Survey on density of states and saturation effect of spectrum for an energy-dependent harmonic interaction

In this article, we test the influence of the modification of the scalar product, found in the problems of the energy-dependent potential, on the physical properties of the harmonic oscillator in one dimension. For this, we first discuss the effect of this change on the thermodynamic properties of this oscillator, and then investigate the parameters of Fisher and Shannon of quantum information. For the second problem, the dependence of these parameters with the density of state, is the main reason of our choice to study the influence of this type of potential on these properties. Finally, the well-know uncertainty relation of Cramer-Rao is well recovered.