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

Zhijie Chen; Ting-Jung Kuo; Chang-Shou Lin

doi:10.4171/jst/529

JournalsjstVol. 14, No. 4pp. 1475–1512

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

Zhijie Chen
Tsinghua University, Beijing, P. R. China
Ting-Jung Kuo
National Taiwan Normal University, Taipei City, Taiwan
Chang-Shou Lin
National Taiwan University, Taipei City, Taiwan

Download PDF

This article is published open access under our Subscribe to Open model.

Abstract

We study the general elliptic KdV potentials, which can be expressed (up to adding a constant) as

q_{p} (z) : - j = 1 \sum n m_{j} (m_{j} + 1) ℘ (z - p_{j}), m_{j} \in N .

We give an elementary proof of the theorem that the singularity

p = (p_{1}, \dots, p_{1} m_{1} (m_{1} + 1) /2, \dots, p_{n}, \dots, p_{n} m_{n} (m_{n} + 1) /2)

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.

Cite this article

Zhijie Chen, Ting-Jung Kuo, Chang-Shou Lin, On the closure of the Airault–Mckean–Moser locus for elliptic KdV potentials via Darboux transformations. J. Spectr. Theory 14 (2024), no. 4, pp. 1475–1512

DOI 10.4171/JST/529