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Register a new userLocalization for a one-dimensional split-step quantum walk with bound states robust against perturbations
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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.
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This paper studies the spectrum of a multi-dimensional split-step quantum walk with a defect that cannot be analysed in the previous papers. To this end, we have developed a new technique which allow us to use a spectral mapping theorem for the one-defect model. We also derive the time-averaged limit measure for one-dimensional case as an application of the spectral analysis.
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.
Quantum walks are promising for information processing tasks because on regular graphs they spread quadratically faster than random walks. Static disorder, however, can turn the tables: unlike random walks, quantum walks can suffer Anderson localization, whereby the spread of the walker stays within a finite region even in the infinite time limit. It is therefore important to understand when we can expect a quantum walk to be Anderson localized and when we can expect it to spread to infinity even in the presence of disorder. In this work we analyze the response of a generic one-dimensional quantum walk -- the split-step walk -- to different forms of static disorder. We find that introducing static, symmetry-preserving disorder in the parameters of the walk leads to Anderson localization. In the completely disordered limit, however, a delocalization sets in, and the walk spreads subdiffusively. Using an efficient numerical algorithm, we calculate the bulk topological invariants of the disordered walk, and interpret the disorder-induced Anderson localization and delocalization transitions using these invariants.
We study multipoint scatterers with zero-energy bound states in three dimensions. We present examples of such scatterers with multiple zero eigenvalue or with strong multipole localization of zero-energy bound states.
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.
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