2009/06/10 by L. Marin, L. MARIN
Materials Science · Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Bounded function #Dimension (graph theory) #Diophantine equation #Exponent #Fibonacci number #Irrational number #Operator (biology) #Quasicrystal Structures and Properties #Spectral Theory in Mathematical Physics #Upper and lower bounds #math-ph #math.MP #msc:47B36 #msc:81Q10
paper · pdf · doi:10.1142/s0129055x10004090
arxiv created 2009/06/10 · openalex publication_date 2010/09/01 · arxiv updated 2015/05/13 · openalex created_date 2017/10/20 · openalex updated_date 2026/08/05
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. Damanik and Tcheremchantsev developed a method in [10] 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.