Constrained energy minimization and orbital stability for the NLS equation on a star graph

We consider a nonlinear Schr\"odinger equation with focusing nonlinearity of power type on a star graph ${\mathcal G}$, written as $i \partial_t \Psi (t) = H \Psi (t) - |\Psi (t)|^{2\mu}\Psi (t)$, where $H$ is the selfadjoint operator which defines the linear dynamics on the graph with an attractive $\delta$ interaction, with strength $\alpha < 0$, at the vertex. The mass and energy functionals are conserved by the flow. We show that for $0<\mu<2$ the energy at fixed mass is bounded from below and that for every mass $m$ below a critical mass $m^*$ it attains its minimum value at a certain \hat \Psi_m \in H^1(\GG) , while for $m>m^*$ there is no minimum. Moreover, the set of minimizers has the structure {\mathcal M}={e^{i\theta}\hat \Psi_m, \theta\in \erre}. Correspondingly, for every $m<m^*$ there exists a unique $\omega=\omega(m)$ such that the standing wave $\hat\Psi_{\omega}e^{i\omega t}$ is orbitally stable. To prove the above results we adapt the concentration-compactness method to the case of a star graph. This is non trivial due to the lack of translational symmetry of the set supporting the dynamics, i.e. the graph. This affects in an essential way the proof and the statement of concentration-compactness lemma and its application to minimization of constrained energy. The existence of a mass threshold comes from the instability of the system in the free (or Kirchhoff's) case, that in our setting corresponds to \al=0.Comment: 26 pages, 1 figur

Reduction criterion for separability

We introduce a separability criterion based on the positive map Γ:ρ→(Tr ρ)-ρ, where ρ is a trace-class Hermitian operator. Any separable state is mapped by the tensor product of Γ and the identity into a non-negative operator, which provides a simple necessary condition for separability. This condition is generally not sufficient because it is vulnerable to the dilution of entanglement. In the special case where one subsystem is a quantum bit, Γ reduces to time reversal, so that this separability condition is equivalent to partial transposition. It is therefore also sufficient for 2×2 and 2×3 systems. Finally, a simple connection between this map for two qubits and complex conjugation in the “magic” basis [Phys. Rev. Lett. 78, 5022 (1997)] is displayed

Variational properties and orbital stability of standing waves for NLS equation on a star graph

We study standing waves for a nonlinear Schr\"odinger equation on a star graph {$\mathcal{G}$} i.e. $N$ half-lines joined at a vertex. At the vertex an interaction occurs described by a boundary condition of delta type with strength $\alpha\leqslant 0$. The nonlinearity is of focusing power type. The dynamics is given by an equation of the form $i \frac{d}{dt}\Psi_t = H \Psi_t - | \Psi_t |^{2\mu} \Psi_t$, where $H$ is the Hamiltonian operator which generates the linear Schr\"odinger dynamics. We show the existence of several families of standing waves for every sign of the coupling at the vertex for every $\omega > \frac{\alpha^2}{N^2}$. Furthermore, we determine the ground states, as minimizers of the action on the Nehari manifold, and order the various families. Finally, we show that the ground states are orbitally stable for every allowed $\omega$ if the nonlinearity is subcritical or critical, and for $\omega<\omega^\ast$ otherwise.Comment: 36 pages, 2 figures, final version appeared in JD

On the nature of faint Low Surface Brightness galaxies in the Coma cluster

This project is the continuation of our study of faint Low Surface Brightness Galaxies (fLSBs) in one of the densest nearby galaxy regions known, the Coma cluster. Our goal is to improve our understanding of the nature of these objects by comparing the broad band spectral energy distribution with population synthesis models. The data were obtained with the MEGACAM and CFH12K cameras at the CFHT. We used the resulting photometry in 5 broad band filters (u*, B, V, R, and I), that included new u*-band data, to fit spectral models. With these spectral fits we inferred a cluster membership criterium, as well as the ages, dust extinctions, and photometric types of these fLSBs. We show that about half of the Coma cluster fLSBs have a spectral energy distribution well represented in our template library while the other half present a flux deficit at ultraviolet wavelengths. Among the well represented, ~80% are probably part of the Coma cluster based on their spectral energy distribution. They are relatively young (younger than 2.3 Gyrs for 90% of the sample) non-starburst objects. The later their type, the younger fLSBs are. A significant part of the fLSBs are quite dusty objects. fLSBs are low stellar mass objects (the later their type the less massive they are), with stellar masses comparable to globular clusters for the faintest ones. Their characteristics are correlated with infall directions, confirming the disruptive origin for part of them.Comment: Accepted for publication in A&A, 10 pages, 10 figure

Photometric redshifts as a tool to study the Coma cluster galaxy populations

We investigate the Coma cluster galaxy luminosity function (GLF) at faint magnitudes, in particular in the u* band by applying photometric redshift techniques applied to deep u*, B, V, R, I images covering a region of ~1deg2 (R 24). Global and local GLFs in the B, V, R and I bands obtained with photometric redshift selection are consistent with our previous results based on a statistical background subtraction. In the area covered only by the u* image, the GLF was also derived after applying a statistical background subtraction. The GLF in the u* band shows an increase of the faint end slope towards the outer regions of the cluster (from alpha~1 in the cluster center to alpha~2 in the cluster periphery). This could be explained assuming a short burst of star formation in these galaxies when entering the cluster. The analysis of the multicolor type spatial distribution reveals that late type galaxies are distributed in clumps in the cluster outskirts, where X-ray substructures are also detected and where the GLF in the u* band is steeper.Comment: 14 pages, 2 figures in jpeg format, accepted in A&

Forming Clusters of Galaxies as the Origin of Unidentified GeV Gamma-Ray Sources

Over half of GeV gamma-ray sources observed by the EGRET experiment have not yet been identified as known astronomical objects. There is an isotropic component of such unidentified sources, whose number is about 60 in the whole sky. Here we calculate the expected number of dynamically forming clusters of galaxies emitting gamma-rays by high energy electrons accelerated in the shock wave when they form, in the framework of the standard theory of structure formation. We find that a few tens of such forming clusters should be detectable by EGRET and hence a considerable fraction of the isotropic unidentified sources can be accounted for, if about 5% of the shock energy is going into electron acceleration. We argue that these clusters are very difficult to detect in x-ray or optical surveys compared with the conventional clusters, because of their extended angular size of about 1 degree. Hence they define a new population of ``gamma-ray clusters''. If this hypothesis is true, the next generation gamma-ray telescopes such as GLAST will detect more than a few thousands of gamma-ray clusters. It would provide a new tracer of dynamically evolving structures in the universe, in contrast to the x-ray clusters as a tracer of hydrodynamically stabilized systems. We also derive the strength of magnetic field required for the extragalactic gamma-ray background by structure formation to extend up to 100 GeV as observed, that is about 10^{-5} of the shock-heated baryon energy density.Comment: Accepted by ApJ after minor revisions. Received May 9, Accepted August 3. 8 pages including 2 figure

Coma cluster object populations down to M_R~-9.5

This study follows a recent analysis of the galaxy luminosity functions and colour-magnitude red sequences in the Coma cluster (Adami et al. 2007). We analyze here the distribution of very faint galaxies and globular clusters in an east-west strip of $\sim 42 \times 7$ arcmin$^2$ crossing the Coma cluster center (hereafter the CS strip) down to the unprecedented faint absolute magnitude of M$_R \sim -9.5$. This work is based on deep images obtained at the CFHT with the CFH12K camera in the B, R, and I bands. The analysis shows that the observed properties strongly depend on the environment, and thus on the cluster history. When the CS is divided into four regions, the westernmost region appears poorly populated, while the regions around the brightest galaxies NGC 4874 and NGC 4889 (NGC 4874 and NGC 4889 being masked) are dominated by faint blue galaxies. They show a faint luminosity function slope of -2, very significantly different from the field estimates. Results are discussed in the framework of galaxy destruction (which can explain part of the very faint galaxy population) and of structures infalling on to Coma.Comment: To be published in A&