Fuchsian bispectral operators (Q1599929)

An ordinary differential operator \(L=L(x, \partial_x)\) is called bispectral if it has an eigenfunction \(\psi =\psi (x,z)\) which is at the same time an eigenfunction of another differential operator \(\wedge =\wedge (z,\partial_z)\), i.e., one has \(L\psi =f(z)\psi\) and \(\wedge \psi =\theta (x)\psi\). Here, the authors consider bispectral operators \(L\) of the form \(L=\sum_{k=0}^N V_k (x) \partial_x^k\) with \(V_N=1\), \(V_{N-1}=0\) and \(V_j (x)\to 0\) for \(x\to \infty\) and \(j<N-1\). Examples for such operators are the generalized Bessel operators \(L_\beta =x^{-N} (D-\beta_1)\dots (D-\beta_N)\) with \(D=x\partial_x\) and \(\beta_i \in \mathbb{C}\). The main result in this paper states that bispectral operators satisfying the above conditions can be obtained from Bessel operators by monomial Darboux transformations. This result is shown by the analysis of the point at infinity and the study of algebras of differential operators.