Super-extensions of energy dependent Schr�dinger operators

Abstract

We consider the energy dependent super Schrodinger linear problem $$\sum\limits_{i = 0}^N {\lambda ^i [(\varepsilon _i \partial ^2 + u_i )\psi + \eta _i \phi ] = 0, \sum\limits_{i = 0}^N {\lambda ^i (\varepsilon _i \partial \phi + \eta _i \psi )} } = 0$$ which is a direct generalization of the purely even, energy dependent Schrodinger equation discussed in [1]. We show that the isospectral flows of that problem possess (N+1) compatible Hamiltonian structures. We also extend a generalised factorisation approach of [2] to this case and derive a sequence ofN modifications for the 2N component systems. Then th such modification possesses (N−n+1) compatible Hamiltonian structures.