Abstract
In this paper, we study the saturation effect in the energy or mass spectra of three quantum models with energy-dependent potentials: the harmonic oscillator, the hydrogen atom, and the heavy quarkonia. We used the method proposed in \cite{garcia2009exactly}, which is based on studying various canonical point and gauge transformations applied to a function, g(x), multiplied by a given differential equation of known solutions as special orthogonal functions, that convert it into a Schr{{\"o}}dinger-like equation. The first two models stem from implementing the method on the confluent hypergeometric differential of the well-known solutions ₁ F₁, while the third model (heavy quarkonia) stems possibly from the hypergeometric differential of the well-known solutions ₂ F₁. In particular, the heavy quarkonia mass spectra for both cc and bb are produced at different values of the saturation parameter λ and compared with the available experimental data. It is found that these systems may exhibit saturation effect when the energy-dependent effect is included.