Strictly ergodic subshifts and associated operators. (English) Zbl 1130.82017
Gesztesy, Fritz (ed.) et al., Spectral theory and mathematical physics. A festschrift in honor of Barry Simon’s 60th birthday. Ergodic Schrödinger operators, singular spectrum, orthogonal polynomials, and inverse spectral theory. Based on the SimonFest conference, Pasadena, CA, USA, March 27–31, 2006. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4249-2/pt.2; 978-0-8218-3783-2/set). Proceedings of Symposia in Pure Mathematics 76, Pt. 2, 505-538 (2007).
The author considers ergodic families of Schrödinger operators over base dynamics given by strictly ergodic subshifts on finite alphabets. For a large class of operators of this type have established that they have a purely singular continuous spectrum supported on a Cantor set of zero Lebesgue measure.
The author reviews the mechanisms leading to these results and discusses analogues for Cantero-Moval-Velázquez (CMV) matrices [see B. Simon, Orthogonal polynomials on the unit circle. Part 1: Classical theory. Providence, RI: American Mathematical Society (AMS) (2005; Zbl 1082.42020), Part 2: Spectral theory (2005; Zbl 1082.42021)].
For the entire collection see [Zbl 1110.00015].
The author reviews the mechanisms leading to these results and discusses analogues for Cantero-Moval-Velázquez (CMV) matrices [see B. Simon, Orthogonal polynomials on the unit circle. Part 1: Classical theory. Providence, RI: American Mathematical Society (AMS) (2005; Zbl 1082.42020), Part 2: Spectral theory (2005; Zbl 1082.42021)].
For the entire collection see [Zbl 1110.00015].
Reviewer: Nasir N. Ganikhodjaev (Kuantan)
MSC:
| 82B44 | Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics |
| 47B80 | Random linear operators |
| 47B36 | Jacobi (tridiagonal) operators (matrices) and generalizations |
| 81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |
