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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.

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A scattering zipper is a system obtained by concatenation of scattering events with equal even number of incoming and out going channels. The associated scattering zipper operator is the unitary equivalent of Jacobi matrices with matrix entries and generalizes BlatterBrowne and Chalker-Coddington models and CMV matrices. Weyl discs are analyzed and used to prove a bijection between the set of semi-infinite scattering zipper operators and matrix valued probability measures on the unit circle. Sturm-Liouville oscillation theory is developed as a tool to calculate the spectra of finite and periodic scattering zipper operators.

We define coined Quantum Walks on the infinite rooted binary tree given by unitary operators $U(C)$ on an associated infinite dimensional Hilbert space, depending on a unitary coin matrix $C\in U(3)$, and study their spectral properties. For circulant unitary coin matrices $C$, we derive an equation for the Carath\'eodory function associated to the spectral measure of a cyclic vector for $U(C)$. This allows us to show that for all circulant unitary coin matrices, the spectrum of the Quantum Walk has no singular continuous component. Furthermore, for coin matrices $C$ which are orthogonal circulant matrices, we show that the spectrum of the Quantum Walk is absolutely continuous, except for four coin matrices for which the spectrum of $U(C)$ is pure point

We consider the trace map associated with the silver ratio Schrödinger operator as a diffeomorphism on the invariant surface associated with a given coupling constant and prove that the non-wandering set of this map is hyperbolic if the coupling is sufficiently large. As a consequence, for this values of the coupling constant, the local and global Hausdorff dimension and the local and global box counting dimension of the spectrum of this operator all coincide and are smooth functions of the coupling constant.

We study two versions of quasicrystal model, both subcases of Jacobi matrices. For off-diagonal model, we show an upper bound of dynamical exponent and the norm of the transfer matrix. We apply this result to the off-diagonal Fibonacci Hamiltonian and obtain a sub-ballistic bound for coupling large enough. In diagonal case, we improve previous lower bounds on the fractal box-counting dimension of the spectrum.

Absolutely continuous spectrum implies ballistic transport for quantum particles in a random potential on tree graphs Local vibrational modes in crystal lattices with a simply connected region of the quasi-continuous phonon spectrum Low Temp.