Spectral characteristics of Schrodinger operators generated by product systems
Damanik D, Fillman J, Gohlke P (2022)
Journal of Spectral Theory 12(4): 1659-1718.
Zeitschriftenaufsatz
| Veröffentlicht | Englisch
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Autor*in
Damanik, David;
Fillman, Jake;
Gohlke, PhilippUniBi
Einrichtung
Abstract / Bemerkung
Motivated by the question of what spectral properties of dynamically defined Schrodinger operators may be preserved under periodic perturbations, we study ergodic Schrodinger operators defined over product dynamical systems in which one factor is periodic and the other factor is either a subshift over a finite alphabet or an irrational rotation of the circle. The scenario given by a periodic background potential corresponds to a separable structure in which the sampling function is the sum of two pieces, each of which depends only on a single factor of the product system. However, in each case that we study, our methods apply more generally to sampling functions that allow non-trivial dependencies between the product factors.In the case in which one factor is a Boshernitzan subshift, we prove that either the resulting operators are periodic or the resulting spectra must be Cantor sets. The main ingredient is a suitable stability result for the Boshernitzan condition under taking products. We also discuss the stability of purely singular continuous spectrum, which, given the zero-measure spectrum result, amounts to stability results for eigenvalue exclusion. In particular, we examine situations in which the existing criteria for the exclusion of eigenvalues are stable under periodic perturbations. As a highlight of this, we show that any simple Toeplitz subshift over a binary alphabet exhibits uniform absence of eigenvalues on the hull for any periodic perturbation whose period is commensurate with the coding sequence. This is new, even in the case in which the periodic background vanishes entirely. In the case of a full shift, we give an effective criterion to compute exactly the spectrum of a random Anderson model perturbed by a potential of period two, and we further show that the naive generalization of this criterion does not hold for period three. Next, we consider quasi-periodic potentials with potentials generated by trigonometric polynomials with periodic background. We show that the quasiperiodic cocycle induced by passing to blocks of period length is subcritical when the coupling constant is small and supercritical when the coupling constant is large. Thus, the spectral type is absolutely continuous for small coupling and pure point (for a.e. frequency and phase) when the coupling is large.
Stichworte
Schrodinger operators;
dynamical systems;
product systems;
Cantor;
spectrum
Erscheinungsjahr
2022
Zeitschriftentitel
Journal of Spectral Theory
Band
12
Ausgabe
4
Seite(n)
1659-1718
ISSN
1664-039X
eISSN
1664-0403
Page URI
https://pub.uni-bielefeld.de/record/2980539
Zitieren
Damanik D, Fillman J, Gohlke P. Spectral characteristics of Schrodinger operators generated by product systems. Journal of Spectral Theory. 2022;12(4):1659-1718.
Damanik, D., Fillman, J., & Gohlke, P. (2022). Spectral characteristics of Schrodinger operators generated by product systems. Journal of Spectral Theory, 12(4), 1659-1718. https://doi.org/10.4171/JST/445
Damanik, David, Fillman, Jake, and Gohlke, Philipp. 2022. “Spectral characteristics of Schrodinger operators generated by product systems”. Journal of Spectral Theory 12 (4): 1659-1718.
Damanik, D., Fillman, J., and Gohlke, P. (2022). Spectral characteristics of Schrodinger operators generated by product systems. Journal of Spectral Theory 12, 1659-1718.
Damanik, D., Fillman, J., & Gohlke, P., 2022. Spectral characteristics of Schrodinger operators generated by product systems. Journal of Spectral Theory, 12(4), p 1659-1718.
D. Damanik, J. Fillman, and P. Gohlke, “Spectral characteristics of Schrodinger operators generated by product systems”, Journal of Spectral Theory, vol. 12, 2022, pp. 1659-1718.
Damanik, D., Fillman, J., Gohlke, P.: Spectral characteristics of Schrodinger operators generated by product systems. Journal of Spectral Theory. 12, 1659-1718 (2022).
Damanik, David, Fillman, Jake, and Gohlke, Philipp. “Spectral characteristics of Schrodinger operators generated by product systems”. Journal of Spectral Theory 12.4 (2022): 1659-1718.
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