Exponential instability in an inverse problem for the Schrödinger equation
نویسندگان
چکیده
منابع مشابه
Exponential instability in an inverse problem for Schrödinger equation
We consider the problem of the determination of the potential from the Dirichlet to Neumann map of the Schrödinger operator. We show that this problem is severely ill posed. The results extend to the electrical impedance tomography. They show that the logarithmic stability results of Alessandrini are optimal.
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Let (− k 2)u = −u + q(x)u − k 2 u = δ(x), x ∈ R, ∂u ∂|x| − iku → 0, |x| → ∞. Assume that the potential q(x) is real-valued and compactly supported: q(x) = q(x), q(x) = 0 for |x| ≥ 1, 1 −1 |q|dx < ∞, and that q(x) produces no bound states. Let u(−1, k) and u(1, k) ∀k > 0 be the data. Theorem.Under the above assumptions these data determine q(x) uniquely.
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15 صفحه اولExponential instability in inverse problems for the Schrödinger equation
We give an overview of known results on stability and instability in two problems: the Gel’fand inverse boundary value problem and the inverse scattering problem(3D). In particular, we present our new instability estimates, see [M. Isaev, Exponential instability in the Gel’fand inverse problem on the energy intervals, J. Inverse Ill-Posed Probl., Vol. 19(3), 2011, 453-473] and [M. Isaev, Expone...
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ژورنال
عنوان ژورنال: Inverse Problems
سال: 2001
ISSN: 0266-5611,1361-6420
DOI: 10.1088/0266-5611/17/5/313