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Generalized Gaudin models and Riccatians

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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 $𝓛_r$ of rank r coincide with the orders of corresponding Casimir invariants. In the paper we present an explicit form of Riccatians associated with algebras $A_1, A_2, B_2, G_2, A_3, B_3, C_3$. Another important result is that functions $c_{n_i}(λ)$ satisfying the system of generalized Riccati equations constructed from Riccatians of the type $𝓛_r$ exactly coincide with eigenvalues of the Gaudin spectral problem associated with algebra $𝓛_r$. This result suggests that the generalized Gaudin models admit a total separation of variables. $(1)$ The exact meaning of the adjective "spectral" will be clarified in subsection 1.1. Here we only note that the class of ordinary spectral Riccati equations contains, for example, the delinearized version of Lame equation.
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  • Department of Theoretical Physics, University of Łódź, Pomorska 149/153, PL-90-236 Łódź, Poland
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