Sharp exponential localization for eigenfunctions of the Dirac Operator

We determine the fastest possible rate of exponential decay at infinity for eigenfunctions of the Dirac operator $\mathcal D_n + \mathbb V$, being $\mathcal D_n$ the massless Dirac operator in dimensions $n=2,3$ and $\mathbb V$ a matrix-valued perturbation such that $|\mathbb V(x)| \sim |x|^{-\epsilon}$ at infinity, for $\epsilon < 1$. Moreover, we provide explicit examples of solutions that have the prescripted decay, in presence of a potential with the related behaviour at infinity, proving that our results are sharp. This work is a result of unique continuation from infinity.INDAM - Istituto Italiano di Alta Matematic

Boundary Triples for the Dirac Operator with Coulomb-Type Spherically Symmetric Perturbations

We determine explicitly a boundary triple for the Dirac operator $H:=-i\alpha\cdot \nabla + m\beta + \mathbb V(x)$ in $\mathbb R^3$, for $m\in\mathbb R$ and $\mathbb V(x)= |x|^{-1} ( \nu \mathbb{I}_4 +\mu \beta -i \lambda \alpha\cdot{x}/|x|\,\beta)$, with $\nu,\mu,\lambda \in \mathbb R$. Consequently we determine all the self-adjoint realizations of $H$ in terms of the behaviour of the functions of their domain in the origin. When $\sup_{x} |x| |\mathbb V(x)| \leq 1$, we discuss the problem of selecting the distinguished extension requiring that its domain is included in the domain of the appropriate quadratic form

Self-Adjoint Extensions for the Dirac Operator with Coulomb-Type Spherically Symmetric Potentials

We describe the self-adjoint realizations of the operator $H:=-i\alpha\cdot \nabla + m\beta + \mathbb V(x)$, for $m\in\mathbb R$, and $\mathbb V(x)= |x|^{-1} ( \nu \mathbb{I}_4 +\mu \beta -i \lambda \alpha\cdot{x}/{|x|}\,\beta)$, for $\nu,\mu,\lambda \in \mathbb R$. We characterize the self-adjointness in terms of the behaviour of the functions of the domain in the origin, exploiting Hardy-type estimates and trace lemmas. Finally, we describe the distinguished extension.Istituto Italiano di Alta Matematica "F. Severi

A Hardy-type inequality and some spectral characterizations for the Dirac-Coulomb operator

We prove a sharp Hardy-type inequality for the Dirac operator. We exploit this inequality to obtain spectral properties of the Dirac operator perturbed with Hermitian matrix-valued potentials V of Coulomb type: we characterise its eigenvalues in terms of the Birman–Schwinger principle and we bound its discrete spectrum from below, showing that the ground-state energy is reached if and only if V verifies some rigidity conditions. In the particular case of an electrostatic potential, these imply that V is the Coulomb potential

Anxious to see you: Neuroendocrine mechanisms of social vigilance and anxiety during adolescence.

Social vigilance is a behavioral strategy commonly used in adverse or changing social environments. In animals, a combination of avoidance and vigilance allows an individual to evade potentially dangerous confrontations while monitoring the social environment to identify favorable changes. However, prolonged use of this behavioral strategy in humans is associated with increased risk of anxiety disorders, a major burden for human health. Elucidating the mechanisms of social vigilance in animals could provide important clues for new treatment strategies for social anxiety. Importantly, during adolescence the prevalence of social anxiety increases significantly. We hypothesize that many of the actions typically characterized as anxiety behaviors begin to emerge during this time as strategies for navigating more complex social structures. Here, we consider how the social environment and the pubertal transition shape neural circuits that modulate social vigilance, focusing on the bed nucleus of the stria terminalis and prefrontal cortex. The emergence of gonadal hormone secretion during adolescence has important effects on the function and structure of these circuits, and may play a role in the emergence of a notable sex difference in anxiety rates across adolescence. However, the significance of these changes in the context of anxiety is still uncertain, as not enough studies are sufficiently powered to evaluate sex as a biological variable. We conclude that greater integration between human and animal models will aid the development of more effective strategies for treating social anxiety

The connection between radio halos and cluster mergers and the statistical properties of the radio halo population

We discuss the statistical properties of the radio halo population in galaxy clusters. Radio bi-modality is observed in galaxy clusters: a fraction of clusters host giant radio halos while the majority of clusters do not show evidence of diffuse cluster-scale radio emission. The radio bi-modality has a correspondence in terms of dynamical state of the hosting clusters showing that merging clusters host radio halos and follow the well known radio--X-ray correlation, while more relaxed clusters do not host radio halos and populate a region well separated from that correlation. These evidences can be understood in the framework of a scenario where merger-driven turbulence re-accelerate the radio emitting electrons. We discuss the main statistical expectations of this scenario underlining the important role of upcoming LOFAR surveys to test present models.Comment: 11 pages, 2 figures, Invited talk at the conference "Diffuse Relativistic Plasmas", Bangalore, 1-4 March 2011; accepted for publication in the Journal of Astrophysics and Astronom

Comparative analysis of the diffuse radio emission in the galaxy clusters A1835, A2029, and Ophiuchus

We recently performed a study of a sample of relaxed, cooling core galaxy clusters with deep Very Large Array observations at 1.4 GHz. We find that in the central regions of A1835, A2029, and Ophiuchus the dominant radio galaxy is surrounded by a diffuse low-brightness radio emission that takes the form of a mini-halo. Here we present the results of the analysis of the extended diffuse radio emission in these mini-halos. In order to investigate the morphological properties of the diffuse radio emission in clusters of galaxies we propose to fit their azimuthally averaged brightness profile with an exponential, obtaining the central brightness and the e-folding radius from which the radio emissivity can be calculated. We investigate the radio properties of the mini-halos in A1835, A2029, and Ophiuchus in comparison with the radio properties of a representative sample of mini-halos and halos already known in the literature. We find that radio halos can have quite different length-scales but their emissivity is remarkably similar from one halo to the other. In contrast, mini-halos span a wide range of radio emissivity. Some of them, like the Perseus mini-halos, are characterized by a radio emissivity which is more than 100 times greater than that of radio halos. On the other hand, the new mini-halos in cooling core clusters analyzed in this work, namely A2029, Ophiuchus, and A1835, have a radio emissivity which is much more typical of halos in merging clusters rather than similar to that of the other mini-halos previously known.Comment: 17 pages, 11 figures, A&A in press. For a version with high quality figures, see http://erg.ca.astro.it/preprints/mini_halo_2