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=⟨αˣᵢ𝑝ᵢⱼ(𝑥), i = 1,…, 𝑛, j = 1,…, 𝑛ᵢ⟩, where αᵢ ∈ ℂ* and 𝑝ᵢⱼ(𝑥) are polynomials, we consider the formal conjugate Š†𝑊 of the quotient difference operator ŠW satisfying Ŝ = Š𝑊S𝑊. Here, S𝑊 is a linear difference operator of order dim𝑊 annihilating 𝑊, and Ŝ is a linear difference operator with constant coefficients depending on αᵢ and deg𝑝ᵢⱼ(𝑥) only. We construct a space of quasi-exponentials of dimension ord Š†𝑊, which is annihilated by Š†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 𝑥ᶻ𝑞(𝑥), where 𝓏 ∈ 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 (𝖌𝔩ₖ, 𝖌𝔩ₙ)-duality of the trigonometric Gaudin Hamiltonians and trigonometric dynamical Hamiltonians acting on the space of polynomials in 𝑘𝑛 anticommuting variables.