Spectral Theory of the Nazarov-Sklyanin Lax Operator

• Veröffentlicht in: Symmetry, Integrability and Geometry: Methods and Applications
• Datum: 2023
• Hauptverfasser: Mickler, Ryan, Moll, Alexander
• Format: Artikel
• Sprache: Englisch
• Veröffentlicht: Інститут математики НАН України 2023
• Online Zugang: https://nasplib.isofts.kiev.ua/handle/123456789/212021
• Назва журналу: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Zitieren: 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 𝓛: 𝐹[𝓌] → 𝐹[𝓌] where 𝐹 is the ring of symmetric functions, and w is a variable. In this paper, we (1) establish a cyclic decomposition 𝐹[𝓌] ≅ ⨁λ 𝑍(𝑗λ, 𝓛) into finite-dimensional 𝓛-cyclic subspaces in which Jack polynomials 𝑗λ may be taken as cyclic vectors and (2) prove that the restriction of 𝓛 to each 𝑍(jλ, 𝓛) has simple spectrum given by the anisotropic contents [𝑠] of the addable corners 𝑠 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 𝓛, both established by Nazarov-Sklyanin. Finally, we conjecture that the 𝓛-eigenfunctions 𝜓ˢλ ∈ 𝐹[𝓌] {with eigenvalue [𝑠] and constant term} 𝜓ˢλ|𝓌₌₀ = 𝑗λ are polynomials in the rescaled power sum basis 𝑉μ𝓌ˡ of 𝐹[𝓌] with integer coefficients.