Osaka Journal of Mathematics

Invariants of the trace map and uniform spectral properties for discrete Sturmian Dirac operators

Roberto A. Prado and Ruy C. Charão

Full-text: Open access

Abstract

We establish invariants for the trace map associated to a family of 1D discrete Dirac operators with Sturmian potentials. Using these invariants we prove that the operators have purely singular continuous spectrum of zero Lebesgue measure, uniformly on the mass and parameters that define the potentials. For rotation numbers of bounded density we prove that these Dirac operators have purely $\alpha$-continuous spectrum, as to the Schrödinger case, for some $\alpha \in (0,1)$. To the Sturmian Schrödinger and Dirac models we establish a comparison between invariants of the trace maps, which allows to compare the numbers $\alpha$'s and lower bounds on transport exponents.

Article information

Source
Osaka J. Math., Volume 56, Number 2 (2019), 391-416.

Dates
First available in Project Euclid: 3 April 2019

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1554278430

Mathematical Reviews number (MathSciNet)
MR3934981

Zentralblatt MATH identifier
07080090

Subjects
Primary: 81Q10: Selfadjoint operator theory in quantum theory, including spectral analysis
Secondary: 47B99: None of the above, but in this section

Citation

Prado, Roberto A.; Charão, Ruy C. Invariants of the trace map and uniform spectral properties for discrete Sturmian Dirac operators. Osaka J. Math. 56 (2019), no. 2, 391--416. https://projecteuclid.org/euclid.ojm/1554278430


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References

  • J. Bellissard, A. Bovier and J.-M. Ghez: Spectral Properties of a Tight Binding Hamiltonian with Period Doubling Potential, Comm. Math. Phys. 135 (1991), 379–399.
  • J. Bellissard, B. Iochum, E. Scoppola and D. Testard: Spectral Properties of One-Dimensional Quasi-crystals, Comm. Math. Phys. 125 (1989), 527–543.
  • P. Bougerol, J. Lacroix: Products of Random Matrices with Applications to Schrödinger Operators, Birkhäuser, Boston, 1985.
  • A. Bovier and J.-M. Ghez: Spectral Properties of One-Dimensional Schrödinger Operators with Potentials Generated by Substitutions, Comm. Math. Phys. 158 (1993), 45–66.
  • S.L. Carvalho, C.R. de Oliveira and R.A. Prado: Sparse One-dimensional Discrete Dirac Operators II: Spectral Properties, J. Math. Phys. 52 (2011), 073501, 21pp.
  • D. Damanik: $\alpha$-continuity properties of one-dimensional quasicrystals, Comm. Math. Phys. 192 (1998), 169–182.
  • D. Damanik: Uniform singular continuous spectrum for the period doubling Hamiltonian, Ann. Henri Poincaré, 2 (2001), 101–108.
  • D. Damanik: Substitution Hamiltonians with bounded trace map orbits, J. Math. Anal. Appl. 249 (2000), 393–411.
  • D. Damanik: Singular continuous spectrum for a class of substitution Hamiltonians, Lett. Math. Phys. 46 (1998), 303–311.
  • D. Damanik, R. Killip and D. Lenz: Uniform spectral properties of one-dimensional quasicrystals, III. $\alpha$-continuity, Comm. Math. Phys. 212 (2000), 191–204.
  • D. Damanik and D. Lenz: Uniform spectral properties of one-dimensional quasicrystals, I. Absence of eigenvalues, Comm. Math. Phys. 207 (1999), 687–696.
  • D. Damanik and D. Lenz: Uniform spectral properties of one-dimensional quasicrystals, II. The Lyapunov exponent, Lett. Math. Phys. 50 (1999), 245–257.
  • D. Damanik, A. Sütő and S. Tcheremchantsev: Power-law bounds on transfer matrices and quantum dynamics in one dimension. II, J. Funct. Anal. 216 (2004), 362–387.
  • F. Delyon and D. Petritis: Absence of localization in a class of Schrödinger operators with quasiperiodic potential, Comm. Math. Phys. 103 (1986), 441–444.
  • C.R. de Oliveira and R.A. Prado: Spectral and localization properties for the one-dimensional Bernoulli discrete Dirac operator, J. Math. Phys. 46 (2005), 072105, 17pp.
  • B. Iochum, L. Raymond and D. Testard: Resistance of one-dimensional quasicrystals, Phy A, 187 (1992), 353–368.
  • S. Jitomirskaya and Y. Last: Power-law subordinacy and singular spectra, II. Line operators, Comm. Math. Phys. 211 (2000), 643–658.
  • A.Ya. Khinchin: Continued Fractions, Dover Publications, Mineola, 1997.
  • S. Kotani: Jacobi matrices with random potentials taking finitely many values, Rev. Math. Phys. 1 (1989), 129–133.
  • Y. Last: Quantum dynamics and decomposition of singular continuous spectra, J. Funct. Anal. 142 (1996), 406–445.
  • V. Oseledec: A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc. 19 (1968), 197–231.
  • R.A. Prado and C.R. de Oliveira: Dynamical lower bounds for 1D Dirac operators, Math. Z. 259 (2008), 45–60.
  • R.A. Prado and C.R. de Oliveira: Sparse 1D discrete Dirac operators I: quantum transport, J. Math. Anal. Appl. 385 (2012), 947–960.
  • M. Reed and B. Simon: Methods of Modern Mathematical Physics, Vol. I, Functional Analysis, Academic Press, New York, 1972.
  • M. Reed and B. Simon: Methods of Modern Mathematical Physics, Vol. IV, Analysis of Operators, Academic Press, New York, 1977.
  • C.A. Rogers: Hausdorff Measures, Cambridge Univ. Press, London, 1970.
  • C.A. Rogers and S.J. Taylor: The analysis of additive set functions in Euclidean space, Acta Math. 101 (1959), 273–302.
  • A. Sütő : The spectrum of a quasiperiodic Schrödinger operator, Comm. Math. Phys. 111 (1987), 409–415.