limiting absorption principle by energy estimates , J. Funct. Anal. 254
2008
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C. Gordon, P. Guerini, T. Kappeler, and D. Webb, Inverse spectral results on even dimensional tori , Ann. Inst. Fourier (Grenoble) 58
2008
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J. D. Joannopoulos, S. Johnson, R. D. Meade, and J. N. Winn, Photonic crystals: Molding the flow of light , 2nd ed., Princeton University Press, Princeton, N.J., 2008
2008
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Y. Karpeshina and Young-Ran Lee, Absolutely continuous spectrum of a polyharmonic operator with a limit periodic potential in dimension two , Comm. Partial Differential Equations 33
2008
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F. Lledó and O. Post, Existence of spectral gaps, covering manifolds and residually finite groups , Rev. Math. Phys. 20
2008
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L. Parnovski, Bethe-Sommerfeld conjecture , Ann. Henri Poincaré 9
2008
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Z. Shen and P. Zhao, Uniform Sobolev inequalities and absolute continuity of periodic operators , Trans. Amer. Math. Soc. 360
2008
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by same author, Discrete geometric analysis , Analysis on graphs and its applications, Proc. Sympos. Pure Math., vol. 77, Amer. Math. Soc., Providence, RI, 2008, pp. 51–83. MR MR2459864
2008
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G. Barbatis and L. Parnovski, Bethe-Sommerfeld conjecture for pseudodifferential perturbation , Comm. Partial Differential Equations 34
2009
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by same author, On absolute continuity of the spectrum of a periodic magnetic Schrödinger operator , J. Phys. A 42
2009
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by same author, Spectral gaps of Schrödinger operators with periodic singular potentials , Dyn. Partial Differ. Equ. 6
2009
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V. V. Grushin, Application of the multiparameter theory of perturbations of Fredholm operators to Bloch functions , Mat. Zametki 86
2009
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I. Kachkovskiĭ and N. Filonov, Absolute continuity of the spectrum of a periodic Schrödinger operator in a multidimensional cylinder , Algebra i Analiz 21
2009
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by same author, Tight frames of exponentially decaying Wannier functions , J. Phys. A 42
2009
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L. Parnovski and R. Shterenberg, Asymptotic expansion of the ibull-lntegrated density of states of a two-dimensional periodic Schrödinger operator , Invent. Math. 176
2009
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by same author, Asymptotic expansion of the integrated density of states of a two-dimensional periodic Schrödinger operator , Invent. Math. 176
2009
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T. A. Suslina and A. A. Kharin, Homogenization with corrector for a periodic elliptic operator near an edge of inner gap , J. Math. Sci. (N. Y.) 159
2009
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by same author, Lectures on elliptic boundary value problems , AMS Chelsea Publishing, Providence, RI, 2010. MR 2589244 (2011c:35004)
2010
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by same author, On absolute continuity of the spectrum of three- and four-dimensional periodic Schrödinger operators , J. Phys. A 43
2010
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P. Exner, P. Kuchment, and B. Winn, On the location of spectral edges in ℤ \mathbb{Z} -periodic media , J. Phys. A 43
2010
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by same author, Absolute continuity of the spectrum of the periodic Schrödinger operator in a layer and in a smooth cylinder , Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 385
2010
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F. Klopp, Absolute continuity of the spectrum of a Landau Hamiltonian perturbed by a generic periodic potential , Math. Ann. 347
2010
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S. Morozov and I. Parnovski, L.and Pchelintseva, Lower bound on the density of states for periodic Schrödinger operators , Operator theory and its applications, Amer. Math. Soc. Transl. Ser. 2, vol. 231, Amer. Math. Soc., Providence, RI, 2010, pp. 161–171. MR 2757529 (2012e:35179)
2010
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S. Morozov, L. Parnovski, and I. Pchelintseva, Lower bound on the density of states for periodic Schrödinger operators , Operator theory and its applications, Amer. Math. Soc. Transl. Ser. 2, vol. 231, Amer. Math. Soc., Providence, RI, 2010, pp. 161–171. MR 2757529 (2012e:35179)
2010
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L. Parnovski and N. Sidorova, Critical dimensions for counting lattice points in Euclidean annuli , Math. Model. Nat. Phenom. 5
2010
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by same author, Bethe-Sommerfeld conjecture for periodic operators with strong perturbations , Invent. Math. 181
2010
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B. M. Brown, V. Hoang, M. Plum, and I. G. Wood, Floquet-Bloch theory for elliptic problems with discontinuous coefficients , Spectral theory and analysis, Oper. Theory Adv. Appl., vol. 214, Birkhäuser/Springer Basel AG, Basel, 2011, pp. 1–20. MR 2808460 (2012i:35067)
2011
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by same author, On absolute continuity of the spectrum of a 3D periodic magnetic Dirac operator , Integral Equations Operator Theory 71
2011
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A. Enyanshin and A. Ivanovskii, Graphene allotropes: stability, structural and electronic properties from df-tb calculations , Phys. Status Solidi (B) 248
2011
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by same author, Szego’s theorem and its descendants, spectral theory for l 2 l^{2} perturbations of orthogonal polynomials , Princeton University Press, Princeton, NJ, 2011
2011
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by same author, Homogenization with corrector for a multidimensional periodic elliptic operator near an edge of an inner gap , J. Math. Sci. (N. Y.) 177
2011
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M. Vorobets, On the Bethe-Sommerfeld conjecture for certain periodic Maxwell operators , J. Math. Anal. Appl. 377
2011
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R. V. Craster, J. Kaplunov, E. Nolde, and S. Guenneau, Bloch dispersion and high frequency homogenization for separable doubly-periodic structures , Wave Motion 49
2012
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C. Fefferman and M. Weinstein, Honeycomb lattice potentials and Dirac points , J. Amer. Math. Soc. 25
2012
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T. Kappeler, B. Schaad, and P. Topalov, Asymptotics of spectral quantities of Schrödinger operators , Spectral geometry, Proc. Sympos. Pure Math., vol. 84, Amer. Math. Soc., Providence, RI, 2012, pp. 243–284. MR 2985321
2012
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M. I. Katsnelson, Graphene. carbon in two dimensions , Cambridge Univ. Press, 2012
2012
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P. Kuchment and A. Raich, Green’s function asymptotics near the internal edges of spectra of periodic elliptic operators. Spectral edge case , Math. Nachr. 285
2012
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by same author, Geometric analysis , Cambridge Studies in Advanced Mathematics, vol. 134, Cambridge University Press, Cambridge, 2012. MR 2962229
2012
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by same author, Complete asymptotic expansion of the integrated density of states of multidimensional almost-periodic Schrödinger operators , Ann. of Math. (2) 176
2012
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G. Berkolaiko and P. Kuchment, Introduction to quantum graphs , Mathematical Surveys and Monographs, vol. 186, American Mathematical Society, Providence, RI, 2013. MR 3013208
2013
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B. A. Bernevig, Topological insulators and topological superconductors , Princeton Univ. Press, 2013
2013
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D. Borisov and K. Pankrashkin, On the extrema of band functions in periodic waveguides , Funktsional. Anal. i Prilozhen. 47
2013
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by same author, Quantum waveguides with small periodic perturbations: gaps and edges of Brillouin zones , J. Phys. A 46
2013
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B. M. Brown, M. S. P. Eastham, and K. M. Schmidt, Periodic differential operators , Operator Theory: Advances and Applications, vol. 230, Birkhäuser/Springer Basel AG, Basel, 2013. MR 2978285
2013
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Ngoc T. Do and P. Kuchment, Quantum graph spectra of a graphyne structure , Nanoscale Systems: Mathematical Modeling, Theory and Applications 2
2013
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A. Khrabustovskyi, Periodic elliptic operators with asymptotically preassigned spectrum , Asymptot. Anal. 82
2013
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G. Panati and A. Pisante, Bloch bundles, Marzari-Vanderbilt functional and maximally localized Wannier functions , Comm. Math. Phys. 322
2013
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G. Berkolaiko and A. Comech, Symmetry and Dirac points in graphene spectrum , arXiv:1412.8096 (2014)
2014
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Ngoc T. Do, On the quantum graph spectra of graphyne nanotubes , Analysis and Mathematical Physics (2014) 5
2014
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C. Fefferman, J. Lee-Thorp, and M. Weinstein, Topologically protected states in one-dimensional continuous systems and Dirac points , Proc. Natl. Acad. Sci. USA 111
2014
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by same author, Wave packets in honeycomb structures and two-dimensional Dirac equations , Comm. Math. Phys. 326
2014
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Vu Hoang and M. Radosz, Absence of bound states for waveguides in two-dimensional periodic structures , J. Math. Phys. 55
2014
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by same author, Opening up and control of spectral gaps of the Laplacian in periodic domains , J. Math. Phys. 55
2014
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2014
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S. Morozov, L. Parnovski, and R. Shterenberg, Complete asymptotic expansion of the integrated density of states of multidimensional almost-periodic pseudo-differential operators , Ann. Henri Poincaré 15
2014
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S. Shipman, Eigenfunctions of unbounded support for embedded eigenvalues of locally perturbed periodic graph operators , Comm. Math. Phys. 332
2014
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D. Auckly and P. Kuchment, On parseval frames of Wannier functions , in preparation (2015)
2015
Closest in time.
Ngoc T. Do, P. Kuchment, and F. Sottile, On the generic structure of spectral edges for periodic difference operators , In preparation (2015)
2015
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2015
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by same author, Edge states in honeycomb structures , arXiv:1506.06111 (2015)
2015
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2015
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2015
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2015
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A. Khrabustovskyi and E. Khruslov, Gaps in the spectrum of the Neumann Laplacian generated by a system of periodically distributed traps , Math. Methods Appl. Sci. 38
2015
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by same author, Introduction to periodic operators, video at http://www.newton.ac.uk/event/pepw01/timetable , 2015
2015
Closest in time.
B. Mityagin, The zero set of a real analytic function , arXiv:1512.07276 (2015)
2015
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by same author, Some results on ordinary differential operators with periodic coefficients , Complex Analysis and Operator Theory 9
2015
Closest in time.
L. Parnovskii and R. Shterenberg, Personal communication , Unpublished (2015)
2015
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by same author, The gauge transform method in the theory of periodic operators , Lectures at the Isaac Newton Institute, Cambridge (January 2015)
2015
Closest in time.
by same author, Multidimensional periodic Schrödinger operator , Springer Tracts in Modern Physics, vol. 263, Springer, Cham, 2015, Perturbation theory and applications. MR 3328527
2015
Closest in time.
A. Waters, Isospectral periodic Torii in dimension 2 , Ann. Inst. H. Poincaré Anal. Non Linéaire 32
2015
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2016
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J. Zak, Magnetic translation group. II. Irreducible representations , Phys. Rev. (2) 134
2032
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