An explicit characterization of Calogero–Moser systems

Abstract

Abstract. Combining theorems of Halphen, Floquet, and Picard and a Frobenius type analysis, we characterize rational, meromorphic simply periodic, and elliptic KdV potentials. In particular, we explicitly describe the proper extension of the Airault-McKean-Moser locus associated with these three classes of algebro-geometric solutions of the KdV hierarchy with special emphasis on the case of multiple collisions between the poles of solutions. This solves a problem left open since the mid-1970s.

On the poles of Picard potentials

“…All that provides motivation for a more detailed study of Picard potentials with a goal of a more explicit and complete description of them. In [3], such a description was obtained in the case when v(x) ≡ 1, and u(x) is either elliptic or periodic and bounded near the ends of the period strip or rational and holomorphic at ∞: in that case the pair u(x), 1 is Picard if and only if u(x) satisfies a suitable variant of the Calogero-Moser conditions. However, these examples are practically all the known results up to date about an explicit description of Picard potentials.…”

A necessary and sufficient condition for the Darboux-Treibich-Verdier potential with its spectrum contained in $\mathbb{R}$

“…Recalling Section 2, we denote the Hill's discriminants of (4.3) Ly(x) = y ′′ (x) + q(x; τ)y(x) = Ey(x) and (4.4) y ′′ (x) + q T (x)y(x) = Ey(x) by ∆(E; τ) and ∆ T (E) respectively. Now we apply the following key fact about ∆ T (E): Since q n T (z) can be generated from C n T by finite times of Darboux transformations (see [12,Remark 2.7]), it is known (see e.g. [9, Remark 1.3]) that ∆ T (E) coincides with the Hill's discriminant of y ′′ (x) + C n T y(x) = Ey(x) with respect to the period 1, i.e.…”

On Algebraically Integrable Differential Operators on an Elliptic Curve

“…In this subsection we give a proof of the classification theorem of finite-gap potentials on elliptic curves with arbitrary number of poles [10,4,11]; this proof is based on differential Galois theory and follows [4, Section 4.1].…”

“…In this subsection we review the basics on algebraically integrable ordinary differential operators. Most of this material is well known; we refer the reader to [13,16,15,17,10,4,5,11] and references therein.…”

“…where m is an integer; its algebraic integrability was discovered by Hermite. In the case n = 2 and multiple poles, the answer is much more interesting (see [4,Section 4.1], as well as [10,11] and references therein).…”