Quasi-Exactly-Solvable Differential Equations

1994/09/12 by Turbiner, Alexander
#FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #High Energy Physics - Theory (hep-th)

paper · doi:10.48550/arxiv.hep-th/9409068

Abstract

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 sl₂(\bold R) (for differential operators in \bold R) and sl₂(\bold R)_q (for finite-difference operators in \bold R), osp(2,2) (operators in one real and one Grassmann variable, or equivalently, 2 × 2 matrix operators in \bold R) and gl₂ (\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.

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