Spectral Curves of Operators with Elliptic Coefficients⋆
نویسندگان
چکیده
The classical theory of reduction, initiated by Weierstrass, has found modern applications to the fields of Integrable Systems and Number Theory, to name but two. In this short note we only address specific cases, providing minimal historical references, listing the steps that we devised, and exhibiting some new explicit formulas. A full-length discussion of original motivation, theoretical advancements and modern applications would take more than one book to present fairly, and again, we choose to provide one (two-part) reference only, which is recent and captures our point of view [3, 4]. Our point of departure is rooted in the classical theory of Ordinary Differential Equations (ODEs). At a time when activity in the study of elliptic functions was most intense, Halphen, Hermite and Lamé (among many others) obtained deep results in the spectral theory of linear differential operators with elliptic coefficients. Using differential algebra, Burchnall and Chaundy described the spectrum (by which we mean the joint spectrum of the commuting operators) of those operators that are now called algebro-geometric, and some non-linear Partial Differential Equations (PDEs) satisfied by their coefficients under isospectral deformations along ‘time’ flows, t1, . . . , tg, where g is the genus of the spectral ring. We seek algorithms that, starting with an Ordinary Differential Operator (ODO) with elliptic coefficients, produce an
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