Generalized Gaudin Models and Riccatians

Abstract

The systems of differential equations whose solutions coincide with Bethe ansatz solutions of generalized Gaudin models are constructed. These equations we call the {\it generalized spectral Riccati equations}, because the simplest equation of this class has a standard Riccatian form. The general form of these equations is $R_{n_i}[z_1(\lambda),\ldots, z_r(\lambda)] = c_{n_i}(\lambda), \ i=1,\ldots, r$, where $R_{n_i}$ denote some homogeneous polynomials of degrees $n_i$ constructed from functional variables $z_i(\lambda)$ and their derivatives. It is assumed that $\deg \partial^k z_i(\lambda) = k+1$. The problem is to find all functions $z_i(\lambda)$ and $c_{n_i}(\lambda)$ satisfying the above equations under $2r$ additional constraints $P \ z_i(\lambda)=F_i(\lambda)$ and $(1-P) \ c_{n_i}(\lambda)=0$, where $P$ is a projector from the space of all rational functions onto the space of rational functions having their singularities at {\it a priori} given points. It turns out that this problem has solutions only for very special polynomials $R_{n_i}$ called {\it Riccatians}. There exist a one-to-one correspondence between systems of Riccatians and simple Lie algebras. Functions $c_{n_i}(\lambda)$ satisfying the system of equations constructed from Riccatians of the type ${\cal L}_r$ exactly coincide with eigenvalues of the Gaudin spectral problem associated with algebra ${\cal L}_r$. This result suggests that the generalized Gaudin models admit a total separation of variables.