Generalized Schrödinger equations with energy-dependent potentials: Formalism and applications

The quantum mechanical formalism for systems featuring energy-dependent potentials is extended to systems governed by generalized Schrödinger equations that include the position-dependent mass case. Modified versions of the probability density and the probability current lead to adjustments in the scalar product and the norm. Our results are applied to several systems, including energy-dependent modifications of the Mathews-Lakshmanan oscillator and the generalized Swanson system.