Generalized primitive potentials

In our previous work, we introduced a new class of bounded potentials of the one-dimensional Schr\"odinger operator on the real axis, and a corresponding family of solutions of the KdV hierarchy. These potentials, which we call primitive, are obtained as limits of rapidly decreasing reflectionless potentials, or multisoliton solutions of KdV. In this note, we introduce generalized primitive potentials, which are obtained as limits of all rapidly decreasing potentials of the Schr\"odinger operator. These potentials are constructed by solving a contour problem, and are determined by a pair of positive functions on a finite interval and a functional parameter on the real axis.