Abstract
We study the measure of the spectrum of a class of one-dimensional discrete Schrödinger operators H v, ω with potential v(ω) generated by any primitive substitutions. It is well known that the spectrum of H v, ω is singular continuous.(1) We will give a more exact result that the spectrum of H v, ω is a set of Lebesgue measure zero, by removing two hypotheses (the semi-primitive of a certain induced substitution and the existence of square word) from a theorem due to Bovier and Ghez.(2)
Similar content being viewed by others
REFERENCES
A. Hof, O. Knill, B. Simon, Singular continuous spectrum for palindromic Schrödinger operators, Commun. Math. Phys. 174:149-159 (1995).
A. Bovier and J.-M. Ghez, Spectrum properties of one-dimensional Schrödinger operators with potentials generated by substitutions, Commun. Math. Phys. 158:45-66 (1993).
D. Schechtman, I. Blech, D. Gratias, and J. Cahn, Metallic phase with long-range orientational order and no translational symmetry, Phys. Rev. Lett. 53:1951-1954 (1984).
J. Peyrière, Z.-X. Wen, et Z.-Y. Wen, Polynomês associées aux endomorphismes de groupes libres, L'Enseig. Math. t. 39:153-175 (1993).
A. Hof, Some remarks on discrete aperiodic Schrödinger operator, J. Statist. Phys. 72:1353-1374 (1993).
M. Queffélec, Substitution Dynamical System-Spectral Analysis, Lecture Notes in Mathematics, Vol. 1294 (New York, Springer-Verlag, 1987).
S. Kotani, Jacobi matrices with random potentials taking finitely many values, Rev. Math. Phys. 1:129-133 (1990).
J. Bellissard, B. Iochum, E. Scoppola, and D. Testart, Spectral properties of one dimensional quasi-crystals, Commun. Math. Phys. 125:527-543 (1989).
A. Sütö, Schrödinger difference equation with deterministic ergodic potentials, Beyond Quasicrystals, F. Alex and D. Gratias, eds. (Springer-Verlag, 1994).
D. Damanik and D. Lenz, Uniform spectral properties of one-dimensional quasicrystals, I. Absence of eigenvalues, Commun. Math. Phys. 207:687-696 (1999).
M. Casdagli, Symbolic dynamics for the renormalization map of quasipeoriodic Schrödinger equation, Commun. Math. Phys. 107:295-318 (1986).
A. Süto, The spectrum of a quasipeoriodic Schödinger operator, Commun. Math. Phys. 111:409-415 (1987).
A. Bovier and J.-M. Ghez, Erratum Spectrum properties of one-dimensional Schrödinger operators with potentials generated by substitutions, Commun. Math. Phys. 166:431-432 (1994).
M. Kolá? and F. Nori, Trace maps of general substitutional sequences, Phys. Rev. B42:1062-1065(1990).
M. Lothaire, Combinatorics on Words, Encyclopedia of Mathematics and Its Applications, Vol. 17.
Z.-X. Wen, Relations polynomiales entre les traces de produits de matrices, Série I, C. R. Acad. Sci. Paris, t. 318:99-104 (1994).
Z.-X. Wen and Z.-Y. Wen, On the leading term and the degree of the polynomial trace mapping associated with a substitution, J. Statist. Phys. 75:627-641 (1994).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Liu, QH., Tan, B., Wen, ZX. et al. Measure Zero Spectrum of a Class of Schrödinger Operators. Journal of Statistical Physics 106, 681–691 (2002). https://doi.org/10.1023/A:1013718624572
Issue Date:
DOI: https://doi.org/10.1023/A:1013718624572