Semiclassical limit of the scattering cross section as a distribution

We consider quantum scattering from a compactly supported potential $q$. The semiclassical limit amounts to letting the wavenumber $k \to \infty$ while rescaling the potential as $k^2 q$ (alternatively, one can scale Planck's constant $\hbar \searrow 0$). It is well-known that, under appropriate conditions, for \om \in \bbS_{n-1} such that there is exactly one outgoing ray with direction \om (in the sense of geometric optics), the differential scattering cross section |f(\om,k)|^{2} tends to the classical differential cross section |f_{cl}(\om)|^2 as $k \uparrow \infty$. It is also clear that the same can not be true if there is more than one outgoing ray with direction \om or for \emph{nonregular} directions (including the forward direction $\theta_0$). However, based on physical intuition, one could conjecture $|f|^2 \to |f_{cl}|^2 + \sigma_{cl} \delta_{\theta_0}$ where $|f_{cl}|^2$ is the classical cross section and $\delta_{\theta_0}$ is the Dirac measure supported at the forward direction $\theta_0$. The aim of this paper is to prove this conjecture

Uniqueness in potential scattering with reduced near field data

We consider inverse potential scattering problems where the source of the incident waves is located on a smooth closed surface outside of the inhomogeneity of the media. The scattered waves are measured on the same surface at a fixed value of the energy. We show that this data determines the bounded potential uniquely.Comment: arXiv admin note: substantial text overlap with arXiv:1501.0374

Solution of the initial value problem for the focusing Davey-Stewartson II system

We consider a focusing Davey-Stewartson system and construct the solution of the Cauchy problem in the possible presence of exceptional points (and/or curves)

Classification of Singularities and Bifurcations of Critical Points of Even Functions

Singularities of even smooth functions are studied. A classification of singular points which appear in typical parametric families of even functions with at most five parameters is given. Bifurcations of singular points near a caustic value of the parameter are also studied. A determinant for singularity types and conditions for versal deformations are given in terms of partial derivatives (not requiring a preliminary reduction to a canonical form)

Applications of elliptic operator theory to the isotropic interior transmission eigenvalue problem

The paper concerns the isotropic interior transmission eigenvalue (ITE) problem. This problem is not elliptic, but we show that, using the Dirichlet-to-Neumann map, it can be reduced to an elliptic one. This leads to the discreteness of the spectrum as well as to certain results on possible location of the transmission eigenvalues. If the index of refraction $\sqrt{n(x)}$ is real, we get a result on the existence of infinitely many positive ITEs and the Weyl type lower bound on its counting function. All the results are obtained under the assumption that $n(x)-1$ does not vanish at the boundary of the obstacle or it vanishes identically, but its normal derivative does not vanish at the boundary. We consider the classical transmission problem as well as the case when the inhomogeneous medium contains an obstacle. Some results on the discreteness and localization of the spectrum are obtained for complex valued $n(x)$.Comment: A small correction is made in formulas (11), (12) after the paper was published in "Inverse Problems", 29, 201

A priori estimates for high frequency scattering by obstacles of arbitrary shape

High frequency estimates for the Dirichlet-to-Neumann and Neumann-to-Dirichlet operators are obtained for the Helmholtz equation in the exterior of bounded obstacles. These a priori estimates are used to study the scattering of plane waves by an arbitrary bounded obstacle and to prove that the total cross section of the scattered wave does not exceed four geometrical cross sections of the obstacle in the limit as the wave number $k\to \infty$. This bound of the total cross section is sharp.Comment: We corrected a couple of essential misprint

Explicit representation of Green function for 3Dimensional exterior Helmholtz equation

We have constructed a sequence of solutions of the Helmholtz equation forming an orthogonal sequence on a given surface. Coefficients of these functions depend on an explicit algebraic formulae from the coefficient of the surface. Moreover, for exterior Helmholtz equation we have constructed an explicit normal derivative of the Dirichlet Green function. In the same way the Dirichlet-to-Neumann operator is constructed. We proved that normalized coefficients are uniformly bounded from zero

Examples of Admissible Simplification of Mathematical Theories

"Mathematicians, like physicists, are pushed by a strong fascination. Research in mathematics is hard, it is intellectually painful even if it is rewarding, and you would not do it without some strong urge." [D. Ruelle]. We shall give some examples from our experience, when we were able to simplify some serious mathematical models to make them understandable by children, preserving both aesthetic and intellectual value. The latter is in particularly measured by whether a given simplification allows setting a sufficient list of problems feasible for school students.Comment: This article based on the poster presented at the XVI International Congress on Mathematical Physics: August 3-8, 2009; Prague, Czech Republi

Remarks on stochastic automatic adjoint differentiation and financial models calibration

In this work, we discuss the Automatic Adjoint Differentiation (AAD) for functions of the form $G=\frac{1}{2}\sum_1^m (Ey_i-C_i)^2$, which often appear in the calibration of stochastic models. { We demonstrate that it allows a perfect SIMD\footnote{Single Input Multiple Data} parallelization and provide its relative computational cost. In addition we demonstrate that this theoretical result is in concordance with numeric experiments.

Perturbative estimates on the transport cross section in quantum scattering by hard obstacles

The quantum scattering by smooth bodies is considered for small and large values of $kd$, with $k$ the wavenumber and $d$ the scale of the body. In both regimes, we prove that the forward scattering exceeds the backscattering. For high $k$, we need to assume that the body is strictly convex.Comment: 10 pages, 1 figur