Spectrum and eigenstates for a model describing optical second harmonic generation

Without using the assumption of the Bethe ansatz, we have obtained exact explicit analytical expressions for the energy spectrum and eigenstates in terms of a parameter λ for a model of optical second harmonic generation. We have shown that the parameter λ is determined by the roots of a polynomial of order [(N+1)/4]. Here [y] denotes y' s integer part and N = na+2nb with na and nb being photon numbers of the fundamental and the second harmonic modes respectively. We also obtain explicit analytical expressions for λ for N ≤ 17, and explicit analytical expressions for all the eigenstates without showing any interactions between the two modes