Generalized Gaudin models and Riccatians
Volume 37 / 1996
Abstract
The systems of differential equations whose solutions exactly coincide with Bethe ansatz solutions for generalized Gaudin models are constructed. These equations are called the generalized spectral $(^1)$ Riccati equations, because the simplest equation of this class has a standard Riccatian form. The general form of these equations is $R_{n_i}[z_1(λ),..., z_r(λ)] = c_{n_i}(λ)$, i=1,..., r, where $R_{n_i}$ denote some homogeneous polynomials of degrees $n_i$ constructed from functional variables $z_i(λ)$ and their derivatives. It is assumed that $deg ∂^{k} z_i(λ) = k+1$. The problem is to find all functions $z_i(λ)$ and $c_{n_i}(λ)$ satisfying the above equations under 2r additional constraints $P z_i(λ)=F_i(λ)$ and $(1-P)c_{n_i}(λ)=0$, where P is a projector from the space of all rational functions onto the space of rational functions having their singularities at a priori} given points. It turns out that this problem has solutions only for very special polynomials $R_{n_i}$. Simplest polynomials of such sort are called Riccatians}. One of most important results of the paper is the observation that there exist one-to-one correspondence between the systems of Riccatians and simple Lie algebras. In particular, the degrees of Riccatians associated with a given simple Lie algebra $
