On symmetric primitive potentials. Issue 1 (2nd August 2019) - Nabelek, Patrik; Zakharov, Dmitry; Zakharov, Vladimir
Abstract: 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.