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

Superstatistics of the one-dimensional Klein-Gordon oscillator with energy-dependent potentials

“…In this context, we note that the observation of peaks in the curves of specific heat indicates that there are a small number of discrete energy levels dominating the behavior of our system in question. This behavior is due essentially to the dependence of the potential of our oscillator with the energy: as argued in [5], this situation is very similar to the case of Schottky peak (Schottky anomaly) which is a broad maximum in the specific heat observed in systems with several discrete energy levels, and not a phase transition.…”

“…• Contrarily to the non-relativistic limit( see Ref. [5]), the limit of specific heat is equal zero only in very high-temperatures (β → 0). In very low-temperatures (β → ∞), for different values of γ, this limit depends on the value of the parameter q.…”

“…Tto calculate this function, we adopt two approaches: (i) we first fixed the parameter γ, and then varied the values of n until we obtain the saturation phenomena in our spectrum of the energy (Eq. 7) and so on, and (ii) via the method used by the authors [5] which is well described in their work. Following this, Eq.…”

“…To obtain the partition function from the obtained energy spectrum, we adopt the following way, which the partition function Z is defined by [5]:…”

“…Many authors have investigated the Schrödinger, Klein Gordon, and Dirac equation with energydependent potentials. In recent times, Boumali et al, [5] studied the influence of energy-dependent potentials on the thermal properties of the one-dimensional harmonic oscillator using the Euler-Maclaurin approach. To the best of our knowledge, no relativistic case has been reported on the influence of energy-dependent potentials on the thermal properties of the one-dimensional harmonic oscillator.…”