Analyticity and the NcN_c counting rule of SS matrix poles

By studying $\pi\pi$ scattering amplitudes in the large $N_c$ limit, we clarify the $N_c$ dependence of the $S$ matrix pole position. It is demonstrated that analyticity and the $N_c$ counting rule exclude the existence of $S$ matrix poles with ${\cal M}, \Gamma\sim O(1)$. Especially the properties of $\sigma$ and $f_0(980)$ with respect to the $1/N_c$ expansion are discussed. We point out that in general tetra-quark resonances do not exist.Comment: This paper replaces hep-ph/0412175. The latter is withdraw

Levinson's theorem for the Schr\"{o}dinger equation in two dimensions

Levinson's theorem for the Schr\"{o}dinger equation with a cylindrically symmetric potential in two dimensions is re-established by the Sturm-Liouville theorem. The critical case, where the Schr\"{o}dinger equation has a finite zero-energy solution, is analyzed in detail. It is shown that, in comparison with Levinson's theorem in non-critical case, the half bound state for $P$ wave, in which the wave function for the zero-energy solution does not decay fast enough at infinity to be square integrable, will cause the phase shift of $P$ wave at zero energy to increase an additional $\pi$.Comment: Latex 11 pages, no figure and accepted by P.R.A (in August); Email: [email protected], [email protected]

The Relativistic Levinson Theorem in Two Dimensions

In the light of the generalized Sturm-Liouville theorem, the Levinson theorem for the Dirac equation in two dimensions is established as a relation between the total number $n_{j}$ of the bound states and the sum of the phase shifts $\eta_{j}(\pm M)$ of the scattering states with the angular momentum $j$: $$\eta_{j}(M)+\eta_{j}(-M)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$$ $$~~~=\left\{\begin{array}{ll} (n_{j}+1)\pi &{\rm when~a~half~bound~state~occurs~at}~E=M ~~{\rm and}~~ j=3/2~{\rm or}~-1/2\\ (n_{j}+1)\pi &{\rm when~a~half~bound~state~occurs~at}~E=-M~~{\rm and}~~ j=1/2~{\rm or}~-3/2\\ n_{j}\pi~&{\rm the~rest~cases} . \end{array} \right.$$ \noindent The critical case, where the Dirac equation has a finite zero-momentum solution, is analyzed in detail. A zero-momentum solution is called a half bound state if its wave function is finite but does not decay fast enough at infinity to be square integrable.Comment: Latex 14 pages, no figure, submitted to Phys.Rev.A; Email: [email protected], [email protected]

Hierarchical build-up of galactic bulges and the merging rate of supermassive binary black holes

The hierarchical build-up of galactic bulges should lead to the build-up of present-day supermassive black holes by a mixture of gas accretion and merging of supermassive black holes. The tight relation between black hole mass and stellar velocity dispersion is thereby a strong argument that the supermassive black holes in merging galactic bulges do indeed merge. Otherwise the ejection of supermassive black holes by gravitational slingshot would lead to excessive scatter in this relation. At high redshift the coalescence of massive black hole binaries is likely to be driven by the accretion of gas in the major mergers signposted by optically bright QSO activity. If massive black holes only form efficiently by direct collapse of gas in deep galactic potential wells with v_c > 100 km/s as postulated in the model of Kauffmann & Haehnelt (2000) LISA expects to see event rates from the merging of massive binary black holes of about 0.1-1 yr^{-1} spread over the redshift range 0 < z < 5. If, however, the hierarchical build-up of supermassive black holes extends to pre-galactic structures with significantly shallower potential wells event rates may be as high as 10-100 yr^{-1} and will be dominated by events from redshift z > 5.Comment: 8 pages, 4 postscript figures. Proceedings of the 4th International LISA Symposium, Penn State University, 19-24 July 2002, ed. L S Fin

Electronic properties of silica nanowires

Thin nanowires of silicon oxide were studied by pseudopotential density functional electronic structure calculations using the generalized gradient approximation. Infinite linear and zigzag Si-O chains were investigated. A wire composed of three-dimensional periodically repeated Si4O8 units was also optimized, but this structure was found to be of limited stability. The geometry, electronic structure, and Hirshfeld charges of these silicon oxide nanowires were computed. The results show that the Si-O chain is metallic, whereas the zigzag chain and the Si4O8 nanowire are insulators

Levinson's Theorem for the Klein-Gordon Equation in Two Dimensions

The two-dimensional Levinson theorem for the Klein-Gordon equation with a cylindrically symmetric potential $V(r)$ is established. It is shown that $N_{m}\pi=\pi (n_{m}^{+}-n_{m}^{-})= [\delta_{m}(M)+\beta_{1}]-[\delta_{m}(-M)+\beta_{2}]$, where $N_{m}$ denotes the difference between the number of bound states of the particle $n_{m}^{+}$ and the ones of antiparticle $n_{m}^{-}$ with a fixed angular momentum $m$, and the $\delta_{m}$ is named phase shifts. The constants $\beta_{1}$ and $\beta_{2}$ are introduced to symbol the critical cases where the half bound states occur at $E=\pm M$.Comment: Revtex file 14 pages, submitted to Phys. Rev.

Structure formation in a colliding flow: The Herschel view of the Draco nebula

The Draco nebula is a high Galactic latitude interstellar cloud likely to have been formed by the collision of a Galactic halo cloud entering the disk of the Milky Way. Such conditions are ideal to study the formation of cold and dense gas in colliding flows of warm gas. We present Herschel-SPIRE observations that reveal the fragmented structure of the interface between the infalling cloud and the Galactic layer. This front is characterized by a Rayleigh-Taylor instability structure. From the determination of the typical length of the periodic structure (2.2 pc) we estimated the gas kinematic viscosity and the turbulence dissipation scale (0.1 pc) that is compatible with that expected if ambipolar diffusion is the main mechanism of energy dissipation in the WNM. The small-scale structures of the nebula are typical of that seen in some molecular clouds. The gas density has a log-normal distribution with an average value of $10^3$ cm$^{-3}$. The size of the structures is 0.1-0.2 pc but this estimate is limited by the resolution of the observations. The mass ranges from 0.2 to 20 M$_{\odot}$ and the distribution of the more massive clumps follows a power law $dN/d\log(M) \sim M^{-1.4}$. We identify a mass-size relation with the same exponent as that found in GMCs ($M\sim L^{2.3}$) but only 15% of the mass of the cloud is in gravitationally bound structures. We conclude that the increase of pressure in the collision is strong enough to trigger the WNM-CNM transition caused by the interplay between turbulence and thermal instability as self-gravity is not dominating the dynamics.Comment: 16 pages, A&A, in pres

A general condition of inflationary cosmology on trans-Planckian physics

We consider a more general initial condition satisfying the minimal uncertainty relationship. We calculate the power spectrum of a simple model in inflationary cosmology. The results depend on perturbations generated below a fundamental scale, e.g. the Planck scale.Comment: 7 pages, References adde