  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.
