The bispectral problem, the Darboux process, monodromy and the Hermite operator

The complete solution of the bispectral problem for the Schr\"odinger operator $L=-\tfrac{d^2}{dx^2}+V(x)$ in [DG] (J. J. Duistermaat and F. A. Gr\"unbaum, Differential equations in the spectral parameter, Comm. Math. Phys. 103 (1986), 177-240) 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 [DG]. 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 [DG] do not apply verbatim in this case.