Spectral Theory of the Nazarov-Sklyanin Lax Operator

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2023
• Main Authors: Mickler, Ryan, Moll, Alexander
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2023
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/212021
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Spectral Theory of the Nazarov-Sklyanin Lax Operator. Ryan Mickler and Alexander Moll. SIGMA 19 (2023), 063, 22 pages

In their study of Jack polynomials, Nazarov-Sklyanin introduced a remarkable new graded linear operator : [] → [] where is the ring of symmetric functions, and w is a variable. In this paper, we (1) establish a cyclic decomposition [] ≅ ⨁λ (λ, ) into finite-dimensional -cyclic subspaces in which Jack polynomials λ may be taken as cyclic vectors and (2) prove that the restriction of to each (jλ, ) has simple spectrum given by the anisotropic contents [] of the addable corners of the Young diagram of λ. Our proofs of (1) and (2) rely on the commutativity and spectral theorem for the integrable hierarchy associated with , both established by Nazarov-Sklyanin. Finally, we conjecture that the -eigenfunctions ˢλ ∈ [] {with eigenvalue [] and constant term} ˢλ|₌₀ = λ are polynomials in the rescaled power sum basis μˡ of [] with integer coefficients.