On the closure of the Airault–Mckean–Moser locus for elliptic KdV potentials via Darboux transformations

We study the general elliptic KdV potentials, which can be expressed (up to adding a constant) as qₚ(z)∑ⱼ₌₁ⁿmⱼ(mⱼ+1)(z-pⱼ), mⱼ. We give an elementary proof of the theorem that the singularity p=( {m₁( m₁+1) /2}{{p₁, ,p₁}},,{mₙ(mₙ+1) /2}{{pₙ,,pₙ}% }) is contained in the closure of the elliptic Airault–Mckean–Moser locus, which was proved previously by Treibich and Verdier in the late 1980s using purely algebro-geometric methods. Our proof is based on Darboux transformations and does not use algebraic geometry. This solves an open problem posed by Gesztesy, Unterkofler, and Weikard [Trans. Amer. Math. Soc. 358 (2006), 603–656]. Some applications are also given.