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 n q_p.z/ ´ X m_j .m_j C 1/}.z - p_j /; m_j 2 N: j D1 We give an elementary proof of the theorem that the singularity m1.m1C1/=2 mn.mnC1/=2 ‚ …„ ƒ ‚ …„ ƒ p D . p₁;:::; p₁ ;:::; p_n;:::; p_n / 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.