On the density of states of the one dimensional quasi-periodic Schödinger operators. (Sur la densité d’état de l’opérateur de Schrödinger quasi-périodique unidimensionnel.) (French. English summary) Zbl 1108.47032
The author considers a discrete one-dimensional Schrödinger operator with an analytic potential. A Diophantine condition upon the frequencies is assumed. The first result is a reduction theorem for the first-order system obtained from the initial Schrödinger equation. This result resembles recent theorems for the continuous time equations [cf.L. H.Eliasson, Proc.Symp.Pure Math.69, 679–705 (2001; Zbl 1015.34028)]. Then the author considers the density of states, proves its Hölder-\(\frac{1}2\) property, and shows that the length of the gaps has a sub-exponential estimate connected with the gap-labelling theorem.
Reviewer: Anatoly N. Kochubei (Kyïv)
MSC:
| 47B39 | Linear difference operators |
| 39A70 | Difference operators |
| 47N50 | Applications of operator theory in the physical sciences |
| 82B44 | Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics |
Citations:
Zbl 1015.34028References:
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