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

Recently A. G. Ramm (1999) has shown that a subset of phase shifts $\delta_l$, $l=0,1,\ldots$, determines the potential if the indices of the known shifts satisfy the Müntz condition $\sum_{l\neq0,l\in L}\frac{1}{l}=\infty$. 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.