Abstract
Using the trace mapping and the reduced trace mapping associated with a substitution, one obtaines the spectral properties of one-dimensional Schrödinger operators of the formH=−Δ+V onl 2ℤ, where Δ is the discrete Laplacian andV is a diagonal operator with elements derived from a substitution rule. In particular, the reduced trace mapping is closely related to the leading term of the original trace mapping. In this paper, the explicit expression of the leading term is given and its properties are discussed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. P. Allouche and J. Peyrière, Sur une formule de récurrence sur les traces de produits de matrices associés à certaines substitutions,C. R. Acad. Sci. Paris 302(II):1135–1136 (1986).
F. Axel and J. Peyrière, Spectrum and extended states in a harmonic chain with controlled disorder: Effects of the Thue-Morse symmetry,J. Stat. Phys. 57(5/6) (1989).
F. Axel, J. P. Allouche, M. Kléman, M. Mendes France, and J. Peyrière, Vibrational modes in a one-dimensional “quasi-alloy” the Morse case,J. Phys. (Paris)C3:181–187 (1986).
J. Bellissard, A. Bovier, and J. M. Chez, Spectral properties of a tight binding Hamiltonian with periodic doubling potential,Commun. Math. Phys. 135:379–399 (1991).
A. Bovier and J. M. Ghez, Spectral properties of one-dimensional Schrödinger operators with potentials generated by substitutions, Centre de Physique Théorique, Marseille (1993).
E. Bombieri and J. E. Taylor, Which distributions of matter diffract? An initial investigation,J. Phys. (Paris)C3:19–28 (1986).
D. Gratias and L. Michel, eds., International Workshop on Aperiodic Crystals,J. Phys. (Paris)C 3(7) (1986).
B. Iochum and D. Testard, Power law growth for the resistance in the Fibonacci model,J. Stat. Phys. 65:715–723 (1991).
M. Kohmoto and J. R. Banavar, Quasiperiodic lattice: Electronic properties, phonon properties and diffusion,Phys. Rev. B 34:563–566 (1986).
M. Kohmoto, L. P. Kadanoff, and C. Tang, Localization problem in one dimension: Mapping and escape,Phys. Rev. Lett. 50:1870–1872 (1983).
M. Kohmoto and Y. Oono, Center spectrum for an almost periodic Schrödinger equation and a dynamical map,Phys. Lett. 102A:145–148 (1984).
M. Kolar, B. Iochum, and L. Raymond, Structure factor for 1D systems (superlattices) based on two-letter substitution rule I, preprint, Centre de Physique Théorique, Marseille (1993).
S. Ostlun, R. Paudit, D. Rand, H. J. Schnellnhuber, and E. D. Siggia, Schrödinger equation with an almost periodic potential,Phys. Rev. Lett. 50:1873–1876 (1983).
J. Peyrière, On the trace map for products associated with substitutive sequences,J. Stat. Phys. 62:411–414 (1991).
J. Peyrière, Z. X. Wen, and Z. Y. Wen, Polynômes associés aux endomorphismes de groupes libres,Enseignement Math. 39:153–175 (1993).
D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, Metallic phase with long-range orientational order and no translation symmetry,Phys. Rev. Lett. 53:1951–1953 (1984).
P. J. Steinhardt and S. Ostlund,The Physics of Quasicrystals (World Scientific, Singapore, 1987).
M. J. Yacaman, D. Rameu, V. Castano and A. Gomes, eds.,Quasicrystals and Incommensurate Structures in Condensed Matter (World Scientific, Singapore, 1990).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Zhi-Xiong, W., Zhi-Ying, W. On the leading term and the degree of the polynomial trace mapping associated with a substitution. J Stat Phys 75, 627–641 (1994). https://doi.org/10.1007/BF02186874
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02186874