## Osaka Journal of Mathematics

### On the spectral Hausdorff dimension of 1D discrete Schrödinger operators under power decaying perturbations

#### Abstract

We show that spectral Hausdorff dimensional properties of discrete Schrödinger operators with (1) Sturmian potentials of bounded density and (2) a class of sparse potentials are preserved under suitable polynomial decaying perturbations, when the spectrum of these perturbed operators have some singular continuous component.

#### Article information

Source
Osaka J. Math., Volume 54, Number 2 (2017), 273-285.

Dates
First available in Project Euclid: 1 June 2017

https://projecteuclid.org/euclid.ojm/1496282424

Mathematical Reviews number (MathSciNet)
MR3657230

Zentralblatt MATH identifier
1375.39038

#### Citation

Bazao, V.R.; Carvalho, S.L.; de Oliveira, C.R. On the spectral Hausdorff dimension of 1D discrete Schrödinger operators under power decaying perturbations. Osaka J. Math. 54 (2017), no. 2, 273--285. https://projecteuclid.org/euclid.ojm/1496282424

#### References

• D. Damanik: $\alpha$-Continuity Properties of one-dimensional quasicrystals, Commun. Math. Phys. 192 (1998), 169–182.
• D. Damanik and A. Gorodetski: Spectral and quantum dynamical properties of the weakly Fibonacci Hamiltonian, Commun. Math. Phys. 305 (2011), 221–277.
• D. Damanik, R. Killip and D. Lenz: Uniform spectral properties of one-dimensional quasicrystals, III. $\alpha$-Continuity, Commun. Math. Phys. 212 (2000), 191–204.
• D. Damanik and D. Lenz: Half-line eigenfunction estimates and purely singular continuous spectrum of zero Lebesgue measure, Forum Math. 16 (2004), 109–128.
• R. del Rio, S. Jitomirskaya, Y. Last and S. Simon: Operators with singular continuous spectrum. IV. Hausdorff dimensions, rank one perturbations, and localization, J. Anal. Math. 69 (1996), 153–200.
• R. del Rio, M. Makarov and B. Simon: Operators with singular continuous spectrum II. Rank one operators, Commun. Math. Phys. 165 (1994), 59–67.
• K. Falconer: The Geometry of Fractal Sets, Cambridge U. Press, Cambridge,1985.
• D.J. Gilbert: On subordinacy and analysis of the spectrum of Schrödinger operators with two singular endpoints, Proc. Roy. Soc. Edinburgh 112 A (1989), 213–229.
• D.J. Gilbert: Asymptotic methods in the spectral analysis of Sturm-Liouville operators, in W.O. Amrein, A.M. Hinz and D.B. Pearson, Sturm-Liouville: Past and Present, Birkhäuser, Basel, 121–136, 2005.
• D.J. Gilbert and D.B. Pearson: On Subordinacy and analysis of the spectrum of one-dimensional Schrödinger operators, J. Math. Anal. Appl. 128 (1987), 30–56.
• A.Ya. Gordon: Pure point spectrum under 1-parameter perturbations and instability of Anderson localization, Commun. Math. Phys. 164 (1994), 489–505.
• B. Iochum, L. Raymond and D. Testard: Resistance of One-Dimensional Quasicrystals, Physica A 187 (1992), 353–368.
• S. Jitomirskaya and Y. Last: Power-law subordinacy and singular spectra, I. Half line operators, Acta Math. 183 (1999), 171–189.
• S. Jitomirskaya and Y. Last: Power-law subordinacy and singular spectra, II. Line operators, Commun. Math. Phys. 211 (2000), 643–658.
• S. Khan and D.B. Pearson: Subordinacy and spectral theory for infinite matrices, Helv. Phys. Acta 65 (1992), 505–527.
• A.Ya. Khinchin: Continued Fractions, Noordhoff, Groningen, 1963.
• A. Kiselev, Y. Last and B. Simon: Stability of singular spectral types under decaying perturbations, J. Funct. Anal. 198 (2002), 1–27.
• Y. Last: Quantum dynamics and decompositions of singular continuous spectra, J. Funct. Anal. 142 (1996), 406–445.
• C.A. Rogers: Hausdorff Measures, 2nd ed., Cambridge Univ. Press, Cambridge, 1998.
• B. Simon and T. Wolff: Singular continuous spectrum under rank one perturbations and localization for random Hamiltonians, Commun. Pure Appl. Math. 39 (1986), 75–90.
• S. Tcheremchantsev: Dynamical analysis of Schrödinger operators with growing sparse potentials, Commun. Math. Phys. 253 (2005), 221–252.