Quasi-Exactly-Solvable Differential Equations
Abstract
Description
A general classification of linear differential and finite-difference operators possessing a finite-dimensional invariant subspace with a polynomial basis 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. In one-dimensional case 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 ${\bold R}$) and $gl_2 ({\bold R})_K$ ( for the operators containing the differential operators and the parity operator). A classification of linear operators possessing infinitely many finite-dimensional invariant subspaces with a basis in polynomials is presented.
33 pages, to appear as Chapter 12 in CRC Handbook of Lie Group Analysis of Differential Equations, Vol. 3 : New Trends in Theoretical Developments and Computational Methods, ed. N. H. Ibragimov, CRC Press, 1995
33 pages, to appear as Chapter 12 in CRC Handbook of Lie Group Analysis of Differential Equations, Vol. 3 : New Trends in Theoretical Developments and Computational Methods, ed. N. H. Ibragimov, CRC Press, 1995