Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOne-dimensional Discrete Anderson Model in a Decaying Random Potential: from a.c. Spectrum to Dynamical Localization
86
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We consider a one-dimensional Anderson model where the potential decays in average like $n^{-alpha}$, $alpha>0$. This simple model is known to display a rich phase diagram with different kinds of spectrum arising as the decay rate $alpha$ varies. We review an article of Kiselev, Last and Simon where the authors show a.c. spectrum in the super-critical case $alpha>frac12$, a transition from singular continuous to pure point spectrum in the critical case $alpha=frac12$, and dense pure point spectrum in the sub-critical case $alpha<frac12$. We present complete proofs of the cases $alphagefrac12$ and simplify some arguments along the way. We complement the above result by discussing the dynamical aspects of the model. We give a simple argument showing that, despite of the spectral transition, transport occurs for all energies for $alpha=frac12$. Finally, we discuss a theorem of Simon on dynamical localization in the sub-critical region $alpha<frac12$. This implies, in particular, that the spectrum is pure point in this regime.
rate research
Read More
We consider a one-dimensional continuum Anderson model where the potential decays in average like $|x|^{-alpha}$, $alpha>0$. We show dynamical localization for $0<alpha<frac12$ and provide control on the decay of the eigenfunctions.
We study the spectrum and dynamics of a one-dimensional discrete Dirac operator in a random potential obtained by damping an i.i.d. environment with an envelope of type $n^{-alpha}$ for $alpha>0$. We recover all the spectral regimes previously obtained for the analogue Anderson model in a random decaying potential, namely: absolutely continuous spectrum in the super-critical region $alpha>frac12$; a transition from pure point to singular continuous spectrum in the critical region $alpha=frac12$; and pure point spectrum in the sub-critical region $alpha<frac12$. From the dynamical point of view, delocalization in the super-critical region follows from the RAGE theorem. In the critical region, we exhibit a simple argument based on lower bounds on eigenfunctions showing that no dynamical localization can occur even in the presence of point spectrum. Finally, we show dynamical localization in the sub-critical region by means of the fractional moments method and provide control on the eigenfunctions.
We consider the large-time behavior of the solution $ucolon [0,infty)timesZto[0,infty)$ to the parabolic Anderson problem $partial_t u=kappaDelta u+xi u$ with initial data $u(0,cdot)=1$ and non-positive finite i.i.d. potentials $(xi(z))_{zinZ}$. Unlike in dimensions $dge2$, the almost-sure decay rate of $u(t,0)$ as $ttoinfty$ is not determined solely by the upper tails of $xi(0)$; too heavy lower tails of $xi(0)$ accelerate the decay. The interpretation is that sites $x$ with large negative $xi(x)$ hamper the mass flow and hence screen off the influence of more favorable regions of the potential. The phenomenon is unique to $d=1$. The result answers an open question from our previous study cite{BK00} of this model in general dimension.
For given two unitary and self-adjoint operators on a Hilbert space, a spectral mapping theorem was proved in cite{HiSeSu}. In this paper, as an application of the spectral mapping theorem, we investigate the spectrum of a one-dimensional split-step quantum walk. We give a criterion for when there is no eigenvalues around $pm 1$ in terms of a discriminant operator. We also provide a criterion for when eigenvalues $pm 1$ exist in terms of birth eigenspaces. Moreover, we prove that eigenvectors from the birth eigenspaces decay exponentially at spatial infinity and that the birth eigenspaces are robust against perturbations.
We consider some compact non-selfadjoint perturbations of fibered one-dimensional discrete Schrodinger operators. We show that the perturbed operator exhibits finite discrete spectrum under suitable- regularity conditions.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
