Example of two distinct potentials without point eigenvalues which have the same scattering operator with the reflection coefficientc(0)=−1

Abraham, DeFacio, and Moses and Brownstein have given examples of pairs of potentials, which support no point eigenvalues, but which have the same scattering operator. These potentials all have δ-function contributions to them. In the present paper, a pair of potentials is given, neither of which has a $\delta$-function contribution, such that they both have the same simple scattering operator. Like the earlier examples, the new potentials violate the range condition of Faddeev and Deift and Trubowitz used in the proof of the one-to-one correspondence of potentials to scattering operators. Thus the present example is an addition to the list of long-range potentials with simple eigenfunctions and scattering operators.