Abstract The complete solution of the bispectral problem for the Schrödinger operator L=-d²dx²+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) =-14x² 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) =-14x² 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.
Castro et al. (Sat,) studied this question.