Application of an Inverse Problem for Symmetric Periodic Potentials

This paper gives an algorithm for constructing symmetric periodic potential q(x) of Hill's equation from given roots of $\Delta(\lambda)-2=0$. The problem arises from synthesizing dual-mode ring electrical circuits. In the presented method, the discriminant $\Delta(\lambda)$ of Hill's equation is determined from the roots of $\Delta(\lambda)-2=0$ at first, and then q(x) is constructed from the roots of $\Delta(\lambda)\pm 2=0$ and $\Delta(\lambda)=0$. Examples are provided to verify the algorithm.