Lie-Algebras and Linear Operators with Invariant Subspaces.

摘要

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 the algebra of differential (difference) operators in finite-dimensional representation. A classification of linear operators possessing infinitely many one-dimensional invariant polynomial subspace is presented. A connection to the recently-discovered quasi-exactly-solvable spectral problems is discussed.