Gevrey estimates of the resolvent and sub-exponential time-decay of solutions

In this article, we study a class of non-selfadjoint Schr{\"o}dinger operators H which are perturbation of some model operator H 0 satisfying a weighted coercive assumption. For the model operator H 0 , we prove that the derivatives of the resolvent satisfy some Gevrey estimates at threshold zero. As application, we establish large time expansions of semigroups e --tH and e --itH for t > 0 with subexponential time-decay estimates on the remainder, including possible presence of zero eigenvalue and real resonances

Large-time asymptotics of solutions to the Kramers-Fokker-Planck equation with a short-range potential

In this work, we use scattering method to study the Kramers-Fokker-Planck equation with a potential whose gradient tends to zero at the infinity. For short-range potentials in dimension three, we show that complex eigenvalues do not accumulate at low-energies and establish the low-energy resolvent asymptotics. This combined with high energy pseudospectral estimates valid in more general situations gives the large-time asymptotics of the solution in weighted $L^2$ spaces

Constraints on the Asymptotic Baryon Fractions of Galaxy Clusters at Large Radii

While X-ray measurements have so far revealed an increase in the volume-averaged baryon fractions $f_b(r)$ of galaxy clusters with cluster radii $r$, $f_b(r)$ should asymptotically reach a universal value $f_b(\infty)=f_b$, provided that clusters are representative of the Universe. In the framework of hydrostatic equilibrium for intracluster gas, we have derived the necessary conditions for $f_b(\infty)=f_b$: The X-ray surface brightness profile described by the $\beta$ model and the temperature profile approximated by the polytropic model should satisfy $\gamma\approx2(1-1/3\beta)$ and $\gamma\approx1+1/3\beta$ for $\beta1$, respectively, which sets a stringent limit to the polytropic index: $\gamma<4/3$. In particular, a mildly increasing temperature with radius is required if the observationally fitted $\beta$ parameter is in the range $1/3<\beta<2/3$. It is likely that a reliable determination of the universal baryon fraction can be achieved in the small $\beta$ clusters because the disagreement between the exact and asymptotic baryon fractions for clusters with $\beta>2/3$ breaks down at rather large radii (\ga30r_c) where hydrostatic equilibrium has probably become inapplicable. We further explore how to obtain the asymptotic value $f_b(\infty)$ of baryon fraction from the X-ray measurement made primarily over the finite central region of a cluster. We demonstrate our method using a sample of 19 strong lensing clusters, which enables us to place a useful constraint on $f_b(\infty)$: $0.094\pm0.035 \leq f_b(\infty) \leq 0.41\pm0.18$. An optimal estimate of $f_b(\infty)$ based on three cooling flow clusters with $\beta = 0.142\pm0.007$ or $\Omega_M = 0.35\pm0.09$.Comment: 6 pages + 4 figures; accepted for publication in MNRA

The Effect of radiative cooling on the scale-dependence of global stellar and gas contents of groups and clusters of galaxies

It is widely believed that the global baryon content and mass-to-light ratio of groups and clusters of galaxies are a fair representative of the matter mix of the universe and therefore, can be used to reliably determine the cosmic mass density parameter Omega_M. However, this fundamental assumption is challenged by growing evidence from optical and X-ray observations that the average gas mass fraction and mass-to-light ratio increase mildly with scale from poor groups to rich clusters. Although a number of time-consuming hydrodynamical simulations combined with semi-analytic approaches have been carried out, which permit a sophisticated treatment of some complicated processes in the formation and evolution of cosmic structures, the essential physics behind the phenomenon still remains a subject of intense debate. In this Letter, using a simple analytic model, we show that radiative cooling of the hot intragroup/intracluster gas may allow one to reproduce the observed scale-dependence of the global stellar and gas mass fractions and mass-to-light ratio of groups and clusters, provided that about half of the cooled gas is converted into stars. Together with the recent success in the recovery of the entropy excess and the steepening of the X-ray luminosity-temperature relations detected in groups and clusters, radiative cooling provides a simple, unified scheme for the evolution of hot gas and the formation of stars in the largest virialized systems of the universe.Comment: 7 pages, 4 figures, accepted for publication in the June 10 2002 Issue of ApJ Let