The Effects of the modified scalar product on the properties of the one-dimensional harmonic oscillator with energy-dependent potential

In this article, we try to 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 at first discuss the effect of this change on the thermodynamic properties of this oscillator, and then on the parameter of Fisher, well known in the field of quantum information. For the second problem, we are an obligation to redefine this parameter. Finally, the uncertainly relation of Cramer-Rao is well recovered in our problem in question.