Applying supersymmetry to energy dependent potentials

• Supersymmetry extended to energy dependent potentials.
• Generalization of the concept of superpotential.
• An alternative method used for linear E-dependence leads to the same results as Darboux transform.

We investigate the supersymmetry properties of energy dependent potentials in the D=1D=1 dimensional space. We show the main aspects of supersymmetry to be preserved, namely the factorization of the Hamiltonian, the connections between eigenvalues and wave functions of the partner Hamiltonians. Two methods are proposed. The first one requires the extension of the usual rules via the concept of local equivalent potential. In this case, the superpotential becomes depending on the state. The second method, applicable when the potential depends linearly on the energy, is similar to what has been already achieved by means of the Darboux transform.