TY  - JOUR
A1  - Beckus, Siegfried
A1  - Bellissard, Jean
A1  - De Nittis, Giuseppe
T1  - Spectral continuity for aperiodic quantum systems
BT  - applications of a folklore theorem
JF  - Journal of mathematical physics
N2  - This work provides a necessary and sufficient condition for a symbolic dynamical system to admit a sequence of periodic approximations in the Hausdorff topology. The key result proved and applied here uses graphs that are called De Bruijn graphs, Rauzy graphs, or Anderson-Putnam complex, depending on the community. Combining this with a previous result, the present work justifies rigorously the accuracy and reliability of algorithmic methods used to compute numerically the spectra of a large class of self-adjoint operators. The so-called Hamiltonians describe the effective dynamic of a quantum particle in aperiodic media. No restrictions on the structure of these operators other than general regularity assumptions are imposed. In particular, nearest-neighbor correlation is not necessary. Examples for the Fibonacci and the Golay-Rudin-Shapiro sequences are explicitly provided illustrating this discussion. While the first sequence has been thoroughly studied by physicists and mathematicians alike, a shroud of mystery still surrounds the latter when it comes to spectral properties. In light of this, the present paper gives a new result here that might help uncovering a solution.
Y1  - 2020
U6  - https://doi.org/10.1063/5.0011488
SN  - 0022-2488
SN  - 1089-7658
VL  - 61
IS  - 12
PB  - American Institute of Physics
CY  - Melville, NY
ER  - 
TY  - GEN
A1  - Beckus, Siegfried
A1  - Bellissard, Jean
A1  - De Nittis, Giuseppe
T1  - Corrigendum to:  Spectral continuity for aperiodic quantum systems I. General theory. - [Journal of functional analysis. - 275 (2018), 11, S. 2917 – 2977]
T2  - Journal of functional analysis
N2  - A correct statement of Theorem 4 in [1] is provided. The change does not affect the main results.
KW  - Haar system
Y1  - 2019
U6  - https://doi.org/10.1016/j.jfa.2019.06.001
SN  - 0022-1236
SN  - 1096-0783
VL  - 277
IS  - 9
SP  - 3351
EP  - 3353
PB  - Elsevier
CY  - San Diego
ER  - 
TY  - JOUR
A1  - Beckus, Siegfried
A1  - Bellissard, Jean
A1  - Cornean, Horia
T1  - Holder Continuity of the Spectra for Aperiodic Hamiltonians
JF  - Annales de l'Institut Henri Poincaré
N2  - We study the spectral location of a strongly pattern equivariant Hamiltonians arising through configurations on a colored lattice. Roughly speaking, two configurations are "close to each other" if, up to a translation, they "almost coincide" on a large fixed ball. The larger this ball, the more similar they are, and this induces a metric on the space of the corresponding dynamical systems. Our main result states that the map which sends a given configuration into the spectrum of its associated Hamiltonian, is Holder (even Lipschitz) continuous in the usual Hausdorff metric. Specifically, the spectral distance of two Hamiltonians is estimated by the distance of the corresponding dynamical systems.
Y1  - 2019
U6  - https://doi.org/10.1007/s00023-019-00848-6
SN  - 1424-0637
SN  - 1424-0661
VL  - 20
IS  - 11
SP  - 3603
EP  - 3631
PB  - Springer
CY  - Cham
ER  -