Analytical Results of Eigenstates and Eigenenergies by Mixing Models of Five Bosonic Modes

The analytical expressions of eigenstates and eigenenergies are demonstrated using a parameter λ without the assumption of Bethe ansatz for two kinds of five-mode mixing models. It is found that the parameter can be determined by the roots of a simple polynomial. We also illustrate the exact analytical expressions of eigenstates and energies without any unknown parameter.