Absolutely continuous and pure point spectra of discrete operators with sparse potentials

We consider the discrete Schr\”odinger operator $H=-\Delta+V$ with a sparse potential $V$ and find conditions guaranteeing either existence of wave operators for the pair $H$ and $H_0=-\Delta$, or presence of dense purely point spectrum of the operator $H$ on some interval $[\lambda_0,0]$ with $\lambda_0<0$.

Constraints on pure point diffraction on aperiodic point patterns of finite local complexity

We consider the pure point part of the diffraction on families of aperiodic point sets obeying common local rules. It is shown that imposing such rules results in linear constraints on the partial diffraction amplitudes. These relations can be explicitly derived from the geometry of the prototile space representing the local rules.

Eberlein decomposition for PV inflation systems

The Dirac combs of primitive Pisot–Vijayaraghavan (PV) inflations on the real line or, more generally, in $${\mathbb {R}}^d$$ R d are analysed. We construct a mean-orthogonal splitting for such Dirac combs that leads to the classic Eberlein decomposition on the level of the pair correlation measures, and thus to the separation of pure point versus continuous spectral components in the corresponding diffraction measures. This is illustrated with two guiding examples, and an extension to more general systems with randomness is outlined.

Three variations on a theme by Fibonacci

Several variants of the classic Fibonacci inflation tiling are considered in an illustrative fashion, in one and in two dimensions, with an eye on changes or robustness of diffraction and dynamical spectra. In one dimension, we consider extension mechanisms of deterministic and of stochastic nature, while we look at direct product variations in a planar extension. For the pure point part, we systematically employ a cocycle approach that is based on the underlying renormalization structure. It allows explicit calculations, particularly in cases where one meets regular model sets with Rauzy fractals as windows.

Global multiplicity bounds and spectral statistics for random operators

In this paper, we consider Anderson type operators on a separable Hilbert space where the random perturbations are finite rank and the random variables have full support on [Formula: see text]. We show that spectral multiplicity has a uniform lower bound whenever the lower bound is given on a set of positive Lebesgue measure on the point spectrum away from the continuous one. We also show a deep connection between the multiplicity of pure point spectrum and local spectral statistics, in particular, we show that spectral multiplicity higher than one always gives non-Poisson local statistics in the framework of Minami theory. In particular, for higher rank Anderson models with pure point spectrum, with the randomness having support equal to [Formula: see text], there is a uniform lower bound on spectral multiplicity and in case this is larger than one, the local statistics is not Poisson.

Agnotologie, die Grenzen der Gewissheit und die Theorie der Demokratie

Inspired to some extent by the work of Thomas Kuhn, Ignorance Studies seeks to grapple with the implications that follow from the idea that on a theoretical level we can hardly ever know that we have approached truth. In order to test theories, we would need to have access to empirically raw, theoretically unadorned facts and to confront them with theoretically diverse or contradictory accounts of how to make sense of them. It’s the nature of the case that we are not able to penetrate to such a pure point of origin for things. The words that we use to describe them are already suffused with biases and distortions that limit what we can say about them. Our relationship to phenomena inside us and to things outside us always subsists in a symbolically mediated, indirect medium. Our efforts to be either before or after this state of symbolic mediation generally land us in a state of deepened awareness of the extent of the prolongation of the middle. The conceptualized entity is for all intents and purposes the entity we relate to. The paper argues that given the limits to certainty highlighted by Ignorance Studies, our political institutions need to be responsive to the priorities enshrined in democratic theory to ensure that the policies pursued by any given government are reflective of the needs and the wishes of the majority of the people.