1,232 research outputs found

    Uniqueness theorem for inverse scattering problem with non-overdetermined data

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    Let q(x)q(x) be real-valued compactly supported sufficiently smooth function, q∈H0β„“(Ba)q\in H^\ell_0(B_a), Ba:={x:∣xβˆ£β‰€a,x∈R3B_a:=\{x: |x|\leq a, x\in R^3 . It is proved that the scattering data A(βˆ’Ξ²,Ξ²,k)A(-\beta,\beta,k) βˆ€Ξ²βˆˆS2\forall \beta\in S^2, βˆ€k>0\forall k>0 determine qq uniquely. here A(Ξ²,Ξ±,k)A(\beta,\alpha,k) is the scattering amplitude, corresponding to the potential qq

    Uniqueness of the solution to inverse scattering problem with scattering data at a fixed direction of the incident wave

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    Let q(x)q(x) be real-valued compactly supported sufficiently smooth function. It is proved that the scattering data A(Ξ²,Ξ±0,k)A(\beta,\alpha_0,k) βˆ€Ξ²βˆˆS2\forall \beta\in S^2, βˆ€k>0,\forall k>0, determine qq uniquely. Here Ξ±0∈S2\alpha_0\in S^2 is a fixed direction of the incident plane wave

    Creating materials with a desired refraction coefficient

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    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 aa. The number of particles per unit volume around any point is prescribed, the distance between neighboring particles is O(a2βˆ’ΞΊ3)O(a^{\frac{2-\kappa}{3}}) as aβ†’0a\to 0, 0<ΞΊ<10<\kappa<1 is a fixed parameter. The total number of the embedded particle is O(aΞΊβˆ’2)O(a^{\kappa-2}). The physical properties of the particles are described by the boundary impedance ΞΆm\zeta_m of the mβˆ’thm-th particle, ΞΆm=O(aβˆ’ΞΊ)\zeta_m=O(a^{-\kappa}) as aβ†’0a\to 0. The refraction coefficient is the coefficient n2(x)n^2(x) in the wave equation [βˆ‡2+k2n2(x)]u=0[\nabla^2+k^2n^2(x)]u=0

    Creating desired potentials by embedding small inhomogeneities

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    The governing equation is [βˆ‡2+k2βˆ’q(x)]u=0[\nabla^2+k^2-q(x)]u=0 in R3\R^3. It is shown that any desired potential q(x)q(x), vanishing outside a bounded domain DD, can be obtained if one embeds into D many small scatterers qm(x)q_m(x), vanishing outside balls Bm:={x:∣xβˆ’xm∣<a}B_m:=\{x: |x-x_m|<a\}, such that qm=Amq_m=A_m in BmB_m, qm=0q_m=0 outside BmB_m, 1≀m≀M1\leq m \leq M, M=M(a)M=M(a). It is proved that if the number of small scatterers in any subdomain Ξ”\Delta is defined as N(Ξ”):=βˆ‘xmβˆˆΞ”1N(\Delta):=\sum_{x_m\in \Delta}1 and is given by the formula N(Ξ”)=∣V(a)βˆ£βˆ’1βˆ«Ξ”n(x)dx[1+o(1)]N(\Delta)=|V(a)|^{-1}\int_{\Delta}n(x)dx [1+o(1)] as aβ†’0a\to 0, where V(a)=4Ο€a3/3V(a)=4\pi a^3/3, then the limit of the function uM(x)u_{M}(x), lim⁑aβ†’0UM=ue(x)\lim_{a\to 0}U_M=u_e(x) does exist and solves the equation [βˆ‡2+k2βˆ’q(x)]u=0[\nabla^2+k^2-q(x)]u=0 in R3\R^3, where q(x)=n(x)A(x)q(x)=n(x)A(x),and A(xm)=AmA(x_m)=A_m. The total number MM of small inhomogeneities is equal to N(D)N(D) and is of the order O(aβˆ’3)O(a^{-3}) as aβ†’0a\to 0. A similar result is derived in the one-dimensional case

    Recovery of a quarkonium system from experimental data

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    For confining potentials of the form q(r)=r+p(r), where p(r) decays rapidly and is smooth for r>0, it is proved that q(r) can be uniquely recovered from the data {E_j,s_j}, where E_j are the bound states energies and s_j are the values of u'_j(0), and u_j(r) are the normalized eigenfunctions of the problem -u_j" +q(r)u_j=E_ju_j, r>0, u_j(0)=0, ||u_j||=1, where the norm is L^2(0, \infty) norm. An algorithm is given for recovery of p(r) from few experimental data
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