  An algebraic method is introduced for an analytical solution of the
eigenvalue problem of the Tavis-Cummings (TC) Hamiltonian, based on
polynomially deformed su(2), i.e. su_n(2), algebras. In this method the
eigenvalue problem is solved in terms of a specific perturbation theory,
developed here up to third order. Generalization to the N-atom case of the Rabi
frequency and dressed states is also provided. A remarkable enhancement of
spontaneous emission of N atoms in a resonator is found to result from
collective effects.
