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PACKING SUBORDINACY WITH APPLICATION TO SPECTRAL CONTINUITY

  • V. R. BAZAO (a1), S. L. CARVALHO (a2) and C. R. DE OLIVEIRA (a3)

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.

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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).

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[1] 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), 273285.
[2] Bellissard, J., Iochum, B., Scoppola, E. and Testard, D., ‘Spectral properties of one-dimensional quasicrystals’, Commun. Math. Phys. 125 (1989), 527543.10.1007/BF01218415
[3] Carvalho, S. L. and de Oliveira, C. R., ‘Spectral packing dimensions through power-law subordinacy’, Ann. Henri Poincaré 14 (2012), 775792.10.1007/s00023-012-0194-8
[4] 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), 107123.10.1142/9789812794598_0006
[5] Damanik, D., ‘𝛼-continuity properties of one-dimensional quasicrystals’, Commun. Math. Phys. 192 (1998), 169182.10.1007/s002200050295
[6] Damanik, D., Killip, R. and Lenz, D., ‘Uniform spectral properties of one-dimensional quasicrystals, III. 𝛼-continuity’, Commun. Math. Phys. 212 (2000), 191204.10.1007/s002200000203
[7] Damanik, D. and Lenz, D., ‘Uniform spectral properties of one-dimensional quasicrystals, I. Absence of eigenvalues’, Commun. Math. Phys. 207 (1999), 687696.10.1007/s002200050742
[8] Damanik, D. and Lenz, D., ‘Uniform spectral properties of one-dimensional quasicrystals, II. The Lyapunov exponent’, Lett. Math. Phys. 50 (1999), 245257.10.1023/A:1007614218486
[9] Falconer, K., The Geometry of Fractal Sets (Cambridge University Press, Cambridge, 1985).10.1017/CBO9780511623738
[10] 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), 213229.10.1017/S0308210500018680
[11] 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), 3056.10.1016/0022-247X(87)90212-5
[12] Iochum, B., Raymond, L. and Testard, D., ‘Resistance of one-dimensional quasicrystals’, Phys. A 187 (1992), 353368.10.1016/0378-4371(92)90426-Q
[13] Iochum, B. and Testard, D., ‘Power law growth for the resistance in the Fibonacci model’, J. Stat. Phys. 65 (1991), 715723.10.1007/BF01053750
[14] Jitomirskaya, S. and Last, Y., ‘Power-law subordinacy and singular spectra, I. Half line operators’, Acta Math. 183 (1999), 171189.10.1007/BF02392827
[15] Jitomirskaya, S. and Last, Y., ‘Power-law subordinacy and singular spectra, II. Line operators’, Commun. Math. Phys. 211 (2000), 643658.10.1007/s002200050830
[16] Jitomirskaya, S. and Zhang, S., Quantitative continuity of singular continuous spectral measures and arithmetic criteria for quasiperiodic Schrödinger operators, 2015, arXiv:1510.07086.
[17] Khan, S. and Pearson, D. B., ‘Subordinacy and spectral theory for infinite matrices’, Helv. Phys. Acta 65 (1992), 505527.
[18] Khintchin, A.Ya., Continued Fractions (Noordhoff, Groningen, 1963).
[19] Kiselev, A., Last, Y. and Simon, B., ‘Stability of singular spectral types under decaying perturbations’, J. Funct. Anal. 198 (2002), 127.10.1016/S0022-1236(02)00053-8
[20] Last, Y., ‘Quantum dynamics and decompositions of singular continuous spectra’, J. Funct. Anal. 142 (1996), 406445.10.1006/jfan.1996.0155
[21] Liu, Q., Qu, Y. and Wen, Z., ‘The fractal dimensions of the spectrum of Sturm Hamiltonian’, Adv. Math. 257 (2014), 285336.10.1016/j.aim.2014.02.019
[22] Liu, Q. and Wen, Z., ‘Hausdorff dimension of spectrum of one-dimensional Schrödinger operator with Sturmian potentials’, Potential Anal. 20 (2004), 3359.10.1023/A:1025537823884
[23] Matilla, P., Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability (Cambridge University Press, Cambridge, 1999).
[24] Raymond, L., A constructive gap labelling for the discrete Schrödinger operator on a quasiperiodic chain, Preprint, 1997.
[25] Rogers, C. A., Hausdorff Measures, 2nd edn. (Cambridge University Press, Cambridge, 1998).
[26] Simon, B. and Wolff, T., ‘Singular continuous spectrum under rank one perturbations and localization for random Hamiltonians’, Commun. Pure Appl. Math. 39 (1986), 7590.10.1002/cpa.3160390105
[27] Tcheremchantsev, S., ‘Dynamical analysis of Schrödinger operators with growing sparse potentials’, Commun. Math. Phys. 253 (2005), 221252.10.1007/s00220-004-1153-0
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