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 [García-Martínez (Phys Lett 373:3619, 2009)], 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 Schrödinger-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.