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By Reed M., Simon B.

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The answer is: It does not, in general. , [R44]): if q(x) is unknown on the half of the interval 0 ≤ x ≤ 12 , then the knowledge of all {λj }∀j determines q(x) on the remaining half of the interval 12 < x < 1 uniquely. There is an exceptional result, however, due to Ambarzumian (1929), which says that if u (0) = u (1) = 0, then the set of the corresponding eigenvalues {μj }∀j determines q uniquely ([R44]). A multidimensional generalization of this old result is given in [R44]. Let us define the spectral function of the selfadjoint Dirichlet operator d2 l = − dx 2 + q(x) in L(R+ ), R+ = [0, ∞).

1. BASIC DEFINITIONS. EXAMPLES 17 The error of this algorithm is also given in [R44], see also [R31]. Also a stable estimate of a q ∈ Qa is obtained in [R44] when the noisy data Aδ (α , α) are given, sup |Aδ (α , α) − A(α , α)| ≤ δ. 28) α,α ∈S 2 Recently ([R65]) the author has formulated and solved the following inverse scattering-type problem with fixed k = k0 > 0 and fixed α = α0 data A(β) := A(β, α0 , k0 ), known for all β ∈ S 2 . The problem consists in finding a potential q ∈ L2 (D), such that the corresponding scattering amplitude Aq (β, α0 , k0 ) := A(β) would approximate an arbitrary given function f (β) ∈ L2 (S 2 ) with arbitrary accuracy: ||f (β) − A(β)||L2 (S 2 ) < , where > 0 is an a priori given, arbitrarily small, fixed number.

The inverse obstacle scattering problem consists of finding S and the boundary condition (the Dirichlet, Neumann, or Robin) on S given the scattering amplitude on a subset of S 2 ×S 2 ×R+ . The first basic uniqueness theorem for this inverse problem has been obtained by M. Schiffer did not publish his beautiful proof). 30) holds and that A(α , α, k) is known for a fixed α = α0 , all α ∈ S and all k > 0. 1. BASIC DEFINITIONS. EXAMPLES 19 The second basic uniqueness theorem has been obtained in 1985 ([R13]) by the author, who did not preassume the boundary condition on S and proved the following uniqueness theorem: The scattering data A(α , α), given at an arbitrary fixed k = k0 > 0 for all α ∈ S12 and α ∈ S22 , determine uniquely the surface S and the boundary condition on S of Dirichlet, Neumann, or Robin type.

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Methods of modern mathematical physics, Scattering Theory by Reed M., Simon B.


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