Hausdorff Dimension of Spectrum of One-Dimensional Schrödinger Operator with Sturmian Potentials

Potential Analysis

Volumn 20, Issue 1, 2004, Pages 33-59

  Liu, Qing Hui ^a   Wen, Zhi Ying ^b  

^a BEIJING INSTITUTE OF TECHNOLOGY   (China)

^b TSINGHUA UNIVERSITY   (China)

Author keywords

Hausdorff dimension; One dimensional Schr dinger operators; Sturmian sequence

Indexed keywords

EID: 0042878365     PISSN: 09262601     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1025537823884     Document Type: Article

References (16)

1. Axel, F. and Peyriere, J.: 'Spectrum and extended states in a harmonic chain with controlled disorder: effects of the Thue-Morse symmetry', J. Statist. Phys. 57(5-6) (1989), 1013-1047.
2. Bellissard, J., Iochum, B., Scoppola, E. and Testart, D.: 'Spectral properties of one dimensional quasi crystals', Comm. Math. Phys. 125 (1989), 527-543.
3. Damanik, D., Killip, R. and Lenz, D.: 'Uniform spectral properties of one-dimensional quasicrystals, III. α-continuity', Comm. Math. Phys. 212 (2000), 191-204.
4. del Rio, R., Jitomirskaya, S., Last, Y. and Simon, B.: Phys. Rev. Lett. 75 (1995), 117-119; 'Operators with singular continuous spectrum, IV. Hausdorff dimensions, rank one perturbations, and localization', J. Anal. Math. 69 (1996), 153-200.
5. del Rio, R., Jitomirskaya, S., Last, Y. and Simon, B.: Phys. Rev. Lett. 75 (1995), 117-119; 'Operators with singular continuous spectrum, IV. Hausdorff dimensions, rank one perturbations, and localization', J. Anal. Math. 69 (1996), 153-200.
6. Feng, D.-J., Wen, Z.-Y. and Wu, J.: 'Some dimensional results for homogeneous Moran sets', Science of China (Series A) 40(5) (1997), 475-482.
7. Hof, A., Knill, O. and Simon, B.: 'Singular continuous spectrum for palindromic Schrödinger operators', Comm. Math. Phys. 174 (1995), 149-159.
8. Hua, S., Rao, H., Wen, Z.-Y. and Wu, J.: 'On the structures and dimensions of Moran sets', Science of China (Series A) 43(8) (1997), 836-852.
9. Jitomirskaya, S. and Last, Y.: 'Power law subordinacy and singular spectra, I. Half-line operators', Acta Math. 183(2) (1999), 171-189.
10. Jitomirskaya, S. and Last, Y.: 'Power law subordinacy and singular spectra, II. Line operators', Comm. Math. Phys. 211(3) (2000), 643-658.
11. Khinchin, A.Ya.: Continued Fractions, Dover Publications, Mineola, 1997.
12. Qing-Hui Liu, Bo Tan, Zhi-Xiong Wen and Jun Wu, 'Measure zero spectrum of a calss of Schrödinger operator', to appear in Stat. Phys.
13. Raymond, L.: 'A constructive gap labelling for the discrete Schrödinger operator on a quasiperiodic chain', Preprint, 1997.
14. Schechtman, D., Blech, I., Gratias, D. and Cahn, J.: 'Metallic phase with long-range orientational order and no translational symmetry', Phys. Rev. Lett. 53 (1984), 1951-1954.
15. Süto, A.: 'Schrödinger difference equation with deterministic ergodic potentials', in F. Alex and D. Gratias (eds), Beyond Quasicrystals, Springer-Verlag, 1994.
16. Toda, M.: Theory of Nonlinear Lattices, 2nd enlarged edn, Solid-State Sciences 20, Springer-Verlag, 1989, Chap. 4.