203 research outputs found
Semiclassical limit of the scattering cross section as a distribution
We consider quantum scattering from a compactly supported potential . The
semiclassical limit amounts to letting the wavenumber while
rescaling the potential as (alternatively, one can scale Planck's
constant ). 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 . 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
). However, based on physical intuition, one could conjecture where is the
classical cross section and is the Dirac measure supported
at the forward direction . 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
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 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 .Comment: A small correction is made in formulas (11), (12) after the paper was
published in "Inverse Problems", 29, 201
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
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 .
This bound of the total cross section is sharp.Comment: We corrected a couple of essential misprint
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 , 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 , with the wavenumber and the scale of the body. In both
regimes, we prove that the forward scattering exceeds the backscattering. For
high , we need to assume that the body is strictly convex.Comment: 10 pages, 1 figur
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