Quasiregular heterostructures: An overview of the current situation
Section snippets
Quasiregular sequences and quasiregular heterostructures
Take a regular Cantor set and define it as a spectrum of points on the real axis. Then, such spectrum is (i) singular continuous, with fractal dimension and (ii) selfsimilar. A zoom magnification of any fraction of it reproduces the entire spectrum. Repeated magnifications with the same scale factor yield the same result and this holds equally for all fragments that we choose.
There are, on the other hand, abundant mathematical studies of different quasiregular sequences [1], [2], [3], [4], [5],
Further comments and open questions
A number of mathematical studies have been building up, which provide a basis for a formal analysis of the properties of quasiregular sequences. Yet, research on the physical properties of quasiregular heterostructures has been developed, more or less alongside, which appears to have largely ignored the former. Scaling theory, combined with a careful study of the topology of the spectra, provides an inroad into the morphological aspects. It is clear that the simple concept of selfsimilarity
Acknowledgements
This work has greatly benefitted from contacts and collaborations with R. Pérez-Álvarez, V.R. Velasco, H. Rodríguez-Coppola and C. Trallero-Giner. I also acknowledge gratefully the most efficient help of Pilar Jiménez in the preparation of this article.
References (43)
Cantor spectra and scaling of gap widths in deterministic aperiodic systems
Phys. Rev. B.
(1989)- et al.
Trace maps of general substitutional sequences
Phys. Rev. B.
(1990) - et al.
Spectral Properties of a tight binding Hamiltonian with Period Doubling Potential
Comm. Math. Phys.
(1991) - et al.
Gap labelling Theorems for One Dimensional Discrete Schrödinger Operators
Rev. Math. Phys.
(1992) - et al.
Spectral properties of one-dimensional discrete Schrödinger operators with potentials generated by substitutions
Commun. Math. Phys.
(1993) - et al.
Trace maps, invariants, and some of their applications
Int. J. of Mod. Phys. B.
(1993) - et al.
Discrete Schrödinger operators with potentials generated by substitutions
Some remarks on discrete aperiodic Schrödinger operators
J. Statist. Phys.
(1993)- et al.(1994)
- et al.
Remarks on the spectral properties of tight-binding and Kronig-Penney models with substitution sequences
J. Phys. A: Math. Gen.
(1995)
Singular continuous spectrum for palindromic schrödinger operators
Comm. Math. Phys.
Dimensional hausdorff properties of singular continuous spectra
Phys. Rev. Lett.
Analytical results for multifractal properties of spectra of quasiperiodic Hamiltonians near the periodic chain
J. Phys. A: Math. Gen.
Schrödinger Operators with potentials generated by primitive substitutions: An invitation
Univ. Iagel. Acta Math.
On scaling in relation to singular spectra
Comm. Math. Phys.
Singular continuous spectrum for a class of substitution Hamiltonians
Lett. Math. Phys.
On p-sparse Schrödinger operators with quasiperiodic potentials
Helv. Phys. Acta.
Singular continuous spectrum for the period doubling Hamiltonian on a set of full measure
Comm. Math. Phys.
A symmetry group of a Thue—Morse quasicrystal
J. Phys. A.
Bound on the counting function for the eigenvalues of an infinite multistratified acoustic strip
Math. Methods Appl. Sci.
Electrons, Phonons and Excitons in Low Dimensional Aperiodic Systems
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