On the rational monodromy-free potentials with sextic growth

We study the rational potentials V(x), with sextic growth at infinity, such that the corresponding one-dimensional Schrödinger equation has no monodromy in the complex domain for all values of the spectral parameter. We investigate in detail the subclass of such potentials which can be constructed by the Darboux transformations from the well-known class of quasiexactly solvable potentials V=x6−νx2+l(l+1)/x2. We show that, in contrast with the case of quadratic growth, there are monodromy-free potentials which have quasirational eigenfunctions, but which cannot be given by this construction. We discuss the relations between the corresponding algebraic varieties and present some elementary solutions of the Calogero-Moser problem in the external field with sextic potential.