Hostname: page-component-76fb5796d-r6qrq Total loading time: 0 Render date: 2024-04-25T13:51:54.367Z Has data issue: false hasContentIssue false
  • English
  • Français

PACKING SUBORDINACY WITH APPLICATION TO SPECTRAL CONTINUITY

Part of: Difference and functional equations General theory of linear operators Classical measure theory Difference equations

Published online by Cambridge University Press:  13 June 2019

V. R. BAZAO ,
S. L. CARVALHO  and
C. R. DE OLIVEIRA
V. R. BAZAO
Affiliation:
Faculdade de Ciências Exatas e Tecnologias, UFGD Dourados, MS, 79804-970, Brazil
S. L. CARVALHO
Affiliation:
Departamento de Matemática, UFMG, Belo Horizonte, MG, 30161-970, Brazil
C. R. DE OLIVEIRA*
Affiliation:
Departamento de Matemática, UFSCar, São Carlos, SP, 13560-970, Brazil email oliveira@dm.ufscar.br
*
✉oliveira@dm.ufscar.br
Get access Rights & Permissions [Opens in a new window]

Abstract

By using methods of subordinacy theory, we study packing continuity properties of spectral measures of discrete one-dimensional Schrödinger operators acting on the whole line. Then we apply these methods to Sturmian operators with rotation numbers of quasibounded density to show that they have purely $\unicode[STIX]{x1D6FC}$-packing continuous spectrum. A dimensional stability result is also mentioned.

Keywords

subordinacy theorypacking measuresSturmian potentialsspectral theory

MSC classification

Primary: 47A10: Spectrum, resolvent
Secondary: 39A70: Difference operators 28A78: Hausdorff and packing measures
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 108 , Issue 2 , April 2020 , pp. 226 - 244
Copyright
© 2019 Australian Mathematical Publishing Association Inc.

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.)

Footnotes

V.R.B. thanks CAPES for financial support. S.L.C. thanks the partial support by FAPEMIG (Universal Project CEX-APQ-00554-13). CRdO thanks the partial support by CNPq (Universal Project 41004/2014-8).

References

Bazao, V. R., Carvalho, S. L. and de Oliveira, C. R., ‘On the spectral Hausdorff dimension of 1D discrete Schrödinger operators under power decaying perturbations’, Osaka J. Math. 54 (2017), 273–285.Google Scholar
Bellissard, J., Iochum, B., Scoppola, E. and Testard, D., ‘Spectral properties of one-dimensional quasicrystals’, Commun. Math. Phys. 125 (1989), 527–543.Google Scholar
Carvalho, S. L. and de Oliveira, C. R., ‘Spectral packing dimensions through power-law subordinacy’, Ann. Henri Poincaré 14 (2012), 775–792.Google Scholar
Combes, J.-M. and Mantica, G., ‘Fractal dimensions and quantum evolution associated with sparse potential Jacobi matrices’, in: Long Time Behavior of Classical and Quantum Systems, Series on Concrete and Applicable Mathematics, 1 (eds. Graffi, S. and Martinez, A.) (World Scientific, River Edge, NJ, 2001), 107–123.Google Scholar
Damanik, D., ‘𝛼-continuity properties of one-dimensional quasicrystals’, Commun. Math. Phys. 192 (1998), 169–182.Google Scholar
Damanik, D., Killip, R. and Lenz, D., ‘Uniform spectral properties of one-dimensional quasicrystals, III. 𝛼-continuity’, Commun. Math. Phys. 212 (2000), 191–204.Google Scholar
Damanik, D. and Lenz, D., ‘Uniform spectral properties of one-dimensional quasicrystals, I. Absence of eigenvalues’, Commun. Math. Phys. 207 (1999), 687–696.Google Scholar
Damanik, D. and Lenz, D., ‘Uniform spectral properties of one-dimensional quasicrystals, II. The Lyapunov exponent’, Lett. Math. Phys. 50 (1999), 245–257.Google Scholar
Falconer, K., The Geometry of Fractal Sets (Cambridge University Press, Cambridge, 1985).Google Scholar
Gilbert, D. J., ‘On subordinacy and analysis of the spectrum of Schrödinger operators with two singular endpoints’, Proc. Roy. Soc. Edinburgh, Sect. A 112 (1989), 213–229.Google Scholar
Gilbert, D. J. and Pearson, D. B., ‘On subordinacy and analysis of the spectrum of one-dimensional Schrödinger operators’, J. Math. Anal. Appl. 128 (1987), 30–56.Google Scholar
Iochum, B., Raymond, L. and Testard, D., ‘Resistance of one-dimensional quasicrystals’, Phys. A 187 (1992), 353–368.Google Scholar
Iochum, B. and Testard, D., ‘Power law growth for the resistance in the Fibonacci model’, J. Stat. Phys. 65 (1991), 715–723.Google Scholar
Jitomirskaya, S. and Last, Y., ‘Power-law subordinacy and singular spectra, I. Half line operators’, Acta Math. 183 (1999), 171–189.Google Scholar
Jitomirskaya, S. and Last, Y., ‘Power-law subordinacy and singular spectra, II. Line operators’, Commun. Math. Phys. 211 (2000), 643–658.Google Scholar
Jitomirskaya, S. and Zhang, S., Quantitative continuity of singular continuous spectral measures and arithmetic criteria for quasiperiodic Schrödinger operators, 2015, arXiv:1510.07086.Google Scholar
Khan, S. and Pearson, D. B., ‘Subordinacy and spectral theory for infinite matrices’, Helv. Phys. Acta 65 (1992), 505–527.Google Scholar
Khintchin, A.Ya., Continued Fractions (Noordhoff, Groningen, 1963).Google Scholar
Kiselev, A., Last, Y. and Simon, B., ‘Stability of singular spectral types under decaying perturbations’, J. Funct. Anal. 198 (2002), 1–27.Google Scholar
Last, Y., ‘Quantum dynamics and decompositions of singular continuous spectra’, J. Funct. Anal. 142 (1996), 406–445.Google Scholar
Liu, Q., Qu, Y. and Wen, Z., ‘The fractal dimensions of the spectrum of Sturm Hamiltonian’, Adv. Math. 257 (2014), 285–336.Google Scholar
Liu, Q. and Wen, Z., ‘Hausdorff dimension of spectrum of one-dimensional Schrödinger operator with Sturmian potentials’, Potential Anal. 20 (2004), 33–59.Google Scholar
Matilla, P., Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability (Cambridge University Press, Cambridge, 1999).Google Scholar
Raymond, L., A constructive gap labelling for the discrete Schrödinger operator on a quasiperiodic chain, Preprint, 1997.Google Scholar
Rogers, C. A., Hausdorff Measures, 2nd edn. (Cambridge University Press, Cambridge, 1998).Google Scholar
Simon, B. and Wolff, T., ‘Singular continuous spectrum under rank one perturbations and localization for random Hamiltonians’, Commun. Pure Appl. Math. 39 (1986), 75–90.Google Scholar
Tcheremchantsev, S., ‘Dynamical analysis of Schrödinger operators with growing sparse potentials’, Commun. Math. Phys. 253 (2005), 221–252.Google Scholar