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6
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84926531177
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Besides its relevance for the study of the Fibonacci superlattice, an apparent major limitation of this study is the restriction to 1D, but 2D and 3D QC are defined in terms of 1D QP sequences (Ref. 2). Therefore, some of the qualitative pictures of the solution of the 1D model may be carried over to higher-dimensional cases.
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9
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84926595140
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D. Levine (private communication), B. Grunbaum and G. C. Shepard, Tilings and Patterns (Freeman, San Francisco, in press).
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10
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0000491606
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The eigenvalues and eigenstates were obtained using the QL algorithm
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(1968)
Numer. Math.
, vol.11
, pp. 293
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Bowdler, H.1
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13
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84926576907
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and unpublished.
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14
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0000704572
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Note that Harper's equation [see, for instance, and references therein] models a (incommensurate) system with (a) two superimposed competing periodicities, and (b) a sine-modulated potential. The systems studied here have neither two periodic structures superimposed (as for incommensurate systems) nor a sine-modulated potential. Furthermore, crucial properties (metal-insulator transition, nature of the electronic wave functions, etc.) in Harper's equation are determined by the strength of the potential. This is not the case in the present study based on the Schrödinger equation (2). Moreover, the self-similarity present in the Fibonacci lattice studied here is completely nonexistent in the lattices used in the studies based on Harper's equation. For a review, see J. Sokoloff, Phys. Rep. 126, 4, 189 (1985).
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(1984)
Phys. Rev. B
, vol.29
, pp. 1394
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Ostlund and R. Pandit, S.1
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20
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84926573373
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Gordon Baym (private communication).
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21
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0001723647
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Several conditions must be satisfied in order to actually prove convergence in perturbation theory (e.g., each term in the expansion must be bounded, the magnitude of each term must grow at a geometric rate or more slowly). The Kolmogorov-Arnol'd-Moser procedure shows how to avoid the divergencies arising from the small divisors only for a few particular cases [see, for intance, ], since the general problem is still unsolved. Agreement between our perturbation approach and the numerically exact diagonalization of the phonon and electron problems is encouraging and suggests that the theory converges in the present situation.
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(1983)
Commun. Math. Phys.
, vol.88
, pp. 207
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Bellisard, J.1
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