Difference Operators and Duality for Trigonometric Gaudin and Dynamical Hamiltonians - Uvarov, Filipp (Symmetry, Integrability and Geometry: Methods and Applications, 2022)
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 =⟨αˣᵢᵢⱼ(), i = 1,…, , j = 1,…, ᵢ⟩, where αᵢ ∈ ℂ* and ᵢⱼ() are polynomials, we consider the formal conjugate Š† of the quotient difference operator ŠW satisfying Ŝ = ŠS. Here, S is a linear difference operator of order dim annihilating , and Ŝ is a linear difference operator with constant coefficients depending on αᵢ and degᵢⱼ() only. We construct a space of quasi-exponentials of dimension ord Š†, which is annihilated by Š† 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 ᶻ(), where ∈ ℂ and () 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 (ₖ, ₙ)-duality of the trigonometric Gaudin Hamiltonians and trigonometric dynamical Hamiltonians acting on the space of polynomials in anticommuting variables.