Spectral Theory of the Nazarov-Sklyanin Lax Operator

TL;DR

This paper explores the spectral properties of Nazarov-Sklyanin's Lax operator related to Jack polynomials, establishing a cyclic decomposition and simple spectra, and conjectures about eigenfunctions in symmetric functions.

Contribution

It provides a cyclic decomposition of the operator's action on symmetric functions and proves simple spectra for each component, advancing understanding of its spectral theory.

Findings

Established a cyclic decomposition of the operator on symmetric functions.

Proved that the restriction of the operator has simple spectrum given by anisotropic contents.

Conjectured that eigenfunctions are polynomials with integer coefficients in a rescaled power sum basis.

Abstract

In their study of Jack polynomials, Nazarov-Sklyanin introduced a remarkable new graded linear operator ${\mathcal L} \colon F[w] \rightarrow F[w]$ where $F$ is the ring of symmetric functions and $w$ is a variable. In this paper, we (1) establish a cyclic decomposition $F[w] \cong \bigoplus_{\lambda} Z(j_{\lambda}, {\mathcal L})$ into finite-dimensional ${\mathcal L}$-cyclic subspaces in which Jack polynomials $j_{\lambda}$ may be taken as cyclic vectors and (2) prove that the restriction of ${\mathcal L}$ to each $Z(j_{\lambda}, {\mathcal L})$ has simple spectrum given by the anisotropic contents $[s]$ of the addable corners $s$ of the Young diagram of $\lambda$. Our proofs of (1) and (2) rely on the commutativity and spectral theorem for the integrable hierarchy associated to ${\mathcal L}$, both established by Nazarov-Sklyanin. Finally, we conjecture that the ${\mathcal L}$-eigenfunctions $\psi_{\lambda}^s {\in F[w]}$ {with eigenvalue $[s]$ and constant term} $\psi_{\lambda}^s|_{w=0} = j_{\lambda}$ are polynomials in the rescaled power sum basis $V_{\mu} w^l$ of $F[w]$ with integer coefficients.