Saturation effect in confined quantum systems with energy-dependent potentials

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 ${}_1 F_1$, while the third model (heavy quarkonia) stems possibly from the hypergeometric differential of the well-known solutions ${}_2 F_1$. In particular, the heavy quarkonia mass spectra for both $c\bar c$ and $b\bar b$ are produced at different values of the saturation parameter $\lambda$ and compared with the available experimental data. It is found that these systems may exhibit saturation effect when the energy-dependent effect is included.