Hölder Continuity of the Spectra for Aperiodic Hamiltonians


We study the spectral location of a strongly pattern equivariant Hamiltonians arising through configurations on a colored lattice. Roughly speaking, two configurations are “close to each other” if, up to a translation, they “almost coincide” on a large fixed ball. The larger this ball, the more similar they are, and this induces a metric on the space of the corresponding dynamical systems. Our main result states that the map which sends a given configuration into the spectrum of its associated Hamiltonian, is Hölder (even Lipschitz) continuous in the usual Hausdorff metric. Specifically, the spectral distance of two Hamiltonians is estimated by the distance of the corresponding dynamical systems.

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  1. 1.

    Beckus, S., Bellissard, J.: Continuity of the spectrum of a field of self-adjoint operators. Ann. Henri Poincaré 17, 3425–3442 (2016)

    ADS  MathSciNet  Article  Google Scholar 

  2. 2.

    Beckus, S.: Spectral approximation of aperiodic Schrödinger operators. PhD Thesis, Friedrich-Schiller-Universität Jena (2016)

  3. 3.

    Beckus, S., Bellissard, J., de Nittis, G.: Spectral continuity for aperiodic quantum systems I. General theory. J. Funct. Anal. 275(11), 2917–2977 (2018)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Beckus, S., Bellissard, J., de Nittis, G.: Spectral continuity for aperiodic quantum systems II. Periodic approximations in 1D. arXiv:1803.03099 (2018)

  5. 5.

    Beckus, S., Pogorzelski, F.: Delone dynamical systems and spectral convergence. Ergod. Theory Dyn. Syst. (2018). https://doi.org/10.1017/etds.2018.116

    Article  Google Scholar 

  6. 6.

    Beckus, S., Bellissard, J.: Lipschitz continuity for the Kohmoto model (in preparation)

  7. 7.

    Bellissard, J.: \(K\)-theory of \(C^{\ast }\)-algebras in solid state physics. In: Dorlas, T.C., Hugenholtz, M.N., Winnink, M. (eds.) Statistical Mechanics and Field Theory, Mathematical Aspects, Lecture Notes in Physics, vol. 257, pp. 99–156. Springer Verlag, Berlin, Heidelberg, New York (1986)

  8. 8.

    Bellissard, J., Iochum, B., Testard, D.: Continuity properties of the electronic spectrum of 1D quasicrystals. Commun. Math. Phys. 141, 353–380 (1991)

    ADS  Article  Google Scholar 

  9. 9.

    Bellissard, J.: Lipshitz continuity of gap boundaries for Hofstadter-like spectra. Commun. Math. Phys. 160, 599–613 (1994)

    ADS  MathSciNet  Article  Google Scholar 

  10. 10.

    Benza, V.G., Sire, C.: Electronic spectrum of the octagonal quasicrystal: chaos, gaps and level clustering. Phys. Rev. B 44, 10343–10345 (1991)

    ADS  Article  Google Scholar 

  11. 11.

    Cornean, H.D., Purice, R.: On the regularity of the Hausdorff distance between spectra of perturbed magnetic Hamiltonians. Oper. Theory Adv. Appl. 224, 55–66 (2012). Birkhäuser/Springer Basel AG, Basel

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Cornean, H.D., Purice, R.: Spectral edge regularity of magnetic Hamiltonians. J. Lond. Math. Soc. (2) 92(1), 89–104 (2015)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Castaing, C., Valadier, M.: Convex Analysis and Measurable Multifunctions. Lecture Notes in Mathematics, vol. 580. Springer, Berlin (1977)

    Google Scholar 

  14. 14.

    Chabauty, C.: Limite d’ensembles et géométrie des nombres. Bull. Soc. Math. Fr. 78, 143–151 (1950)

    Article  Google Scholar 

  15. 15.

    Dixmier, J., Douady, A.: Champs continus d’espaces hilbertiens et de \({C}^{\ast }\)-algèbres. Bull. Soc. Math. Fr. 91, 227–284 (1963)

    Article  Google Scholar 

  16. 16.

    Dixmier, J.: Les \(C^{\ast }\)-algèbres et leurs représentations. (French) Deuxième édition. Cahiers Scientifiques, Fasc. XXIX. Gauthier-Villars Éditeur, Paris, (1969); (ii) J. Dixmier, C\(^\ast \)-algebras. North-Holland, Amsterdam (1977)

  17. 17.

    Elliott, G.A.: Gaps in the spectrum of an almost periodic Schrödinger operator. C. R. Math. Rep. Acad. Sci. Can. 4, 255–259 (1982)

    MATH  Google Scholar 

  18. 18.

    Exel, R.: Invertibility in groupoid \(\cal{C}^{\ast }\)-algebras. In: Bastos, M.A., Lebre, A., Samko, S. (eds.) Operator Theory: Advances and Applications, vol. 242, pp. 173–183. Birkhäuser/Springer, Basel (2014)

  19. 19.

    Fell, J.M.G.: A Hausdorff topology for the closed subsets of a locally compact non-Hausdorff space. Proc. Amer. Math. Soc. 13, 472–476 (1962)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Hausdorff, F.: Set Theory, 2nd edn. Chelsea Publishing Co., New York (1957/1962). Republished by AMS-Chelsea (2005)

  21. 21.

    Hofstadter, D.R.: Energy levels and wave functions of Bloch electrons in a rational or irrational magnetic field. Phys. Rev. B 14, 2239–2249 (1976)

    ADS  Article  Google Scholar 

  22. 22.

    Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1995)

    Google Scholar 

  23. 23.

    Kaplansky, I.: The structure of certain algebras of operators. Trans. Am. Math. Soc. 70, 219–255 (1951)

    Article  Google Scholar 

  24. 24.

    Kellendonk, J., Prodan, E.: Bulk-boundary correspondence for Sturmian Kohmoto like models. Ann. Henri Poincaré, https://doi.org/10.1007/s00023-019-00792-5 (2019)

    ADS  MathSciNet  Article  Google Scholar 

  25. 25.

    Kellendonk, J.: Pattern-equivariant functions and cohomology. J. Phys. A 36, 5765–5772 (2003)

    ADS  MathSciNet  Article  Google Scholar 

  26. 26.

    Lenz, D., Stollman, P.: Algebras of random operators associated to Delone dynamical systems. Math. Phys. Anal. Geom. 6(3), 269–290 (2003)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Măntoiu, M., Purice, R.: The algebra of observables in a magnetic field. In: Mathematical Results in Quantum Mechanics (Taxco, 2001), Contemp. Math., vol. 307, pp. 239–245. Amer. Math. Soc., Providence (2002)

  28. 28.

    Măntoiu, M., Purice, R.: Strict deformation quantization for a particle in a magnetic field. J. Math. Phys. 46, 052105-1–052105-15 (2005)

    ADS  MathSciNet  Article  Google Scholar 

  29. 29.

    Măntoiu, M., Purice, R., Richard, S.: Twisted crossed products and magnetic pseudodifferential operators. In: Advances in Operator Algebras and Mathematical Physics, Theta Ser. Adv. Math., vol. 5, pp. 137–172. Theta, Bucharest (2005)

  30. 30.

    Nistor, V., Prudhon, N.: Exhaustive families of representations and spectra of pseudodifferential operators. J. Oper. Theory 78(2), 247–279 (2017)

    MathSciNet  MATH  Google Scholar 

  31. 31.

    Ostlund, S., Kim, S.: Renormalization of quasi periodic mappings. Phys. Scr. 9, 193–198 (1985)

    Article  Google Scholar 

  32. 32.

    Prodan, E.: Quantum transport in disordered systems under magnetic fields: a study based on operator algebras. Appl. Math. Res. Express. AMRX 2, 176–255 (2013)

    MathSciNet  MATH  Google Scholar 

  33. 33.

    Prodan, E.: A Computational Non-commutative Geometry Program for Disordered Topological Insulators. SpringerBriefs in Mathematical Physics, vol. 23. Springer, Cham (2017)

    Google Scholar 

  34. 34.

    Simon, B.: Continuity of the density of states in magnetic field. J. Phys. A 15, 2981–2983 (1982)

    ADS  MathSciNet  Article  Google Scholar 

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This research was supported through the program “Research in Pairs” by the Mathematisches Forschungsinstitut Oberwolfach in 2018. This research has been supported by Grant 8021–00084B Mathematical Analysis of Effective Models and Critical Phenomena in Quantum Transport from The Danish Council for Independent Research Natural Sciences. J.B. thanks the School of Mathematics at the Georgia Institute of Technology and the Fachbereich Mathematik at the Westfälische Wilhelms-Universität Münster.

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Correspondence to Siegfried Beckus.

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Beckus, S., Bellissard, J. & Cornean, H. Hölder Continuity of the Spectra for Aperiodic Hamiltonians. Ann. Henri Poincaré 20, 3603–3631 (2019). https://doi.org/10.1007/s00023-019-00848-6

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