A classification of second-order raising operators for Hamiltonians in two variables

We develop a group theoretic method based on results of Winternitz et al. to compute and classify all first- and second-order raising and lowering operators admitted by Hamiltonians of the form H = - (1/2)Δ2 + V (x, y). The key to our results, which generalize to higher dimensions, is a proof that H admits a second-order raising operator only if the Schrödinger equation separates in Cartesian, polar, or elliptic coordinates.