Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-10T20:51:17.578Z Has data issue: false hasContentIssue false
  • English
  • Français

Note on Spectra of Non-Selfadjoint Operators Over Dynamical Systems

Part of: General theory of linear operators Special classes of linear operators

Published online by Cambridge University Press:  15 February 2018

Siegfried Beckus ,
Daniel Lenz ,
Marko Lindner  and
Christian Seifert
Siegfried Beckus
Affiliation:
Mathematisches Institut, Friedrich-Schiller-Universität, Jena, 07743, Jena, Germany (siegfried.beckus@uni-jena.de; daniel.lenz@uni-jena.de)
Daniel Lenz
Affiliation:
Mathematisches Institut, Friedrich-Schiller-Universität, Jena, 07743, Jena, Germany (siegfried.beckus@uni-jena.de; daniel.lenz@uni-jena.de)
Marko Lindner
Affiliation:
Technische Universität Hamburg-Harburg, Institut für Mathematik, 21073 Hamburg, Germany (marko.lindner@tuhh.de; christian.seifert@tuhh.de)
Christian Seifert*
Affiliation:
Technische Universität Hamburg-Harburg, Institut für Mathematik, 21073 Hamburg, Germany (marko.lindner@tuhh.de; christian.seifert@tuhh.de)
*
*Corresponding author.
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider equivariant continuous families of discrete one-dimensional operators over arbitrary dynamical systems. We introduce the concept of a pseudo-ergodic element of a dynamical system. We then show that all operators associated to pseudo-ergodic elements have the same spectrum and that this spectrum agrees with their essential spectrum. As a consequence we obtain that the spectrum is constant and agrees with the essential spectrum for all elements in the dynamical system if minimality holds.

Keywords

minimal dynamical systempseudo-ergodicityspectrum

MSC classification

Primary: 47A10: Spectrum, resolvent 47A35: Ergodic theory 47B37: Operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 61 , Issue 2 , May 2018 , pp. 371 - 386
Copyright
Copyright © Edinburgh Mathematical Society 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Avron, J. and Simon, B., Almost periodic Schrödinger operators. I. Limit periodic potentials, Comm. Math. Phys. 82(1) (1981/82), 101–120.CrossRefGoogle Scholar
2Avron, J. and Simon, B., Almost periodic Schrödinger operators. II. The integrated density of states, Duke Math. J. 50(1) (1983), 369–391.CrossRefGoogle Scholar
3Bellissard, J., Iochum, B., Scoppola, E. and Testard, D., Spectral properties of one-dimensional quasi-crystals, Comm. Math. Phys. 125 (1989), 527–543.CrossRefGoogle Scholar
4Böttcher, A., Embree, M. and Sokolov, V. I., The spectra of large Toeplitz band matrices with a randomly perturbed entry, Math. Comp. 72 (2003), 1329–1348.CrossRefGoogle Scholar
5Böttcher, A., Grudsky, S. and Spitkovsky, I., On the essential spectrum of Toeplitz operators with semi-almost periodic symbols, In Singular integral operators, factorization and applications, Operatory Theory: Advances and Applications, Volume 142, pp. 59–77 (Birkhäuser, Basel, 2003).CrossRefGoogle Scholar
6Carmona, R. and Lacroix, J., Spectral theory of random Schrödinger operators (Birkhäuser, Boston, MA, 1990).CrossRefGoogle Scholar
7Chandler-Wilde, S. N., Chonchaiya, R. and Lindner, M., On the spectra and pseudospectra of a class of non-self-adjoint random matrices and operators, Oper. Matrices 7 (2013), 739–775.CrossRefGoogle Scholar
8Chandler-Wilde, S. N. and Lindner, M., Sufficiency of Favard's condition for a class of band-dominated operators on the axis. J. Funct. Anal. 254 (2008), 1146–1159.CrossRefGoogle Scholar
9Chandler-Wilde, S. N. and Lindner, M., Limit operators, collective compactness, and the spectral theory of infinite matrices, Memoirs of the American Mathematical Society, Volume 210 (American Mathematical Society, Providence, RI, 2011).Google Scholar
10Corduneanu, C., Almost periodic functions (Chelsea Publishers, New York, 1989).Google Scholar
11Cycon, H. L., Froese, R. G., Kirsch, W. and Simon, B., Schrödinger operators with application to quantum mechanics and global geometry, Texts and Monographs in Physics (Springer, Berlin, 1987).Google Scholar
12Dahmen, H. A., Nelson, D. R. and Shnerb, N. M., Population dynamics and non-hermitian localization, In Statistical mechanics of biocomplexity, Lecture Notes in Physics, Volume 527, pp. 124–151 (Springer, 1999).CrossRefGoogle Scholar
13Damanik, D., Gordon-type arguments in the spectral theory of one-dimensional quasicrystals, in Directions in mathematical quasicrystals, CRM Monograph Series, Volume 13, pp. 277–305 (American Mathematical Society, Providence, RI, 2000).Google Scholar
14Damanik, D., The Spectrum of the Almost Mathieu Operator, (arxiv.org/abs/0908.1093; 2009).Google Scholar
15Damanik, D., Embree, M. and Gorodetski, A., Spectral properties of Schrödinger operators arising in the study of quasicrystals’, In Mathematics of aperiodic order (ed. Kellendonk, J., Lenz, D. and Savinien, J.) Progress in Mathematics, Volume 309, pp. 307–370 (Birkhäuser, 2015).CrossRefGoogle Scholar
16Davies, E. B., Spectral theory of pseudo-ergodic operators, Commun. Math. Phys. 216 (2001), 687–704.CrossRefGoogle Scholar
17Favard, J., Sur les equations differentielles lineaires a coefficients presque-periodiques, Acta Math. 51 (1927), 31–81.CrossRefGoogle Scholar
18Feinberg, J. and Zee, A., Spectral curves of non-Hermitean Hamiltonians, Nucl. Phys. B 552 (1999), 599–623.CrossRefGoogle Scholar
19Goldsheid, I. Ya. and Khoruzhenko, B. A., Regular spacings of complex eigenvalues in the one-dimensional non-Hermitian Anderson model, Comm. Math. Phys. 238 (2003), 505–524.Google Scholar
20Hatano, N. and Nelson, D. R., Localization transitions in non-Hermitian quantum mechanics, Phys. Rev. Lett. 77 (1996), 570–573.CrossRefGoogle ScholarPubMed
21Jitomirskaya, S. Ya., Almost everything about the almost Mathieu operator. II, in XIth International Congress of Mathematical Physics (Paris, 1994), pp. 373–382 (International Press, Cambridge, MA, 1995).Google Scholar
22Johnson, R., Exponential dichotomy, rotation number, and linear differential operators with bounded coefficients, J. Differential Equations 61 (1986), 54–78.CrossRefGoogle Scholar
23Kato, T., Perturbation theory for linear operators (Springer, 1980).Google Scholar
24Kurbatov, V. G., Functional differential operators and equations (Kluwer Academic, 1999).Google Scholar
25Lange, B. V. and Rabinovich, V. S., On the Noether property of multidimensional discrete convolutions, Mat. Zametki 37(3) (1985), 407–421 (in Russian); English translation: Math. Notes 37 (1985), 228–237.Google Scholar
26Last, Y., Almost everything about the Almost Mathieu Operator. I., in XIth International Congress of Mathematical Physics (Paris, 1994), pp. 366–372 (International Press, Cambridge, MA, 1995).Google Scholar
27Last, Y. and Simon, B., The essential spectrum of Schrödinger, Jacobi and CMV operators, J. Anal. Math. 98 (2006), 183–220.CrossRefGoogle Scholar
28Lenz, D., Singular continuous spectrum for certain quasicrystal Schrödinger operators, In Complex analysis and dynamical systems, Contemporary Mathematics, Volume 364, pp. 169–180 (American Mathematical Society, Providence, RI, 2004).CrossRefGoogle Scholar
29Lenz, D., Random operators and crossed products, Math. Phys. Anal. Geom. 2 (1999), 197–220.CrossRefGoogle Scholar
30Lindner, M., Limit operators and applications on the space of essentially bounded functions. Dissertation Thesis, TU Chemnitz, 2003, http://nbn-resolving.de/urn:nbn:de:swb: ch1-200301569.Google Scholar
31Lindner, M., Infinite matrices and their finite sections: an introduction to the limit operator method (Birkhäuser, 2006).Google Scholar
32Lindner, M., Fredholmness and index of operators in the Wiener algebra are independent of the underlying space, Oper. Matrices 2 (2008), 297–306.CrossRefGoogle Scholar
33Lindner, M. and Roch, S., Finite sections of random Jacobi operators, SIAM J. Numer. Anal. 50 (2012), 287–306.CrossRefGoogle Scholar
34Lindner, M. and Seidel, M., An affirmative answer to a core issue on limit operators, J. Funct. Anal. 267 (2014), 901–917.CrossRefGoogle Scholar
35Martinez, C., Spectral estimates for the one-dimensional non-self-adjoint Anderson model, J. Operator Theory 56 (2006), 59–88.Google Scholar
36Nelson, D. R. and Shnerb, N. M., Non-Hermitian localization and population biology, Phys. Rev. E 58, (1998), 1383–1403.CrossRefGoogle Scholar
37Pastur, L. A. and Figotin, A., Spectra of random and almost-periodic operators (Springer, Berlin, 1992).CrossRefGoogle Scholar
38Rabinovich, V. S., Roch, S. and Silbermann, B., Fredholm theory and finite section method for band-dominated operators, Integral Equations Operator Theory 30 (1998), 452–495.CrossRefGoogle Scholar
39Rabinovich, V. S., Roch, S. and Silbermann, B., Limit operators and their applications in operator theory (Birkhäuser, 2004).CrossRefGoogle Scholar
40Reed, M. and Simon, B., Methods of modern mathematical physics I: functional analysis (Academic Press, 1980).Google Scholar
41Seidel, M., Fredholm theory for band-dominated and related operators: A survey, Linear Algebra Appl. 445 (2014), 373–394.CrossRefGoogle Scholar
42Seifert, C., Constancy of spectra of equivariant (non-selfadjoint) operators over minimal dynamical systems, Proc. Appl. Math. Mech. 15 (2015), 697–698.CrossRefGoogle Scholar
43Shubin, M. A., Almost periodic functions and partial differential operators, Russian Math. Surv. 33 (1978), 1–52.CrossRefGoogle Scholar
44Stollmann, P., Caught by disorder, bound states in random media, Progress in Mathematical Physics, Volume 20 (Birkhäuser, Boston, 2001).CrossRefGoogle Scholar