The uniqueness of the solution of the Schrodinger equation with discontinuous coefficients

Consider the Schroeodinger equation: - Du(x) - l(x)u + s(x)u = 0, where D is the Laplacian, l(x) > 0 and s(x) is dominated by l(x). We shall extend the celebrated Kato's result on the asymptotic behavior of the solution to the case where l(x) has unbounded discontinuity. The result will be used to establish the limiting absorption principle for a class of reduced wave operators with discontinuous coefficients.Comment: 29 (twenty-nine) pages; no figures; to appear in Reviews of Mathematical Physic

ASCA Measurements of Metallicity and Temperature Distributions in Three Clusters: A4059, MKW 3s and 2A 0335+096

We present ASCA results on the distributions of metallicity and temperature in 3 bright near-by clusters: A4059, MKW 3s and 2A 0335+096. A significant gradient in the metal abundance is detected in A4059, while other clusters suggest similar gradients with low significance. These features together with recent results on AWM 7 and Perseus clusters suggest that metals injected in the ICM are not effectively mixed in the cluster space. Analysis of the GIS data, without explicitly including the cooling flow model, shows no substantial temperature drop at large radii (at half the virial radii) for the 3 systems. This is contrary to the recent results for 30 clusters by Markevitch et al. (1998) who assume rather strong cooling flows. The gas mass fraction of these clusters are 10-15% within 1 Mpc, which suggests that baryonic fraction of about 20% including the stellar mass is a common level in galaxy groups and clusters.Comment: 25 pages, 21 figures; accepted by PASJ; uses PASJ95.sty, PASJ2.sty, and PASJadd.st

X-ray Diagnostics of Thermal Conditions of the Hot Plasmas in the Centaurus Cluster

X-ray data of the Centaurus cluster, obtained with {\it XMM-Newton} for 45 ksec, were analyzed. Deprojected EPIC spectra from concentric thin shell regions were reproduced equally well by a single-phase plasma emission model, or by a two-phase model developed by {\it ASCA}, both incorporating cool (1.7--2.0 keV) and hot ($\sim 4$ keV) plasma temperatures. However, EPIC spectra with higher statistics, accumulated over 3-dimentional thick shell regions, were reproduced better by the two-phase model than by the singe-phase one. Therefore, hot and cool plasma phases are inferred to co-exist in the cluster core region within $\sim 70$ kpc. The iron and silicon abundances of the plasma were reconfirmed to increase significantly towards the center, while that of oxygen was consistent with being radially constant. The implied non-solar abundance ratios explains away the previously reported excess X-ray absorption from the central region. Although an additional cool ($\sim 0.7$ keV) emission was detected within $\sim 20$ kpc of the center, the RGS data gave tight upper limits on any emission with a tempeartures below $\sim 0.5$ keV. These results are compiled into a magnetosphere model, which interprets the cool phase as confined within closed magnetic loops anchored to the cD galaxy. When combined with so-called Rosner-Tucker-Vaiana mechanism which applies to solar coronae, this model can potentially explain basic properties of the cool phase, including its temperature and thermal stability.Comment: 53 pages, 11 figures, accepted for publication in Astrophysical Journa

Optical Hall Effect in the Integer Quantum Hall Regime

Optical Hall conductivity $\sigma_{xy}(\omega)$ is measured from the Faraday rotation for a GaAs/AlGaAs heterojunction quantum Hall system in the terahertz frequency regime. The Faraday rotation angle ($\sim$ fine structure constant $\sim$ mrad) is found to significantly deviate from the Drude-like behavior to exhibit a plateau-like structure around the Landau-level filling $\nu=2$. The result, which fits with the behavior expected from the carrier localization effect in the ac regime, indicates that the plateau structure, although not quantized, still exists in the terahertz regime.Comment: 4 pages, 4 figure

Detection of an X-Ray Hot Region in the Virgo Cluster of Galaxies with ASCA

Based on mapping observations with ASCA, an unusual hot region with a spatial extent of 1 square degree was discovered between M87 and M49 at a center coordinate of R. A. = 12h 27m 36s and Dec. = $9^\circ18'$ (J2000). The X-ray emission from the region has a 2-10 keV flux of $1 \times 10^{-11}$ ergs s$^{-1}$ cm$^{-2}$ and a temperature of $kT \gtrsim 4$ keV, which is significantly higher than that in the surrounding medium of $\sim 2$ keV. The internal thermal energy in the hot region is estimated to be $V n k T \sim 10^{60}$ ergs with a gas density of $\sim 10^{-4}$ cm$^{-3}$. A power-law spectrum with a photon index $1.7-2.3$ is also allowed by the data. The hot region suggests there is an energy input due to a shock which is probably caused by the motion of the gas associated with M49, infalling toward the M87 cluster with a velocity $\gtrsim 1000$ km s$^{-1}$.Comment: 12 pages, 3 figures, accepted to ApJ

X-ray Spectroscopy of the Cluster of Galaxies Abell 1795 with XMM-Newton

The initial results from XMM-Newton observations of the rich cluster of galaxies Abell 1795 are presented. The spatially-resolved X-ray spectra taken by the European Photon Imaging Cameras (EPIC) show a temperature drop at a radius of $\sim 200$ kpc from the cluster center, indicating that the ICM is cooling. Both the EPIC and the Reflection Grating Spectrometers (RGS) spectra extracted from the cluster center can be described by an isothermal model with a temperature of $\sim 4$ keV. The volume emission measure of any cool component ($<1$ keV) is less than a few % of the hot component at the cluster center. A strong OVIII Lyman-alpha line was detected with the RGS from the cluster core. The O abundance and its ratio to Fe at the cluster center is 0.2--0.5 and 0.5--1.5 times the solar value, respectively.Comment: Accepted: A&A Letters, 2001, 6 page

The Structure of the X-Ray Emitting Gas in the Hydra-A Cluster of Galaxies

The temperature and abundance structure in the intracluster medium (ICM) of the Hydra-A cluster of galaxies is studied with ASCA and ROSAT. The effect of the large extended outskirts in the point-spread function of the X-Ray Telescope on ASCA is included in this analysis. In the X-ray brightness profile, the strong central excess above a single beta-model, identified in the Einstein and ROSAT data, is also found in the harder energy band (>4keV). A simultaneous fit of five annular spectra taken with the GIS instrument shows a radial distribution of the temperature and metal abundance. A significant central enhancement in the abundance distribution is found, while the temperature profile suggests that the ICM is approximately isothermal with the temperature of ~3.5keV. The ROSAT PSPC spectrum in the central 1'.5 region indicates a significantly lower temperature than the GIS result. A joint analysis of the GIS and PSPC data reveals that the spectra can be described by a two temperature model as well as by a cooling flow model. In both cases, the hot phase gas with the temperature of ~3.5keV occupies more than 90% of the total emission measure within 1'.5 from the cluster center. The estimated mass of the cooler (0.5-0.7keV) component is ~2-6 x 10^9 M_solar, which is comparable to the mass of hot halos seen in non-cD ellipticals. The cooling flow model gives the mass deposition rate of 60+-30 M_solar/yr, an order of magnitude lower than the previous estimation.Comment: 27 pages, 14 figures, AAS LATEX macros v4.0, to appear in The Astrophysical Journa

A microscopic derivation of the quantum mechanical formal scattering cross section

We prove that the empirical distribution of crossings of a "detector'' surface by scattered particles converges in appropriate limits to the scattering cross section computed by stationary scattering theory. Our result, which is based on Bohmian mechanics and the flux-across-surfaces theorem, is the first derivation of the cross section starting from first microscopic principles.Comment: 28 pages, v2: Typos corrected, layout improved, v3: Typos corrected. Accepted for publication in Comm. Math. Phy

An evolution equation as the WKB correction in long-time asymptotics of Schrodinger dynamics

We consider 3d Schrodinger operator with long-range potential that has short-range radial derivative. The long-time asymptotics of non-stationary problem is studied and existence of modified wave operators is proved. It turns out, the standard WKB correction should be replaced by the solution to certain evolution equation.Comment: This is a preprint of an article whose final and definitive form has been published in Comm. Partial Differential Equations, available online at http://www.informaworld.co