Advanced Search

We provide a general lower bound on the dynamics of one dimensional Schrödinger operators in terms of transfer matrices. In particular it yields a non trivial lower bound on the transport exponents as soon as the norm of transfer matrices does not grow faster than polynomially on a set of energies of full Lebesgue measure, and regardless of the nature of the spectrum. Applications to Hamiltonians with a) sparse, b) quasi-periodic, c) random decaying potential are provided....

We estimate the spreading of the solution of the Schrödinger equation asymptotically in time, in term of the fractal properties of the associated spectral measures. For this, we exhibit a lower bound for the moments of order $p$ at time $T$ for the state $\psi$ defined by $[\frac{1}{T}{\int }_0^T\parallel |X{|}^{p/2}{\mathrm{e}}^{-itH}\psi {\parallel }^2\mathrm{d}t]$. We show that this lower bound can be expressed in term of the generalized Rényi dimension of the spectral measure ${\mu }_{\psi }$ associated to the hamiltonian $H$ and the state $\psi$. We especially concentrate on continuous models.