Formula for the Radius of the Support of the Potential in Terms of Scattering Data

Let q(r) , r = |x| , x ∈ R3 , be a real-valued square-integrable compactly supported function, and [0, a] be the smallest interval containing the support of q(r) . Let A(α′, α) = A(α′ · α) be the corresponding scattering amplitude at a fixed positive energy, k2 = 1 . Let δ` be the phase shifts at k = 1 . It is proved that lim `→∞ ( 2`+ 1 e |δ`| 1 2` ) = a , provided that q(r) does not change sign in some, arbitrary small, neighborhood of a .