Gelfand-Dikii analysis for N=2 supersymmetric KdV equations

We generalize the resolvent approach of Gelfand and Dikii to the KdV equation to study the N=2 supersymmetric KdV equations of Laberge and Mathieu. For the associated Lax operators, we study the coincidence limits of the resolvent kernel and its derivatives, and obtain differential equations which they satisfy. These allow us to obtain recursion relations for the analogues of the Gelfand-Dikii polynomials and to obtain a proof of Hamiltonian integrability of the supersymmetric KdV equations. We are also able to write the Lax equations for the corresponding hierarchies in terms of these polynomials. (orig.)