On the Schatten-von Neumann properties of some pseudo-differential operators

We obtain a number of explicit estimates for quasi-norms of pseudo-differential operators in the Schatten-von Neumann classes $S_q$ with $0<q\le 1$. The estimates are applied to derive semi-classical bounds for operators with smooth or non-smooth symbols.Comment: 22 page

Wiener-Hopf operators in higher dimensions: the Widom conjecture for piece-wise smooth domains

We prove a two-term quasi-classical trace asymptotic formula for the functions of multi-dimensional Wiener-Hopf operators with discontinuous symbols. The discontinuities occur on the surfaces which are assumed to be piece-wise smooth. Such a two-term formula was conjectured by H. Widom in 1982, and proved by A. V Sobolev for smooth surfaces in 2009.Comment: 15 page

On a coefficient in trace formulas for Wiener-Hopf operators

Let $a = a(\xi), \xi\in\mathbb R,$ be a smooth function quickly decreasing at infinity. For the Wiener-Hopf operator $W(a)$ with the symbol $a$, and a smooth function $g:\mathbb C\to~\mathbb C$, H. Widom in 1982 established the following trace formula: $${\rm tr}\bigl(g\bigl(W(a)\bigr) - W(g\circ a)\bigr) = \mathcal B(a; g),$$ where $\mathcal B(a; g)$ is given explicitly in terms of the functions $a$ and $g$. The paper analyses the coefficient $\mathcal B(a; g)$ for a class of non-smooth functions $g$ assuming that $a$ is real-valued. A representative example of one such function is $g(t) = |t|^{\gamma}$ with some $\gamma\in (0, 1]$.Comment: 21 page

Maser action in methanol transitions

We report the detection with the ATCA of 6.7 GHz methanol emission towards OMC-1. The source has a size between 40'' and 90'', is located to the south-east of Ori-KL and may coincide in position with the 25 GHz masers. The source may be an example of an interesting case recently predicted in theory where the transitions of traditionally different methanol maser classes show maser activity simultaneously. In addition, results of recent search for methanol masers from the 25 and 104.3 GHz transitions are reported.Comment: To appear in the Proceedings of the 2004 European Workshop: "Dense Molecular Gas around Protostars and in Galactic Nuclei", Eds. Y.Hagiwara, W.A.Baan, H.J. van Langevelde, 2004, a special issue of ApSS, Kluwer; author list has been corrected, text is unchange

Stochastic Schrodinger equations as limit of discrete filtering

We consider an open model possessing a Markovian quantum stochastic limit and derive the limit stochastic Schrodinger equations for the wave function conditioned on indirect observations using only the von Neumann projection postulate. We show that the diffusion (Gaussian) situation is universal as a result of the central limit theorem with the quantum jump (Poissonian) situation being an exceptional case. It is shown that, starting from the correponding limiting open systems dynamics, the theory of quantum filtering leads to the same equations, therefore establishing consistency of the quantum stochastic approach for limiting Markovian models.Comment: 21 pages, no figure

DISTRIBUTION OF INTEGER LATTICE POINTS IN A BALL CENTRED AT A DIOPHANTINE POINT

We study the variance of the fluctuations in the number of lattice points in a ball and in a thin spherical shell of large radius centred at a Diophantine point