Lie-algebras and Linear Operators with Invariant Subspaces

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 sl2(R) (for differential operators in R) and sl2(R)q (for finite-difference operators in R), osp(2, 2) (operators in one real and one Grassmann variable, or equivalently, 2×2 matrix operators in R), sl3(R), sl2(R)⊕sl2(R) and gl2(R)⋉R , r a natural number (operators in R). A classification of linear operators possessing infinitely many finitedimensional invariant subspaces with a basis in polynomials is presented. A connection to the recently-discovered quasi-exactly-solvable spectral problems is discussed. S. Bochner (1929) asked about a classification of differential equations Tφ = ǫφ (0) where T is a linear differential operator of kth order in one real variable x ∈ R and ǫ is the spectral parameter, having an infinite sequence of orthogonal polynomial solutions [1] (see also [2]). Definition 0.1. Let us give the name of the generalized Bochner problem to the problem of classification of linear differential (difference) operators, for which the eigenvalue problem (0) has a certain number of eigenfunctions in the form of a finite-order polynomial in some variables. Following this definition the original Bochner problem is simply a particular case. In [3] a general method has been formulated for generating eigenvalue problems for linear differential operators, linear matrix differential operators and linear finitedifference operators in one and several variables possessing polynomial solutions. The method was based on considering the eigenvalue problem for the representation of a polynomial element of the universal enveloping algebra of the Lie algebra in a finite-dimensional, ‘projectivized’ representation of this Lie algebra [3]. Below it is shown that this method provides both necessary and sufficient conditions for the 1991 Mathematics Subject Classification. 81C05, 81C40, 17B15. Supported in Part by the Swiss National Foundation. This paper is in final form and no version will be submitted for publication elsewhere. The manuscript is prepared using AMS-LATEX.. 1 2 ALEXANDER TURBINER existence of polynomial solutions of linear differential equations and a certain class of finite-difference equations. The generalized Bochner problem can be subdivided into two parts: (i) a classification of linear operators possessing an invariant subspace (subspaces) with a basis in polynomials of finite degree and (ii) a description of the conditions under which such operators are symmetrical. This paper will be devoted to a solution of the first problem; as for the second one, only the first step has been done (see below). The plan of the paper is the following: Section 1 is devoted to the case of differential operators in one real variable; in Section 2 finite-difference operators in R are treated; operators in one real and one Grassmann variables are considered in Section 3, while the case of operators in R is given in Section 4. The general situation is described in the Conclusion. 1. Ordinary differential equations Consider the space of all polynomials of order n Pn = 〈1, x, x, . . . , x〉, (1) where n is a non-negative integer and x ∈ R. Definition 1.1. Let us name a linear differential operator of the kth order, Tk quasi-exactly-solvable, if it preserves the space Pn. Correspondingly, the operator Ek, which preserves the infinite flag P0 ⊂ P1 ⊂ P2 ⊂ · · · ⊂ Pn ⊂ . . . of spaces of all polynomials, is named exactly-solvable. Lemma 1.1. (i) Suppose n > (k− 1). Any quasi-exactly-solvable operator Tk, can be represented by a k-th degree polynomial of the operators J = x∂x − nx, J = x∂x − n 2 , (2) J = ∂x , (the operators (2) obey the sl2(R) commutation relations 1 ). If n ≤ (k − 1), the part of the quasi-exactly-solvable operator Tk containing derivatives up to order n can be represented by an nth degree polynomial in the generators (2). (ii) Conversely, any polynomial in (2) is quasi-exactly solvable. (iii) Among quasi-exactly-solvable operators there exist exactly-solvable operators Ek ⊂ Tk. Comment 1. If we define the universal enveloping algebra Ug of a Lie algebra g as the algebra of all ordered polynomials in generators, then Tk at k < n + 1 is simply an element of the universal enveloping algebra Usl2(R) of the algebra sl2(R) taken in representation (2). If k ≥ n + 1, then Tk is represented as an element of Usl2(R) plus B d dxn+1 , where B is any linear differential operator of order not higher than (k − n− 1). Proof. The essential part of the proof is based on the Burnside theorem (see, e.g., [4]): 1The representation (2) is one of the ‘projectivized’ representations (see [3]). This realization of sl2(R) has been derived at the first time by Sophus Lie. 2Burnside theorem is a particular case of more general Jacobson theorem (see [5], Chapter XVII.3). I am grateful to V. Kac for this comment. LIE-ALGEBRAS AND LINEAR OPERATORS . . . 3 Let A1, . . . , Ak be linear operators in a linear space E over real numbers, dimE < ∞. Let us assume that there is no linear space L over real numbers, 0 < dimL < dimE , such that Ai : L 7→ L for all i = 1, 2, . . . , k. Then any linear operator acting in E can be represented as a polynomial in A1, . . . , Ak. The operators J act in Pn irreducibly and therefore the theorem can be applied. Also there exist operators B having Pn as a kernel, B : Pn 7→ 0. Clearly, those operators have a form B(x, ∂x) d dxn+1 , where B(x, ∂x) is any linear differential operator, and they make no contribution in Tk if k < n+1. It completes the proof of part (i). Parts (ii) and (iii) are easy to prove based on part (i). Since sl2(R) is a graded algebra, let us introduce the grading of generators (2): deg(J) = +1 , deg(J) = 0 , deg(J) = −1, (3) hence deg[(J)+(J)0(J)− ] = n+ − n−. (4) The grading allows us to classify the operators Tk in the Lie-algebraic sense. Lemma 1.2. A quasi-exactly-solvable operator Tk ⊂ Usl2(R) has no terms of positive grading, if and only if it is an exactly-solvable operator. Theorem 1.1. Let n be a non-negative integer. Take the eigenvalue problem for a linear differential operator of the kth order in one variable Tkφ = εφ , (5) where Tk is symmetric. The problem (5) has (n + 1) linearly independent eigenfunctions in the form of a polynomial in variable x of order not higher than n, if and only if Tk is quasi-exactly-solvable. The problem (5) has an infinite sequence of polynomial eigenfunctions, if and only if the operator is exactly-solvable. Comment 2. The “ if ” part of the first statement is obvious. The “ only if ” part is a direct corollary of Lemma 1.1 . This theorem gives a general classification of differential equations