A method is developed to determine the eigenvalues and eigenfunction oftwo-boson $2\times 2$ matrix Hamiltonians include a wide class of quantumoptical models. The quantum Hamiltonians have been transformed in the form ofthe one variable differential equation and the conditions for its solvabilityhave been discussed. We present two different transformation procedure and weshow our approach unify various approaches based on Lie algebraic technique. Asan application, solutions of the modified Jaynes-Cummings and two-level JahnTeller Hamiltonians are studied.
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