Uniqueness theorem for inverse scattering problem with non-overdetermined data

Let $q(x)$ be real-valued compactly supported sufficiently smooth function, $q\in H^\ell_0(B_a)$, $B_a:=\{x: |x|\leq a, x\in R^3$ . It is proved that the scattering data $A(-\beta,\beta,k)$ $\forall \beta\in S^2$, $\forall k>0$ determine $q$ uniquely. here $A(\beta,\alpha,k)$ is the scattering amplitude, corresponding to the potential $q$

The vacuum backreaction on a pair creating source

Solution is presented to the simplest problem about the vacuum backreaction on a pair creating source. The backreaction effect is nonanalytic in the coupling constant and restores completely the energy conservation law. The vacuum changes the kinematics of motion like relativity theory does and imposes a new upper bound on the velocity of the source.Comment: 9 pages including 2 figures. Latex 2.09. Figures by Metafont, 300 dpi. Keep all files in a separate director

Creating materials with a desired refraction coefficient

A method is given for creating material with a desired refraction coefficient. The method consists of embedding into a material with known refraction coefficient many small particles of size $a$. The number of particles per unit volume around any point is prescribed, the distance between neighboring particles is $O(a^{\frac{2-\kappa}{3}})$ as $a\to 0$, $0<\kappa<1$ is a fixed parameter. The total number of the embedded particle is $O(a^{\kappa-2})$. The physical properties of the particles are described by the boundary impedance $\zeta_m$ of the $m-th$ particle, $\zeta_m=O(a^{-\kappa})$ as $a\to 0$. The refraction coefficient is the coefficient $n^2(x)$ in the wave equation $[\nabla^2+k^2n^2(x)]u=0$

Unification Theory of Angular Magnetoresistance Oscillations in Quasi-One-Dimensional Conductors

We present a unification theory of angular magnetoresistance oscillations, experimentally observed in quasi-one-dimensional organic conductors, by solving the Boltzmann kinetic equation in the extended Brillouin zone. We find that, at commensurate directions of a magnetic field, resistivity exhibits strong minima. In two limiting cases, our general solution reduces to the results, previously obtained for the Lebed Magic Angles and Lee-Naughton-Lebed oscillations. We demonstrate that our theoretical results are in good qualitative and quantitative agreement with the existing measurements of resistivity in (TMTSF)$_2$ClO$_4$ conductor.Comment: 6 pages, 2 figure

High-Resolution 4.7 Micron Keck/NIRSPEC Spectra of Protostars. II. Detection of the ^(13)CO Isotope in Icy Grain Mantles

The high-resolution (R = 25,000) infrared M-band spectrum of the massive protostar NGC 7538 IRS 9 shows a narrow absorption feature at 4.779 μm (2092.3 cm^(-1)) that we attribute to the vibrational stretching mode of the ^(13)CO isotope in pure CO icy grain mantles. This is the first detection of ^(13)CO in icy grain mantles in the interstellar medium. The ^(13)CO band is a factor of 2.3 narrower than the apolar component of the ^(12)CO band. With this in mind, we discuss the mechanisms that broaden solid-state absorption bands. It is shown that ellipsoidally shaped pure CO grains fit the bands of both isotopes at the same time. Slightly worse but still reasonable fits are also obtained by CO embedded in N_2-rich ices and thermally processed O_2-rich ices. In addition, we report new insights into the nature and evolution of interstellar CO ices by comparing the very high resolution multicomponent solid ^(12)CO spectrum of NGC 7538 IRS 9 with that of the previously studied low-mass source L1489 IRS. The narrow absorption of apolar CO ices is present in both spectra but much stronger in NGC 7538 IRS 9. It is superposed on a smooth broad absorption feature well fitted by a combination of CO_2 and H_2O-rich laboratory CO ices. The abundances of the latter two ices, scaled to the total H_2O ice column, are the same in both sources. We thus suggest that thermal processing manifests itself as evaporation of apolar ices only and not the formation of CO_2 or polar ices. Finally, the decomposition of the ^(12)CO band is used to derive the ^(12)CO/^(13)CO abundance ratio in apolar ices. A ratio of ^(12)CO/^(13)CO = 71 ± 15 (3 σ) is deduced, in good agreement with gas-phase CO studies (~77) and the solid ^(12)CO_2/^(13)CO_2 ratio of 80 ± 11 found in the same line of sight. The implications for the chemical path along which CO_2 is formed are discussed

Creating desired potentials by embedding small inhomogeneities

The governing equation is $[\nabla^2+k^2-q(x)]u=0$ in $\R^3$. It is shown that any desired potential $q(x)$, vanishing outside a bounded domain $D$, can be obtained if one embeds into D many small scatterers $q_m(x)$, vanishing outside balls $B_m:=\{x: |x-x_m|<a\}$, such that $q_m=A_m$ in $B_m$, $q_m=0$ outside $B_m$, $1\leq m \leq M$, $M=M(a)$. It is proved that if the number of small scatterers in any subdomain $\Delta$ is defined as $N(\Delta):=\sum_{x_m\in \Delta}1$ and is given by the formula $N(\Delta)=|V(a)|^{-1}\int_{\Delta}n(x)dx [1+o(1)]$ as $a\to 0$, where $V(a)=4\pi a^3/3$, then the limit of the function $u_{M}(x)$, $\lim_{a\to 0}U_M=u_e(x)$ does exist and solves the equation $[\nabla^2+k^2-q(x)]u=0$ in $\R^3$, where $q(x)=n(x)A(x)$,and $A(x_m)=A_m$. The total number $M$ of small inhomogeneities is equal to $N(D)$ and is of the order $O(a^{-3})$ as $a\to 0$. A similar result is derived in the one-dimensional case