Radiation diffusion in a medium with a strongly elongated scattering indicatrix

Approximation method for calculating radiation diffusion in medium with elongated scattering matri

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

Planetary atmosphere albedo

Computation of plane and spherical albedo of planetary atmosphere

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