Mathematical physics, Spectral theory
Spectral properties of periodic systems cut at an angle
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.

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DOI: 10.5802/crmath.251
Gontier, David 1

1 CEREMADE, University of Paris-Dauphine, PSL University, 75016 Paris, France
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Gontier, David. 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. http://www.numdam.org/articles/10.5802/crmath.251/

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

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

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

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

[5] Davies, Edward B.; Simon, Barry 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] Gontier, David Edge states for second order elliptic operators (2021) | arXiv

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

[8] 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 | DOI | Zbl

[9] 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, Springer, 2012, pp. 421-432 | DOI | Zbl

[10] 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 | DOI | MR | Zbl

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

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

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

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

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

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

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