Adjacency Graphs and Long-Range Interactions of Atoms in Quasi-Degenerate States: Applied Graph Theory

We analyze, in general terms, the evolution of energy levels in quantum mechanics, as a function of a coupling parameter, and demonstrate the possibility of level crossings in systems described by irreducible matrices. In long-range interactions, the coupling parameter is the interatomic distance. We demonstrate the utility of adjacency matrices and adjacency graphs in the analysis of "hidden" symmetries of a problem; these allow us to break reducible matrices into irreducible subcomponents. A possible breakdown of the no-crossing theorem for higher-dimensional irreducible matrices is indicated, and an application to the 2S-2S interaction in hydrogen is briefly described. The analysis of interatomic interactions in this system is important for further progress on optical measurements of the 2S hyperfine splitting.