We investigate the equivalence between dynamical localization and localization properties of eigenfunctions of Schrödinger Hamiltonians. We introduce three classes of equivalent properties and study the relationships between them. These relationships are optimal as shown by counterexamples.
In this note, we prove the equality of the quantum bulk and the edge Hall conductances in mobility edges and in presence of disorder. The bulk and edge perturbations can be either of electric or magnetic nature. The edge conductance is regularized in a suitable way to enable the Fermi level to lie in a region of localized states.
Let H0 be a purely absolutely continuous selfadjoint operator acting on some separable infinite-dimensional Hilbert space and V be a compact non-selfadjoint perturbation. We relate the regularity properties of V to various spectral properties of the perturbed operator H0 + V . The structure of the discrete spectrum and the embedded eigenvalues are analysed jointly with the existence of limiting absorption principles in a unified framework. Our results are based on a suitable combination of complex scaling techniques, resonance theory and positive commutators methods. Various results scattered throughout the literature are recovered and extended. For illustrative purposes, the case of the one-dimensional discrete Laplacian is emphasized.
In this paper, we develop the radial transfer matrix formalism for unitary one-channel operators. This generalizes previous formalisms for CMV matrices and scattering zippers. We establish an analog of Carmona's formula and deduce criteria for absolutely continuous spectrum which we apply to random Hilbert Schmidt perturbations of periodic scattering zippers. Contents 1. Setup and result 1 1.1. One-channel unitary operators 3 1.2. Transfer matrices 6 1.3. Spectral average formula and criteria for a.c. spectrum 11 1.4. Absolutely continuous spectrum for periodic one-channel scattering zippers with random decaying perturbation 12 2. Examples 14 2.1. One-dimensional quantum walks 14 2.2. Generalized one-channel quantum walks 16 2.3. Stroboscopic unitary dynamics on Z 2 17 3. Transfer matrix and Resolvent 18 4. Spectral averaging and weak convergence 21 5. Perturbation of periodic scattering zipper 24 Appendix A. Facts of linear algebra 29 References 33
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