Quasi Exactly Solvable Matrix Models in Sl(n)

We reconsider the quasi exactly solvable matrix models constructed recently by R. Zhdanov. The 2×2 matrix operators representing the algebra sl(2) are generalized to matrices of arbitrary dimension and a similar construction is achieved for the algebra sl(n). ∗† Work supported by grant no KBN 2P03B07610 1. Recently Zhdanov [1],[2] considered the realizations of the sl(2) algebra in terms of first order matrix differential operators. It reads Q− = d dx , Q0 = x d dx + A , Q+ = x 2 d dx + 2xA+B (1) [Q0, Q±] = ±Q± , [Q−, Q+] = 2Q0 (2) where A and B are (in general complex) L× L matices obeying [A,B] = B (3) This realization acts in the tensor product of CI L with the linear space of smooth functions. In the following this vector space is denoted V . Zhdanov posed and solved, in the case L = 2, the problem of characterizing the invariant finitedimensional subspaces of the representation space V . In the present note we give the general characterization of such subspaces. The main result is that they all basically arise from the Clebsch-Gordan decomposition of the tensor product of the differential representation of spin n/2, q− = d dx , q0 = x d dx − n 2 , q+ = x 2 d dx − nx , (4) with the standard matrix representation of some spin s. 2. Let us first consider the general form of two finite-dimensional operators X, Y obeying [X, Y ] = Y (5) It follows from eq. (5) that (X − (λ+m))Y m = Y (X − λ) (6) for all integers m,n ≥ 0. Let us put X in Jordan form. The space R in which X and Y act is spanned by the (generalized) eigenvectors of X. Let ψ be some (generalized) eigenvector of X corresponding to the eigenvalue λ. Then Y ψ, Y ψ, ... correspond to the 1 eigenvalues λ+ 1, λ+ 2, ... and, if non vanishing, are linearly independent. This implies Y ψ = 0 for some k ≤ N , where N is the dimension of the vector space under consideration. Therefore Y is nilpotent, Y K ≡ 0 for some K ≤ N . Note in passing that if we apply this result to X = −Q0, Y = Q−, restricted to some finite-dimensional subspace of V we get at once the conclusion that such a subspace consists of vectors with polynomial entries. In order to get more information about the structure of the operators X and Y , let us define λ to be an isolated (generalized) eigenvalue of X if neither λ− 1 nor λ+1 are (generalized) eigenvalues. Let R0 be the direct sum of all subspaces of R which correspond to the Jordan blocks of X related to isolated eigenvalues. Then, due to eq. (6), R0 is invariant under the action of X and Y , Y |R0 = 0 and the whole space R is a sum of R0 and of the invariant subspace R1 corresponding to the remaining Jordan blocks of X. Due to the definition of isolated eigenvalues the set of remaining eigenvalues is a disjoint sum of finite collection of chains Λα consisting of eigenvalues differing by one, Λα = {λα, λα+1, · · · , λα+nα−1}, nα ∈ N I . The subspace R1 can be again written as R1 = ⊕αRα, where Rα corresponds to the Jordan blocks related to λα, λα + 1, · · · , λα + nα − 1; each Rα is an invariant subspace under the action of X and Y . However, it is obviously not the end of the story. The subspaces corresponding to some λα +L, λα +L+1 can be (but not necessarily are) related by the action of Y . Therefore each chain Λα can further be decomposed into disjoint sum of subchains consisting of the eigenvalues having the property that the subspaces corresponding to consecutive eigenvalues are related by the action of Y operator. Any subspace Rα can be correspondingly decomposed in direct sum of invariant subspace corresponding to such subchains. Therefore we have only to characterize such subspaces. Let S be one of them. One can write, according to the Jordan theorem, S = ⊕Ki=1Si (7)