三角Gaudin算子和动力学Hamiltonian算子的差分算子和对偶性

IF 0.9 3区 物理与天体物理 Q2 MATHEMATICS
F. Uvarov
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引用次数: 1

摘要

我们研究了[Tarasov V.,Uvarov F.,Lett.Math.Phys.110(2020),3375-3400,arXiv:1907.02117]中商微分算子的差分类似^{x}p_{ij}(x),\,i=1,\dots,n,\,j=1,\dots,n_{i}\rangle$,其中$\alpha_{i}\在{\mathbb C}^{*}$和$p_{ij}(x)$中是多项式,我们考虑商差算子$\check的形式共轭{S}_{W} $满足$\widehat{S}=\check{S}_{W}S_{W} $。这里,$S_{W}$是阶为$\dim W$的线性差分算子,湮灭$W$,$\widehat{S}$是具有仅取决于$\alpha_{i}$和$\deg p_{ij}(x)$的常系数的线性差算子。我们构造了一个被$\check{S}^{\digger}_{W}$湮灭的维数为$\cooperorname{ord}\check{S}^{\dagger}_{W}$的拟指数空间,并描述了它的基和离散指数。我们还考虑了与拟多项式空间相关的微分算子的类似构造,拟多项式空间是形式为$x的函数的线性组合^{z}q(x) $,其中$z\in\mathbb C$和$q(x)$是多项式。将我们的结果与[Mukhin E.,Tarasov V.,Varchenko A.,Adv.Math.218(2008),216-265,arXiv:Math.QA/QC 0605172]中获得的关于双谱对偶的结果相结合,我们将商差算子的构造与$(\mathfrak{gl}_{k} ,\mathfrak{gl}_{n} )$-kn$反交换变量中作用于多项式空间的三角Gaudin哈密顿量和三角动力学哈密顿量的$-对偶性。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Difference Operators and Duality for Trigonometric Gaudin and Dynamical Hamiltonians
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 $W=\langle \alpha_{i}^{x}p_{ij}(x),\, i=1,\dots, n,\, j=1,\dots, n_{i}\rangle$, where $\alpha_{i}\in{\mathbb C}^{*}$ and $p_{ij}(x)$ are polynomials, we consider the formal conjugate $\check{S}^{\dagger}_{W}$ of the quotient difference operator $\check{S}_{W}$ satisfying $\widehat{S} =\check{S}_{W}S_{W}$. Here, $S_{W}$ is a linear difference operator of order $\dim W$ annihilating $W$, and $\widehat{S}$ is a linear difference operator with constant coefficients depending on $\alpha_{i}$ and $\deg p_{ij}(x)$ only. We construct a space of quasi-exponentials of dimension $\operatorname{ord} \check{S}^{\dagger}_{W}$, which is annihilated by $\check{S}^{\dagger}_{W}$ 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 $x^{z}q(x)$, where $z\in\mathbb C$ and $q(x)$ 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 $(\mathfrak{gl}_{k},\mathfrak{gl}_{n})$-duality of the trigonometric Gaudin Hamiltonians and trigonometric dynamical Hamiltonians acting on the space of polynomials in $kn$ anticommuting variables.
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来源期刊
CiteScore
1.80
自引率
0.00%
发文量
87
审稿时长
4-8 weeks
期刊介绍: Scope Geometrical methods in mathematical physics Lie theory and differential equations Classical and quantum integrable systems Algebraic methods in dynamical systems and chaos Exactly and quasi-exactly solvable models Lie groups and algebras, representation theory Orthogonal polynomials and special functions Integrable probability and stochastic processes Quantum algebras, quantum groups and their representations Symplectic, Poisson and noncommutative geometry Algebraic geometry and its applications Quantum field theories and string/gauge theories Statistical physics and condensed matter physics Quantum gravity and cosmology.
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