Lie-algebras and linear operators with invariant subspaces
Lie-algebras and linear operators with invariant subspaces
复制标题
李代数和具有不变子空间的线性算子
DOI：
10.1090/conm/160/01576
复制
发表时间：
1993
期刊：
影响因子：
--
通讯作者：
A. Turbiner
中科院分区：
文献类型：
--
作者：
A. Turbiner
给出了具有多项式基（广义博赫纳问题）的有限维不变子空间的线性微分和有限差分算子的一般分类。主要结果是，具有上述性质的任何算子都必须具有作为有限维表示中微分（差）算子的某些代数的通用包络代数的多项式元素的表示加上消灭有限维不变子空间的算子。在低维中，分类由代数 $sl_2({\bold R})$ （对于 ${\bold R}$ 中的微分算子）和 $sl_2({\bold R})_q$ （对于 ${\bold R}$ 中的有限差分算子）、$osp(2,2)$ （一个实数和一个 Grassmann 变量中的算子，或者等效地，$2 \times 2$ 矩阵算子给出） ${\bold R}$)、$sl_3({\bold R})$、$sl_2({\bold R}) \oplus sl_2({\bold R})$ 和 $gl_2 ({\bold R}) \ltimes {\bold R}^{r+1}\ ，r$ 自然数（${\bold R^2}$ 中的运算符）。提出了具有无限多个有限维不变子空间且以多项式为基础的线性算子的分类。讨论了与最近发现的准精确可解光谱问题的联系。
A general classification of linear differential and finite-difference operators possessing a finite-dimensional invariant subspace with a polynomial basis (the generalized Bochner problem) is given. The main result is that any operator with the above property must have a representation as a polynomial element of the universal enveloping algebra of some algebra of differential (difference) operators in finite-dimensional representation plus an operator annihilating the finite-dimensional invariant subspace. In low dimensions a classification is given by algebras $sl_2({\bold R})$ (for differential operators in ${\bold R}$) and $sl_2({\bold R})_q$ (for finite-difference operators in ${\bold R}$), $osp(2,2)$ (operators in one real and one Grassmann variable, or equivalently, $2 \times 2$ matrix operators in ${\bold R}$), $sl_3({\bold R})$, $sl_2({\bold R}) \oplus sl_2({\bold R})$ and $gl_2 ({\bold R}) \ltimes {\bold R}^{r+1}\ , r$ a natural number (operators in ${\bold R^2}$). A classification of linear operators possessing infinitely many finite-dimensional invariant subspaces with a basis in polynomials is presented. A connection to the recently-discovered quasi-exactly-solvable spectral problems is discussed.