The geometry of the Hill equation and of the Neumann system

Abstract

Let there be given a finite-gap operator L = d 2 /d x 2 + q and its Baker function ψ( x, p ), which is analytic for p on a certain hyperelliptic curve C . It is shown that a sequence of Bäcklund transformations maps C to a projective space. This embedding can be interpreted as a matrix representation of the Hill equation by the Neumann system of constrained harmonic oscillators. The image curve, C' , lies on a rational ruled surface; the structure of this surface is explained by use of ideas due to Burchnall & Chaundy ( Proc. R. Soc. Lond . A 118, 557-583 (1928)). Baker functions and Bäcklund transformations are then used to define a (many-to-many) correspondence between effective divisors on the curve C and points lying on a quadric, or in the intersection of two or more quadrics. This relates the theory of the Hill equation to earlier work of Knörrer, Moser and Reid. It is then shown that the Kummer image of the Jacobian of C can be realized as a hypersurface in the space of momentum variables of the Neumann system. Further projects, such as extensions to non-hyperelliptic curves, are outlined.