Spectral Properties of Schrödinger Operators Associated with Almost Minimal Substitution Systems

@article{Eichinger2021SpectralPO,
  title={Spectral Properties of Schr{\"o}dinger Operators Associated with Almost Minimal Substitution Systems},
  author={Benjamin Eichinger and Philipp Gohlke},
  journal={Annales Henri Poincare},
  year={2021},
  volume={22},
  pages={1377 - 1427}
}
We study the spectral properties of ergodic Schrödinger operators that are associated with a certain family of non-primitive substitutions on a binary alphabet. The corresponding subshifts provide examples of dynamical systems that go beyond minimality, unique ergodicity and linear complexity. In some parameter region, we are naturally in the setting of an infinite ergodic measure. The almost sure spectrum is singular and contains an interval. We show that under certain conditions, eigenvalues… 
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References

SHOWING 1-10 OF 53 REFERENCES
Singular continuous spectrum for palindromic Schrödinger operators
We give new examples of discrete Schrödinger operators with potentials taking finitely many values that have purely singular continuous spectrum. If the hullX of the potential is strictly ergodic,
Spectral Properties of Limit-Periodic Operators
We survey results concerning the spectral properties of limit-periodic operators. The main focus is on discrete one-dimensional Schr\"odinger operators, but other classes of operators, such as Jacobi
Singular Spectrum of Lebesgue Measure Zero¶for One-Dimensional Quasicrystals
Abstract: The spectrum of one-dimensional discrete Schrödinger operators associated to strictly ergodic dynamical systems is shown to coincide with the set of zeros of the Lyapunov exponent if and
The Fibonacci Hamiltonian
We consider the Fibonacci Hamiltonian, the central model in the study of electronic properties of one-dimensional quasicrystals, and establish relations between its spectrum and spectral
A condition of Boshernitzan and uniform convergence in the multiplicative ergodic theorem
This paper is concerned with uniform convergence in the multiplicative ergodic theorem on aperiodic subshifts. If such a subshift satisfies a certain condition, originally introduced by Boshernitzan,
Uniform Convergence of Schrödinger Cocycles over Simple Toeplitz Subshift
For locally constant cocycles defined on an aperiodic subshift, Damanik and Lenz (Duke Math J 133(1): 95–123, 2006) proved that if the subshift satisfies a certain condition (B), then the cocycle is
Sur le spectre des opérateurs aux différences finies aléatoires
We study a class of random finite difference operators, a typical example of which is the finite difference Schrödinger operator with a random potential which arises in solid state physics in the
Uniform Convergence of Schrödinger Cocycles over Bounded Toeplitz Subshift
For locally constant cocycle defined on an aperiodic subshift, Damanik and Lenz proved that if the subshift satisfies a certain condition (B), then the cocycle is uniform. For any simple Toeplitz
The absolutely continuous spectrum of Jacobi matrices
I explore some consequences of a groundbreaking result of Breimesser and Pearson on the absolutely continuous spectrum of one-dimensional Schr"odinger operators. These include an Oracle Theorem that
Singular continuous spectrum for a class of nonprimitive substitution Schrodinger operators
We present a class of discrete Schrodinger operators, with potentials derived from nonprimitive substitutions, that has purely singular continuous spectrum. We give sufficient conditions on the
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