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The de Sitter invariant Special Relativity (dS-SR) is a SR with constantcurvature, and a natural extension of usual Einstein SR (E-SR). In this paper,we solved the dS-SR Dirac equation of Hydrogen by means of the adiabaticapproach and the quasi-stationary perturbation calculations of QM. Hydrogenatoms are located on the light cone of the Universe. FRW metric and$\Lambda$CDM cosmological model are used to discuss this issue. To the atom,effects of de Sitter space-time geometry described by Beltrami metric are takeninto account. The dS-SR Dirac equation turns out to be a time dependent quantumHamiltonian system. We revealed that: 1,The fundamental physics constants$m_e,\;\hbar,\;e$ variate adiabatically along with cosmologic time in dS-SR QMframework. But the fine-structure constant $\alpha\equiv e^2/(\hbar c)$ keepsto be invariant; 2,$(2s^{1/2}-2p^{1/2})$-splitting due to dS-SR QM effects: Bymeans of perturbation theory, that splitting $\Delta E(z)$ were calculatedanalytically, which belongs to $\mathcal{O}(1/R^2)$-physics of dS-SR QM.Numerically, we found that when $|R|\simeq \{10^3 Gly,\;10^4 Gly,\;10^5Gly\;\}$, and $z\simeq \{1,\;{\rm or}\;2\}$, the $\Delta E(z)>> 1{\rm (Lamb\;shift)}$. This indicate that for these cases the hyperfine structure effectsdue to QED could be ignored, and the dS-SR fine structure effects are dominant.This effect could be used to determine the universal constant $R$ in dS-SR, andbe thought as a new physics beyond E-SR.

One Electron Atom in Special Relativity with de Sitter Space-Time Symmetry