Effect of Generalized Fractional Derivatives on the Energy Spectrum of the Hydrogen Atom in Plasma

ABSTRACT

The study of atomic systems embedded in plasma environments is critical for expanding our understanding of microelectronics, fusion plasmas, and condensed matter physics. In this investigation, for the first time, the energy spectrum of a hydrogen atom in a dense plasma environment is analyzed within the framework of generalized fractional derivatives. Utilizing the parametric generalized fractional Nikiforov–Uvarov (PGFNU) method in conjunction with the Greene–Aldrich (GA) approximation to treat the centrifugal barrier, approximate analytical expressions for the bound‐state energy spectrum under the Debye screening potential (DSP) are derived for both integer and fractional regimes. According to our results, the effects of Coulombic attraction become weak and the bound states become unstable as the screening parameter defined by increases, particularly for the s ‐states and lower principal quantum numbers. The generalized fractional derivative employed in this study is a strictly local conformable‐type differential operator, distinct from the non‐local formulation found in Laskin's fractional quantum mechanics. Consequently, the fractional order is treated as an effective phenomenological parameter rather than one deduced from microscopic plasma kinetics. Our results reveal that reducing below the conventional integer limit (), the binding strength is considerably increased through local kinetic scaling, effectively counteracting the destabilizing influence of the plasma medium. Comparative analyses, presented both tabularly and graphically across a range of screening parameters ( and 1/), indicate that our analytical eigenvalues are in a good alignment with established results from asymptotic iteration method (AIM), supersymmetric quantum mechanics (SUSYQM), variational, and numerical methods. Furthermore, the critical screening parameter () is computed across various , , and states, revealing an accidental ‐degeneracy in the classical limit (); this degeneracy is naturally lifted in the fractional domain (), where tuning yields good agreement with established numerical plasma benchmarks.