L^p boundedness of the wave operator for the one dimensional Schroedinger operator

Given a one dimensional perturbed Schroedinger operator H=-(d/dx)^2+V(x) we consider the associated wave operators W_+, W_- defined as the strong L^2 limits as s-> \pm\infty of the operators e^{isH} e^{-isH_0} We prove that the wave operators are bounded operators on L^p for all 1<p<\infty, provided (1+|x|)^2 V(x) is integrable, or else (1+|x|)V(x) is integrable and 0 is not a resonance. For p=\infty we obtain an estimate in terms of the Hilbert transform. Some applications to dispersive estimates for equations with variable rough coefficients are given.Comment: 26 page

Heat Kernel Asymptotics on Homogeneous Bundles

We consider Laplacians acting on sections of homogeneous vector bundles over symmetric spaces. By using an integral representation of the heat semi-group we find a formal solution for the heat kernel diagonal that gives a generating function for the whole sequence of heat invariants. We argue that the obtained formal solution correctly reproduces the exact heat kernel diagonal after a suitable regularization and analytical continuation.Comment: 29 pages, Proceedings of the 2007 Midwest Geometry Conference in Honor of Thomas P. Branso

Observational signatures of forming young massive clusters: continuum emission from dense HII regions

Young massive clusters (YMCs) are the most massive star clusters forming in nearby galaxies and are thought to be a young analogue to the globular clusters. Understanding the formation process of YMCs leads to looking into very efficient star formation in high-redshift galaxies suggested by recent JWST observations. We investigate possible observational signatures of their formation stage, particularly when the mass of a cluster is increasing via accretion from a natal molecular cloud. To this end, we study the broad-band continuum emission from ionized gas and dust enshrouding YMCs, whose formation is followed by recent radiation-hydrodynamics simulations. We perform post-process radiative transfer calculations using simulation snapshots and find characteristic spectral features at radio and far-infrared frequencies. We show that a striking feature is long-lasting, strong free-free emission from a $\sim$ 10pc-scale HII region with a large emission measure of $\gtrsim 10^7 \mathrm{cm}^{-6} \ \mathrm{pc}$, corresponding to the mean electron density of $\gtrsim 10^3~\mathrm{cm}^{-3}$. There is a turnover feature below $\sim$ 10 GHz, a signature of the optically-thick free-free emission, often found in Galactic ultra-compact HII regions. These features come from the peculiar YMC formation process, where the cluster's gravity effectively traps photoionized gas for a long duration and enables continuous star formation within the cluster. Such large and dense HII regions show distinct distribution on the density-size diagram, apart from the standard sequence of Galactic HII regions. This is consistent with the observational trend inferred for extragalactic HII regions associated with YMCs.Comment: 12 pages, 10 figures, accepted for publication in MNRA

Hidden Order and Dimerization Transition in S=2S=2 Chains

We study ground state properties of the $S=2$ quantum antiferromagnetic chain with a bond alternation H = \sum_{j} [ 1 + \delta (-1)^j ] \mbox{\boldmath $S$}_{j} \cdot \mbox{\boldmath $S$}_{j+1} by a Quantum Monte Carlo calculation. We find that the hidden $Z_2 \times Z_2$ symmetry is broken for $0.3 < |\delta| < 0.5$ while it is unbroken in the other regions. This confirms the successive dimerization transitions first predicted by Affleck and Haldane. Our result shows that these transitions can be understood in terms of the hidden $Z_2 \times Z_2$ symmetry breaking, as was discussed using the Valence-Bond-Solid states. Furthermore, we find that the behavior of the generalized string correlation is qualitatively very similar to that in the Valence-Bond-Solid states, including the location of zeroes as a function of the angle parameter.Comment: 3 pages (LaTex with jpsj-style files (ftp://ftp.u-tokyo.ac.jp/pub/SOCIETY/JPSJ)) and 1 Postscript figur

Inverse Scattering at a Fixed Quasi-Energy for Potentials Periodic in Time

We prove that the scattering matrix at a fixed quasi--energy determines uniquely a time--periodic potential that decays exponentially at infinity. We consider potentials that for each fixed time belong to $L^{3/2}$ in space. The exponent 3/2 is critical for the singularities of the potential in space. For this singular class of potentials the result is new even in the time--independent case, where it was only known for bounded exponentially decreasing potentials.Comment: In this revised version I give a more detailed motivation of the class of potentials that I consider and I have corrected some typo