The complete solution of the bispectral problem for the Schrödinger operator \(L=-\tfrac{d^2}{dx^2}+V(x)\) in [19] is obtained by the application of the Darboux process to the cases of \(V=0\) and \(V(x)=-\tfrac{1}{4x^2}\) . Both of these cases are trivially bispectral and after repeated applications of the Darboux process one gets either a pair of rank one bundles of bispectral situations (when starting from \(V=0\) ) or a rank two bispectral bundle (when starting from \(V(x)=-\tfrac{1}{4x^2}\) ). In the first case all operators have “trivial monodromy” as defined in [19]. In the second case the monodromy group of all operators is given by the integers. In this paper we start from \(V(x)=x^2\) , use the Darboux process and explore the connection between the rank of certain non-polynomial bispectral families and trivial monodromy by means of examples. The main conclusion is that the results in [19] do not apply verbatim in this case.
