Spectral Theory of Differential Operators and Energy Levels of Subatomic Particles

Abstract

Motivated by the Bohr atomic model, in this article we establish a mathematical 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 subatomic 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 number $N$ of energy levels and the average energy level gradient (for two adjacent energy levels) are rigorously estimated in terms of certain physical parameters. These estimates 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.