Inverse scattering with fixed energy and an inverse eigenvalue problem on the half-line

Abstract

Recently A. G. Ramm (1999) has shown that a subset of phase shifts δ l \delta _l , l = 0 , 1 , … l=0,1,\ldots , determines the potential if the indices of the known shifts satisfy the Müntz condition ∑ l ≠ 0 , l ∈ L 1 l = ∞ \sum _{l\neq 0,l\in L}\frac {1}{l}=\infty . We prove the necessity of this condition in some classes of potentials. The problem is reduced to an inverse eigenvalue problem for the half-line Schrödinger operators.