Spectral Gaps for Periodic Elliptic Operators with High Contrast: an Overview

We discuss the band-gap structure and the integrated density of states for periodic elliptic operators in the Hilbert space $L_2(\R^m)$, for $m \ge 2$. We specifically consider situations where high contrast in the coefficients leads to weak coupling between the period cells. Weak coupling of periodic systems frequently produces spectral gaps or spectral concentration. Our examples include Schr\"odinger operators, elliptic operators in divergence form, Laplace-Beltrami-operators, Schr\"odinger and Pauli operators with periodic magnetic fields. There are corresponding applications in heat and wave propagation, quantum mechanics, and photonic crystals.Comment: 12 pages, 1 eps-figure, LaTe

Spectral Properties of Grain Boundaries at Small Angles of Rotation

We study some spectral properties of a simple two-dimensional model for small angle defects in crystals and alloys. Starting from a periodic potential $V \colon \R^2 \to \R$, we let $V_\theta(x,y) = V(x,y)$ in the right half-plane $\{x \ge 0\}$ and $V_\theta = V \circ M_{-\theta}$ in the left half-plane $\{x < 0\}$, where $M_\theta \in \R^{2 \times 2}$ is the usual matrix describing rotation of the coordinates in $\R^2$ by an angle $\theta$. As a main result, it is shown that spectral gaps of the periodic Schr\"odinger operator $H_0 = -\Delta + V$ fill with spectrum of $R_\theta = -\Delta + V_\theta$ as $0 \ne \theta \to 0$. Moreover, we obtain upper and lower bounds for a quantity pertaining to an integrated density of states measure for the surface states.Comment: 22 pages, 3 figure

On Open Scattering Channels for Manifolds with Ends

In the framework of time-dependent geometric scattering theory, we study the existence and completeness of the wave operators for perturbations of the Riemannian metric for the Laplacian on a complete manifold of dimension $n$. The smallness condition for the perturbation is expressed (intrinsically and coordinate free) in purely geometric terms using the harmonic radius; therefore, the size of the perturbation can be controlled in terms of local bounds on the injectivity radius and the Ricci-curvature. As an application of these ideas we obtain a stability result for the scattering matrix with respect to perturbations of the Riemannian metric. This stability result implies that a scattering channel which interacts with other channels preserves this property under small perturbations.Comment: updated version, now 43 page

Personality and attitudes towards current political topics

Riemann R, Grubich C, Hempel S, Mergl S, Richter M. Personality and attitudes towards current political topics. Personality and Individual Differences. 1993;15(3):313-321.We presented a representative list of 162 political issues currently discussed in Germany and the German NEO-FFI to 184 subjects (45% university students). Principal components analysis of the attitude items reveals four factors which are interpreted as (1) general conservatism, preference for authoritarian punitiveness, (2) social welfare and support of women's equality, (3) liberalism and affirmation of technological progress, and (4) affirmation of increase in taxation for environmental protection and the development of East Europe. The first unrotated factor is identified as general conservatism. The analysis of zero and higher order correlations shows meaningful relationships between political attitudes and personality dimensions. The highest (negative) correlations are found between openness to experience and conservatism. Age and sex effects on political attitudes are reported