# On symmetric primitive potentials

## TL;DR

This paper studies symmetric primitive potentials for the Schrödinger operator, showing how equal dressing functions lead to symmetric and elliptic one-gap potentials, with an analytic power series construction.

## Contribution

It introduces a reduction where dressing functions are equal, demonstrating the resulting symmetry and providing an explicit power series method for potential computation.

## Key findings

- Equal dressing functions produce symmetric primitive potentials.
- The potential reduces to the elliptic one-gap potential when dressing functions are one.
- Analytic power series can be used to compute the potential explicitly.

## Abstract

The concept of a primitive potential for the Schroedinger operator on the line was introduced in [2,3,4]. 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 paper, 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.