# Generalized Gaudin models and Riccatians

• Volume: 37, Issue: 1, page 259-288
• ISSN: 0137-6934

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## Abstract

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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 ${\left(}^{1}\right)$ Riccati equations, because the simplest equation of this class has a standard Riccatian form. The general form of these equations is ${R}_{{n}_{i}}\left[{z}_{1}\left(\lambda \right),...,{z}_{r}\left(\lambda \right)\right]={c}_{{n}_{i}}\left(\lambda \right)$, i=1,..., r, where ${R}_{{n}_{i}}$ denote some homogeneous polynomials of degrees ${n}_{i}$ constructed from functional variables ${z}_{i}\left(\lambda \right)$ and their derivatives. It is assumed that $deg{\partial }^{k}{z}_{i}\left(\lambda \right)=k+1$. The problem is to find all functions ${z}_{i}\left(\lambda \right)$ and ${c}_{{n}_{i}}\left(\lambda \right)$ satisfying the above equations under 2r additional constraints $P{z}_{i}\left(\lambda \right)={F}_{i}\left(\lambda \right)$ and $\left(1-P\right){c}_{{n}_{i}}\left(\lambda \right)=0$, where P is a projector from the space of all rational functions onto the space of rational functions having their singularities at a priorigiven 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 ${}_{r}$ 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 ${c}_{{n}_{i}}\left(\lambda \right)$ satisfying the system of generalized Riccati equations constructed from Riccatians of the type ${}_{r}$ exactly coincide with eigenvalues of the Gaudin spectral problem associated with algebra ${}_{r}$. This result suggests that the generalized Gaudin models admit a total separation of variables. $\left(1\right)$ 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.

## How to cite

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Ushveridze, Aleksander. "Generalized Gaudin models and Riccatians." Banach Center Publications 37.1 (1996): 259-288. <http://eudml.org/doc/208604>.

@article{Ushveridze1996,
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 priorigiven 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 $_r$ 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 $_r$ exactly coincide with eigenvalues of the Gaudin spectral problem associated with algebra $_r$. 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. },
author = {Ushveridze, Aleksander},
journal = {Banach Center Publications},
keywords = {generalized Riccati operators; Gaudin model},
language = {eng},
number = {1},
pages = {259-288},
title = {Generalized Gaudin models and Riccatians},
url = {http://eudml.org/doc/208604},
volume = {37},
year = {1996},
}

TY - JOUR
AU - Ushveridze, Aleksander
TI - Generalized Gaudin models and Riccatians
JO - Banach Center Publications
PY - 1996
VL - 37
IS - 1
SP - 259
EP - 288
AB - 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 priorigiven 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 $_r$ 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 $_r$ exactly coincide with eigenvalues of the Gaudin spectral problem associated with algebra $_r$. 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.
LA - eng
KW - generalized Riccati operators; Gaudin model
UR - http://eudml.org/doc/208604
ER -

## References

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1. [1] M. Gaudin, J. Physique 37 (1976), 1087-98.
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7. [7] E. K. Sklyanin, Preprint of Helsinki University HU-TFT-91-51, Helsinki (see also hep-th/9211111), 1991.
8. [8] E. K. Sklyanin, Preprint of Cambridge University NI-92013, Cambridge, 1992.
9. [9] A. G. Ushveridze, Sov. J. Part. Nucl. 20 (1989), 1185-245.
10. [10] A. G. Ushveridze, Preprint of Georgian Institute of Physics FTT-16, Tbilisi, 1990 (in Russian).
11. [11] A. G. Ushveridze, Sov. J. Part. Nucl. 23 (1992), 25-51.
12. [12] A. G. Ushveridze, Quasi-exactly solvable models in quantum mechanics (Bristol: IOP Publishing), 1994. Zbl0834.58042

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