Geodesic
Generating Operator of XXX or Gaudin Transfer Matrices Has Quasi-Exponential Kernel
Symmetry, integrability and geometry: methods and applications, Tome 3 (2007) Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $M$ be the tensor product of finite-dimensional polynomial evaluation $Y(\mathfrak{gl}_N)$-modules. Consider the universal difference operator $\mathfrak D=\sum\limits_{k=0}^N (-1)^k\mathfrak T_k(u) e^{-k\partial _u }$ whose coefficients $\mathfrak T_k(u)\colon M\to M$ are the XXX transfer matrices associated with $M$. We show that the difference equation $\mathfrak D f=0$ for an $M$-valued function $f$ has a basis of solutions consisting of quasi-exponentials. We prove the same for the universal differential operator $D=\sum\limits_{k=0}^N (-1)^k\mathcal S_k(u)\partial_u^{N-k}$ whose coefficients $\mathcal S_k(u)\colon\mathcal M\to\mathcal M$ are the Gaudin transfer matrices associated with the tensor product $\mathcal M$ of finite-dimensional polynomial evaluation $\mathfrak{gl}_N[x]$-modules.
Keywords: Gaudin model; XXX model; universal differential operator. 
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     author = {Evgeny Mukhin and Vitaly Tarasov and Alexander Varchenko},
     title = {Generating {Operator} of {XXX} or {Gaudin} {Transfer} {Matrices} {Has} {Quasi-Exponential} {Kernel}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2007},
     volume = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2007_3_a59/}
}
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Evgeny Mukhin; Vitaly Tarasov; Alexander Varchenko. Generating Operator of XXX or Gaudin Transfer Matrices Has Quasi-Exponential Kernel. Symmetry, integrability and geometry: methods and applications, Tome 3 (2007). http://geodesic.mathdoc.fr/item/SIGMA_2007_3_a59/

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