Difference Operators and Duality for Trigonometric Gaudin and Dynamical Hamiltonians

Abstract

We study the difference analog of the quotient differential operator from [Tarasov V., Uvarov F., Lett. Math. Phys. 110 (2020), 3375-3400, arXiv:1907.02117]. Starting with a space of quasi-exponentials $W=⟨{\alpha }_i^x{p}_{ij}(x),i=1,\dots ,n,j=1,\dots ,n_i⟩$, where ${\alpha }_i\in {\mathbb{C}}^{\ast }$ and $p_{ij}(x)$ are polynomials, we consider the formal conjugate ${\check{S}}_W^{†}$ of the quotient difference operator ${\check{S}}_W$ satisfying $\hat{S}={\check{S}}_W{S}_W$. Here, $S_W$ is a linear difference operator of order $\dim W$ annihilating $W$, and $\hat{S}$ is a linear difference operator with constant coefficients depending on ${\alpha }_i$ and $°p_{ij}(x)$ only. We construct a space of quasi-exponentials of dimension $\mathrm{ord}{\check{S}}_W^{†}$, which is annihilated by ${\check{S}}_W^{†}$ and describe its basis and discrete exponents. We also consider a similar construction for differential operators associated with spaces of quasi-polynomials, which are linear combinations of functions of the form $x^{z}q(x)$, where $z\in \mathbb{C}$ and $q(x)$ is a polynomial. Combining our results with the results on the bispectral duality obtained in [Mukhin E., Tarasov V., Varchenko A., Adv. Math. 218 (2008), 216-265, arXiv:math.QA/0605172], we relate the construction of the quotient difference operator to the $({\mathfrak{gl}}_k,{\mathfrak{gl}}_n)$-duality of the trigonometric Gaudin Hamiltonians and trigonometric dynamical Hamiltonians acting on the space of polynomials in $kn$ anticommuting variables.

References in corpus (2)

• Bethe eigenvectors of higher transfer matrices
• Quasi-polynomials and the Bethe Ansatz