Comptes Rendus
Physique mathématique, Théorie spectrale
Spectral properties of periodic systems cut at an angle
David Gontier1
1 CEREMADE, University of Paris-Dauphine, PSL University, 75016 Paris, France
Comptes Rendus. Mathématique, Volume 359 (2021) no. 8, pp. 949-958.

We consider a semi-periodic two-dimensional Schrödinger operator which is cut at an angle. When the cut is commensurate with the periodic lattice, the spectrum of the operator has the band-gap Bloch structure. We prove that in the incommensurable case, there are no gaps: the gaps of the bulk operator are filled with edge spectrum.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/crmath.251

David Gontier 1

1 CEREMADE, University of Paris-Dauphine, PSL University, 75016 Paris, France
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{CRMATH_2021__359_8_949_0,
     author = {David Gontier},
     title = {Spectral properties of periodic systems cut at an angle},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {949--958},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {359},
     number = {8},
     year = {2021},
     doi = {10.5802/crmath.251},
     language = {en},
}
TY  - JOUR
AU  - David Gontier
TI  - Spectral properties of periodic systems cut at an angle
JO  - Comptes Rendus. Mathématique
PY  - 2021
SP  - 949
EP  - 958
VL  - 359
IS  - 8
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.251
LA  - en
ID  - CRMATH_2021__359_8_949_0
ER  - 
%0 Journal Article
%A David Gontier
%T Spectral properties of periodic systems cut at an angle
%J Comptes Rendus. Mathématique
%D 2021
%P 949-958
%V 359
%N 8
%I Académie des sciences, Paris
%R 10.5802/crmath.251
%G en
%F CRMATH_2021__359_8_949_0
David Gontier. Spectral properties of periodic systems cut at an angle. Comptes Rendus. Mathématique, Volume 359 (2021) no. 8, pp. 949-958. doi : 10.5802/crmath.251. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.251/

[1] Michael F. Atiyah; Vijay K. Patodi; Isadore M. Singer Spectral asymmetry and Riemannian geometry. III, Math. Proc. Camb. Philos. Soc., Volume 79 (1976) no. 1, pp. 71-99 | DOI | MR | Zbl

[2] Joseph Avron; Barry Simon Almost periodic Schrödinger operators II. The integrated density of states, Duke Math. J., Volume 50 (1983) no. 1, pp. 369-391 | DOI | Zbl

[3] Gilles Carron Déterminant relatif et la fonction Xi, Am. J. Math., Volume 124 (2002) no. 2, pp. 307-352 | DOI | MR | Zbl

[4] Jean-Michel Combes; Lyn Thomas Asymptotic behaviour of eigenfunctions for multiparticle Schrödinger operators, Commun. Math. Phys., Volume 34 (1973) no. 4, pp. 251-270 | DOI | Zbl

[5] Edward B. Davies; Barry Simon Scattering theory for systems with different spatial asymptotics on the left and right, Commun. Math. Phys., Volume 63 (1978) no. 3, pp. 277-301 | DOI | MR | Zbl

[6] David Gontier Edge states for second order elliptic operators (2021) | arXiv

[7] Rainer Hempel; Martin Kohlmann Spectral properties of grain boundaries at small angles of rotation, J. Spectr. Theory, Volume 1 (2011), pp. 197-219 | DOI | MR | Zbl

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

[9] Rainer Hempel; Martin Kohlmann 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, Springer, 2012, pp. 421-432 | DOI | Zbl

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

[11] Tosio Kato Perturbation theory for linear operators, Grundlehren der Mathematischen Wissenschaften, 132, Springer, 2013 | Zbl

[12] John Phillips Self-adjoint Fredholm operators and spectral flow, Can. Math. Bull., Volume 39 (1996) no. 4, pp. 460-467 | DOI | MR | Zbl

[13] Michael Reed; Barry Simon Methods of Modern Mathematical Physics Vol. I: Functional analysis, Academic Press Inc., 1972 | DOI | Zbl

[14] Michael Reed; Barry Simon Methods of Modern Mathematical Physics. IV: Analysis of Operators, Academic Press Inc., 1977 | Zbl

[15] Michael Reed; Barry Simon Methods of Modern Mathematical Physics. III, Scattering Theory, Academic Press Inc., 1978 | Zbl

[16] Barry Simon Schrödinger semigroups, Bull. Am. Math. Soc., Volume 7 (1982) no. 3, pp. 447-526 | DOI | Zbl

  • Camilo Gómez Araya; David Gontier; Hanne Van Den Bosch Edge states for tight-binding operators with soft walls, Communications in Mathematical Physics, Volume 406 (2025) no. 7, p. 47 (Id/No 160) | DOI:10.1007/s00220-025-05326-1 | Zbl:8055622
  • Charles L. Fefferman; Sonia Fliss; Michael I. Weinstein Discrete honeycombs, rational edges, and edge states, Communications on Pure and Applied Mathematics, Volume 77 (2024) no. 3, pp. 1575-1634 | DOI:10.1002/cpa.22141 | Zbl:7793233
  • Bérangère Delourme; Sonia Fliss Guided modes in a hexagonal periodic graph like domain, Multiscale Modeling Simulation, Volume 22 (2024) no. 3, pp. 1196-1245 | DOI:10.1137/23m1600177 | Zbl:7950207
  • David Gontier Edge states for second order elliptic operators in a channel, Journal of Spectral Theory, Volume 12 (2022) no. 3, pp. 1155-1202 | DOI:10.4171/jst/430 | Zbl:1514.35295

Cité par 4 documents. Sources : zbMATH

Commentaires - Politique