Localization on quantum graphs with random vertex couplings

We consider Schr\"odinger operators on a class of periodic quantum graphs with randomly distributed Kirchhoff coupling constants at all vertices. Using the technique of self-adjoint extensions we obtain conditions for localization on quantum graphs in terms of finite volume criteria for some energy-dependent discrete Hamiltonians. These conditions hold in the strong disorder limit and at the spectral edges

The weak localization for the alloy-type Anderson model on a cubic lattice

We consider alloy type random Schr\"odinger operators on a cubic lattice whose randomness is generated by the sign-indefinite single-site potential. We derive Anderson localization for this class of models in the Lifshitz tails regime, i.e. when the coupling parameter $\lambda$ is small, for the energies $E \le -C \lambda^2$.Comment: 45 pages, 2 figures. To appear in J. Stat. Phy

Lifshitz Tails in Constant Magnetic Fields

We consider the 2D Landau Hamiltonian $H$ perturbed by a random alloy-type potential, and investigate the Lifshitz tails, i.e. the asymptotic behavior of the corresponding integrated density of states (IDS) near the edges in the spectrum of $H$. If a given edge coincides with a Landau level, we obtain different asymptotic formulae for power-like, exponential sub-Gaussian, and super-Gaussian decay of the one-site potential. If the edge is away from the Landau levels, we impose a rational-flux assumption on the magnetic field, consider compactly supported one-site potentials, and formulate a theorem which is analogous to a result obtained in the case of a vanishing magnetic field

Understanding the Random Displacement Model: From Ground-State Properties to Localization

We give a detailed survey of results obtained in the most recent half decade which led to a deeper understanding of the random displacement model, a model of a random Schr\"odinger operator which describes the quantum mechanics of an electron in a structurally disordered medium. These results started by identifying configurations which characterize minimal energy, then led to Lifshitz tail bounds on the integrated density of states as well as a Wegner estimate near the spectral minimum, which ultimately resulted in a proof of spectral and dynamical localization at low energy for the multi-dimensional random displacement model.Comment: 31 pages, 7 figures, final version, to appear in Proceedings of "Spectral Days 2010", Santiago, Chile, September 20-24, 201

Spectral extrema and Lifshitz tails for non monotonous alloy type models

In the present note, we determine the ground state energy and study the existence of Lifshitz tails near this energy for some non monotonous alloy type models. Here, non monotonous means that the single site potential coming into the alloy random potential changes sign. In particular, the random operator is not a monotonous function of the random variables

Development of the Jefferson Scale of Empathy for Teachers (JSE-T) to Measure Empathy in Educationally Relevant Situations

The Jefferson Scale of Empathy is one of the most commonly used scales in medical education to measure empathy. It is specific to the field of medical education and geared toward orienting medical students to physician empathy in patient care situations. The scale was transferred to the educational context in teacher education. In doing so, the questionnaire was reduced from the original 20 items to 9 because of content and methodological issues. A CFA showed good model-fit parameters for a three-factor model, and correlations with the German version of the Interpersonal reactivity Index were in line with the magnitudes reported in previous literature. In total, the JST-E scales show evidence for their factorial and convergent validity and their reliability. The JSE-T proves to be a good instrument for measuring empathy in educational contexts and thus closes the gap between trait measurement procedures such as the IRI and concrete-situational judgment tests, so that an economical, multidimensional testing of empathy becomes possible

Further genetic heterogeneity for autosomal dominant human sutural cataracts

A unique sutural cataract was observed in a 4-generation German family to be transmitted as an isolated autosomal, dominant trait. Since mutations in the gamma-crystallin encoding CRYG genes have previously been demonstrated to be the most frequent reason for isolated congenital cataracts, all 4 active CRYG genes have been sequenced. A single base-pair change in the CRYGA gene has been shown, leading to a premature stop codon. This was not observed in 170 control individuals. However, it did not segregate with the disease phenotype. This is the first truncating mutation in an active CRYG gene without a dominant phenotype. As the CRYGA mutation did not explain the cataract, several other candidate loci (CCV, GJA8, CRYBB2, BFSP2, MIP, GJA8, central pouch-like, CRYBA1) were investigated by micro-satellite markers and linkage analysis, but they were excluded based on the combination of haplotype analysis and two-point linkage analysis. The phenotype in this family is due to a mutation in another sutural cataract gene yet to be identified

Low lying spectrum of weak-disorder quantum waveguides

We study the low-lying spectrum of the Dirichlet Laplace operator on a randomly wiggled strip. More precisely, our results are formulated in terms of the eigenvalues of finite segment approximations of the infinite waveguide. Under appropriate weak-disorder assumptions we obtain deterministic and probabilistic bounds on the position of the lowest eigenvalue. A Combes-Thomas argument allows us to obtain so-called 'initial length scale decay estimates' at they are used in the proof of spectral localization using the multiscale analysis.Comment: Accepted for publication in Journal of Statistical Physics http://www.springerlink.com/content/0022-471

Decorrelation estimates for the eigenlevels of the discrete Anderson model in the localized regime

The purpose of the present work is to establish decorrelation estimates for the locally renormalized eigenvalues of the discrete Anderson model near two distinct energies inside the localization region. In dimension one, we prove these estimates at all energies. In higher dimensions, the energies are required to be sufficiently far apart from each other

Persistence of Anderson localization in Schr\"odinger operators with decaying random potentials

We show persistence of both Anderson and dynamical localization in Schr\"odinger operators with non-positive (attractive) random decaying potential. We consider an Anderson-type Schr\"odinger operator with a non-positive ergodic random potential, and multiply the random potential by a decaying envelope function. If the envelope function decays slower than $|x|^{-2}$ at infinity, we prove that the operator has infinitely many eigenvalues below zero. For envelopes decaying as $|x|^{-\alpha}$ at infinity, we determine the number of bound states below a given energy $E<0$, asymptotically as $\alpha\downarrow 0$. To show that bound states located at the bottom of the spectrum are related to the phenomenon of Anderson localization in the corresponding ergodic model, we prove: (a) these states are exponentially localized with a localization length that is uniform in the decay exponent $\alpha$; (b)~ dynamical localization holds uniformly in $\alpha$