Spectral Theory of the Nazarov-Sklyanin Lax Operator

Author:

Mickler Ryan, ,Moll Alexander,

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.

Publisher

SIGMA (Symmetry, Integrability and Geometry: Methods and Application)

Subject

Geometry and Topology,Mathematical Physics,Analysis

同舟云学术

1.学者识别学者识别

2.学术分析学术分析

3.人才评估人才评估

"同舟云学术"是以全球学者为主线,采集、加工和组织学术论文而形成的新型学术文献查询和分析系统,可以对全球学者进行文献检索和人才价值评估。用户可以通过关注某些学科领域的顶尖人物而持续追踪该领域的学科进展和研究前沿。经过近期的数据扩容,当前同舟云学术共收录了国内外主流学术期刊6万余种,收集的期刊论文及会议论文总量共计约1.5亿篇,并以每天添加12000余篇中外论文的速度递增。我们也可以为用户提供个性化、定制化的学者数据。欢迎来电咨询!咨询电话:010-8811{复制后删除}0370

www.globalauthorid.com

TOP

Copyright © 2019-2024 北京同舟云网络信息技术有限公司
京公网安备11010802033243号  京ICP备18003416号-3