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

A family of anisotropic integral operators and behaviour of its maximal eigenvalue

We study the family of compact integral operators $\mathbf K_\beta$ in $L^2(\mathbb R)$ with the kernel K_\beta(x, y) = \frac{1}{\pi}\frac{1}{1 + (x-y)^2 + \beta^2\Theta(x, y)}, depending on the parameter $\beta >0$, where $\Theta(x, y)$ is a symmetric non-negative homogeneous function of degree $\gamma\ge 1$. The main result is the following asymptotic formula for the maximal eigenvalue $M_\beta$ of $\mathbf K_\beta$: M_\beta = 1 - \lambda_1 \beta^{\frac{2}{\gamma+1}} + o(\beta^{\frac{2}{\gamma+1}}), \beta\to 0, where $\lambda_1$ is the lowest eigenvalue of the operator $\mathbf A = |d/dx| + \Theta(x, x)/2$. A central role in the proof is played by the fact that $\mathbf K_\beta, \beta>0,$ is positivity improving. The case $\Theta(x, y) = (x^2 + y^2)^2$ has been studied earlier in the literature as a simplified model of high-temperature superconductivity.Comment: 16 page

On Hankel-type operators with discontinuous symbols in higher dimensions

We obtain an asymptotic formula for the counting function of the discrete spectrum for Hankel-type pseudo-differential operators with discontinuous symbols.Comment: 8 page

The investigation of the kinetics of hydrochemical oxidation of metal sulphides with the aim of determination of the optimal conditions for the selective extraction of molybdenum from ores

The kinetics of the oxidation of molybdenyte, pyrite and sphalerite in solutions of nitric acid, hydrogen peroxide, and sodium hypochlorite was studied by the rotating disk method. The influence of the molar concentration of reagent, pH of solution, temperature, disk rotation frequency, and duration of measurements on the specific rate of hydrochemical oxidation of sulpfides was determined. The kinetic models allowing to calculate the dissolution rate of sulphides when these parameters change simultaneously were obtained. The conditions of kinetically and diffusion-controlled processes were detected. The details of mechanism of the studied processes were revealed. The nature of intermediate solid products, the reasons and the conditions of their formation as well as the character of their influence on the kinetics of dissolution processes were determined. The probable schemes of interactions corresponding to the observable kinetic dependences were offered. The conditions of the effective and selective molybdenum leaching directly from ore without its concentration were found

RadioAstron space-VLBI project: studies of masers in star forming regions of our Galaxy and megamasers in external galaxies

Observations of the masers in the course of RadioAstron mission yielded detections of fringes for a number of sources in both water and hydroxyl maser transitions. Several sources display numerous ultra-compact details. This proves that implementation of the space VLBI technique for maser studies is possible technically and is not always prevented by the interstellar scattering, maser beaming and other effects related to formation, transfer, and detection of the cosmic maser emission. For the first time, cosmic water maser emission was detected with projected baselines exceeding Earth Diameter. It was detected in a number of star-forming regions in the Galaxy and megamaser galaxies NGC 4258 and NGC 3079. RadioAstron observations provided the absolute record of the angular resolution in astronomy. Fringes from the NGC 4258 megamaser were detected on baseline exceeding 25 Earth Diameters. This means that the angular resolution sufficient to measure the parallax of the water maser source in the nearby galaxy LMC was directly achieved in the cosmic maser observations. Very compact features with angular sizes about 20 microarcsec have been detected in star-forming regions of our Galaxy. Corresponding linear sizes are about 5-10 million kilometers. So, the major step from milli- to micro-arcsecond resolution in maser studies is done in the RadioAstron mission. The existence of the features with extremely small angular sizes is established. Further implementations of the space-VLBI maser instrument for studies of the nature of cosmic objects, studies of the interaction of extremely high radiation field with molecular material and studies of the matter on the line of sight are planned.Comment: To be published in Astrophysical Masers: Unlocking the Mysteries of the Universe, IAU Symposium 336, 201

Schr\"odinger operator on homogeneous metric trees: spectrum in gaps

The paper studies the spectral properties of the Schr\"odinger operator $A_{gV} = A_0 + gV$ on a homogeneous rooted metric tree, with a decaying real-valued potential $V$ and a coupling constant $g\ge 0$. The spectrum of the free Laplacian $A_0 = -\Delta$ has a band-gap structure with a single eigenvalue of infinite multiplicity in the middle of each finite gap. The perturbation $gV$ gives rise to extra eigenvalues in the gaps. These eigenvalues are monotone functions of $g$ if the potential $V$ has a fixed sign. Assuming that the latter condition is satisfied and that $V$ is symmetric, i.e. depends on the distance to the root of the tree, we carry out a detailed asymptotic analysis of the counting function of the discrete eigenvalues in the limit $g\to\infty$. Depending on the sign and decay of $V$, this asymptotics is either of the Weyl type or is completely determined by the behaviour of $V$ at infinity.Comment: AMS LaTex file, 47 page