On positronium states in $\mathrm{Q}\mathrm{E}{\mathrm{D}}_3$

In this note, we present a new method to investigate the positronium states in $\mathrm{Q}\mathrm{E}{\mathrm{D}}_3$. According to the $\mathrm{K}\ddot{\mathrm{a}}\mathrm{l}\mathrm{l}\acute{\mathrm{e}}\mathrm{n}-\mathrm{L}\mathrm{e}\mathrm{h}\mathrm{m}\mathrm{a}\mathrm{n}\mathrm{n}$ spectral representation, the energy eigenvalues of bound states are poles of the correlation function. Using the chain approximation, we obtain the energy eigenvalues of the vector positronium states by taking into account the single-valued branches of multi-valued function. Using the same method, we also find the electron physical mass at some single-valued branch of multi-value function. Our results are agreement with the known ones.