155 research outputs found
Spectra of Discrete Schr\"odinger Operators with Primitive Invertible Substitution Potentials
We study the spectral properties of discrete Schr\"odinger operators with
potentials given by primitive invertible substitution sequences (or by Sturmian
sequences whose rotation angle has an eventually periodic continued fraction
expansion, a strictly larger class than primitive invertible substitution
sequences). It is known that operators from this family have spectra which are
Cantor sets of zero Lebesgue measure. We show that the Hausdorff dimension of
this set tends to as coupling constant tends to . Moreover, we
also show that at small coupling constant, all gaps allowed by the gap labeling
theorem are open and furthermore open linearly with respect to .
Additionally, we show that, in the small coupling regime, the density of states
measure for an operator in this family is exact dimensional. The dimension of
the density of states measure is strictly smaller than the Hausdorff dimension
of the spectrum and tends to as tends to
Tridiagonal substitution Hamiltonians
We consider a family of discrete Jacobi operators on the one-dimensional
integer lattice with Laplacian and potential terms modulated by a primitive
invertible two-letter substitution. We investigate the spectrum and the
spectral type, the fractal structure and fractal dimensions of the spectrum,
exact dimensionality of the integrated density of states, and the gap
structure. We present a review of previous results, some applications, and open
problems. Our investigation is based largely on the dynamics of trace maps.
This work is an extension of similar results on Schroedinger operators,
although some of the results that we obtain differ qualitatively and
quantitatively from those for the Schoedinger operators. The nontrivialities of
this extension lie in the dynamics of the associated trace map as one attempts
to extend the trace map formalism from the Schroedinger cocycle to the Jacobi
one. In fact, the Jacobi operators considered here are, in a sense, a test
item, as many other models can be attacked via the same techniques, and we
present an extensive discussion on this.Comment: 41 pages, 5 figures, 81 reference
Selfdual Substitutions in Dimension One
There are several notions of the 'dual' of a word/tile substitution. We show
that the most common ones are equivalent for substitutions in dimension one,
where we restrict ourselves to the case of two letters/tiles. Furthermore, we
obtain necessary and sufficient arithmetic conditions for substitutions being
selfdual in this case. Since many connections between the different notions of
word/tile substitution are discussed, this paper may also serve as a survey
paper on this topic.Comment: 28 pages, 5 figures. Several typos removed, some proofs shortened,
thanks to the referees. The accepted version of this paper is shorter (22
pages, 4 figures), this arxiv version includes more examples, two appendices,
plus a self-contained proof of Theorem 2.
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