Localization for a matrix-valued Anderson model

We study localization properties for a class of one-dimensional, matrix-valued, continuous, random Schr\"odinger operators, acting on L^2(\R)\otimes \C^N, for arbitrary $N\geq 1$. We prove that, under suitable assumptions on the F\"urstenberg group of these operators, valid on an interval $I\subset \R$, they exhibit localization properties on $I$, both in the spectral and dynamical sense. After looking at the regularity properties of the Lyapunov exponents and of the integrated density of states, we prove a Wegner estimate and apply a multiscale analysis scheme to prove localization for these operators. We also study an example in this class of operators, for which we can prove the required assumptions on the F\"urstenberg group. This group being the one generated by the transfer matrices, we can use, to prove these assumptions, an algebraic result on generating dense Lie subgroups in semisimple real connected Lie groups, due to Breuillard and Gelander. The algebraic methods used here allow us to handle with singular distributions of the random parameters

Localization Bounds for Multiparticle Systems

We consider the spectral and dynamical properties of quantum systems of $n$ particles on the lattice $\Z^d$, of arbitrary dimension, with a Hamiltonian which in addition to the kinetic term includes a random potential with iid values at the lattice sites and a finite-range interaction. Two basic parameters of the model are the strength of the disorder and the strength of the interparticle interaction. It is established here that for all $n$ there are regimes of high disorder, and/or weak enough interactions, for which the system exhibits spectral and dynamical localization. The localization is expressed through bounds on the transition amplitudes, which are uniform in time and decay exponentially in the Hausdorff distance in the configuration space. The results are derived through the analysis of fractional moments of the $n$-particle Green function, and related bounds on the eigenfunction correlators

Quantum harmonic oscillator systems with disorder

We study many-body properties of quantum harmonic oscillator lattices with disorder. A sufficient condition for dynamical localization, expressed as a zero-velocity Lieb-Robinson bound, is formulated in terms of the decay of the eigenfunction correlators for an effective one-particle Hamiltonian. We show how state-of-the-art techniques for proving Anderson localization can be used to prove that these properties hold in a number of standard models. We also derive bounds on the static and dynamic correlation functions at both zero and positive temperature in terms of one-particle eigenfunction correlators. In particular, we show that static correlations decay exponentially fast if the corresponding effective one-particle Hamiltonian exhibits localization at low energies, regardless of whether there is a gap in the spectrum above the ground state or not. Our results apply to finite as well as to infinite oscillator systems. The eigenfunction correlators that appear are more general than those previously studied in the literature. In particular, we must allow for functions of the Hamiltonian that have a singularity at the bottom of the spectrum. We prove exponential bounds for such correlators for some of the standard models

Spectra of soft ring graphs

We discuss of a ring-shaped soft quantum wire modeled by $\delta$ interaction supported by the ring of a generally nonconstant coupling strength. We derive condition which determines the discrete spectrum of such systems, and analyze the dependence of eigenvalues and eigenfunctions on the coupling and ring geometry. In particular, we illustrate that a random component in the coupling leads to a localization. The discrete spectrum is investigated also in the situation when the ring is placed into a homogeneous magnetic field or threaded by an Aharonov-Bohm flux and the system exhibits persistent currents.Comment: LaTeX 2e, 17 pages, with 10 ps figure

Localization on a quantum graph with a random potential on the edges

We prove spectral and dynamical localization on a cubic-lattice quantum graph with a random potential. We use multiscale analysis and show how to obtain the necessary estimates in analogy to the well-studied case of random Schroedinger operators.Comment: LaTeX2e, 18 page

Defining pathways to healthy sustainable urban development

Goals and pathways to achieve sustainable urban development have multiple interlinkages with human health and wellbeing. However, these interlinkages have not been examined in depth in recent discussions on urban sustainability and global urban science. This paper fills that gap by elaborating in detail the multiple links between urban sustainability and human health and by mapping research gaps at the interface of health and urban sustainability sciences. As researchers from a broad range of disciplines, we aimed to: 1) define the process of urbanization, highlighting distinctions from related concepts to support improved conceptual rigour in health research; 2) review the evidence linking health with urbanization, urbanicity, and cities and identify cross-cutting issues; and 3) highlight new research approaches needed to study complex urban systems and their links with health. This novel, comprehensive knowledge synthesis addresses issue of interest across multiple disciplines. Our review of concepts of urban development should be of particular value to researchers and practitioners in the health sciences, while our review of the links between urban environments and health should be of particular interest to those outside of public health. We identify specific actions to promote health through sustainable urban development that leaves no one behind, including: integrated planning; evidence-informed policy-making; and monitoring the implementation of policies. We also highlight the critical role of effective governance and equity-driven planning in progress towards sustainable, healthy, and just urban development

An Improved Combes-Thomas Estimate of Magnetic Schr\"{o}dinger Operators

In the present paper, we prove an improved Combes-Thomas estimate, i.e., the Combes-Thomas estimate in trace-class norms, for magnetic Schr\"{o}dinger operators under general assumptions. In particular, we allow unbounded potentials. We also show that for any function in the Schwartz space on the reals the operator kernel decays, in trace-class norms, faster than any polynomial.Comment: 25 pages, some errors correcte

Generalized eigenfunctions and spectral theory for strongly local Dirichlet forms

We present an introduction to the framework of strongly local Dirichlet forms and discuss connections between the existence of certain generalized eigenfunctions and spectral properties within this framework. The range of applications is illustrated by a list of examples

Wegner Estimate and Disorder Dependence for Alloy-Type Hamiltonians with Bounded Magnetic Potential

We consider non-ergodic magnetic random Sch\"odinger operators with a bounded magnetic vector potential. We prove an optimal Wegner estimate valid at all energies. The proof is an adaptation of the arguments from [Kle13], combined with a recent quantitative unique continuation estimate for eigenfunctions of elliptic operators from [BTV15]. This generalizes Klein's result to operators with a bounded magnetic vector potential. Moreover, we study the dependence of the Wegner-constant on the disorder parameter. In particular, we show that above the model-dependent threshold $E_0(\infty) \in (0, \infty]$, it is impossible that the Wegner-constant tends to zero if the disorder increases. This result is new even for the standard (ergodic) Anderson Hamiltonian with vanishing magnetic field