On symmetric primitive potentials

The concept of a primitive potential for the Schrödinger operator on the line was introduced in Dyachenko et al. (2016, Phys. D, 333, 148–156), Zakharov, Dyachenko et al. (2016, Lett. Math. Phys., 106, 731–740) and Zakharov, Zakharov et al. (2016, Phys. Lett. A, 380, 3881–3885). Such a potential is determined by a pair of positive functions on a finite interval, called the dressing functions, which are not uniquely determined by the potential. The potential is constructed by solving a contour problem on the complex plane. In this article, we consider a reduction where the dressing functions are equal. We show that in this case, the resulting potential is symmetric, and describe how to analytically compute the potential as a power series. In addition, we establish that if the dressing functions are both equal to one, then the resulting primitive potential is the elliptic one-gap potential.