The Bispectral Problem, the Darboux Process, Monodromy and the Hermite Operator

The complete solution of the bispectral problem for the Schrödinger operator L=-d²/dx²+V(x) L = - d 2 d x 2 + V ( x ) in [19] is obtained by the application of the Darboux process to the cases of $$V=0$$ V = 0 and V(x)=-1/4x² V ( x ) = - 1 4 x 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 $$V=0$$ V = 0 ) or a rank two bispectral bundle (when starting from V(x)=-1/4x² V ( x ) = - 1 4 x 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)=x² 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.