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Motivated by the Bohr atomic model, in this article we establish a mathe- matical theory to study energy levels, corresponding to bounds states, for subatomic particles. We show that the energy levels of each subatomic particle are finite and discrete, and corresponds to negative eigenvalues of the related eigenvalue problem. Consequently there are both upper and lower bounds of the energy levels for all sub- atomic particles. In particular, the energy level theory implies that the frequencies of mediators such as photons and gluons are also discrete and finite. Both the total num- ber N of energy levels and the average energy level gradient (for two adjacent energy levels) are rigorously estimated in terms of certain physical parameters. These esti- mates show that the energy level gradient is extremely small, consistent with the fact that it is hard to notice the discrete behavior of the frequency of subatomic particles. AMS subject classifications: 35Q75, 37N20, 83C, 83F

Spectral Theory of Differential Operators and Energy Levels of Subatomic Particles