Dynamical bounds for Sturmian Schrödinger operators. (English) Zbl 1201.81052
Summary: The Fibonacci Hamiltonian, that is a Schrödinger operator associated to a quasiperiodical Sturmian potential with respect to the golden mean has been investigated intensively in recent years. D. Damanik and S. Tcheremchantsev developed a method in [J. Anal. Math. 97, 103–131 (2006; Zbl 1132.81018)] and used it to exhibit a non trivial dynamical upper bound for this model. In this paper, we use this method to generalize to a large family of Sturmian operators dynamical upper bounds and show at sufficently large coupling anomalous transport for operators associated to irrational number with a generic Diophantine condition. As a counterexample, we exhibit a pathological irrational number which does not verify this condition and show its associated dynamic exponent only has ballistic bound. Moreover, we establish a global lower bound for the lower box counting dimension of the spectrum that is used to obtain a dynamical lower bound for bounded density irrational numbers.
MSC:
81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |
47B36 | Jacobi (tridiagonal) operators (matrices) and generalizations |
35Q41 | Time-dependent Schrödinger equations and Dirac equations |
82D20 | Statistical mechanics of solids |
37C55 | Periodic and quasi-periodic flows and diffeomorphisms |
Citations:
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