2013/02/02 by A. G. Ramm, А. Г. Рамм, Ramm, A. G.
Engineering · Mathematics · Physics and Astronomy · #35J10 #35R30 #81Q05 #FOS: Physical sciences #Mathematical Analysis and Transform Methods #Mathematical Physics (math-ph) #Microwave Imaging and Scattering Analysis #Numerical methods in inverse problems #math-ph #math.MP #msc:35J10 #msc:35R30 #msc:81Q05
paper · pdf · doi:10.48550/arxiv.1302.5000
arxiv created 2013/02/02 · openalex publication_date 2013/02/02 · arxiv updated 2013/02/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Consider the Schrödinger operator -∇2+q q, q=q(x), x ∈ R3. Let A(β,α, k) be the corresponding scattering amplitude, k2 be the energy, α∈ S2 be the incident direction, β∈ S2 be the direction of scattered wave, S2 be the unit sphere in R3. Assume that k=k0 >0 is fixed, and α=α0 is fixed. Then the scattering data are A(β)= A(β,α0, k0)=Aq(β) is a function on S2. The following invers \textitIP: Given an arbitrary f ∈ L2(S2) and an arbitrary small number q ∈ C0∞(D), where D ∈ R3 is an arbitrary fixed domai ||Aq(β)-f(β)||L2(S2) < ε? A positive answer to this question is given. A method for constructing such a q is proposed. There are infinitely many such q, not necessarily real-valued.