We focus on the definition of the unitary transformation leading to aneffective second order Hamiltonian, inside degenerate eigensubspaces of thenon-perturbed Hamiltonian. We shall prove, by working out in detail theSu-Schrieffer-Heeger Hamiltonian case, that the presence of degenerate states,including fermions and bosons, which might seemingly pose an obstacle towardsthe determination of such "Froehlich-transformed" Hamiltonian, in fact doesnot: we explicitly show how degenerate states may be harmlessly included in thetreatment, as they contribute with vanishing matrix elements to the effectiveHamiltonian matrix. In such a way, one can use without difficulty theeigenvalues of the effective Hamiltonian to describe the renormalized energiesof the real excitations in the interacting system. Our argument applies also tofew-body systems where one may not invoke the thermodynamic limit to get rid ofthe "dangerous" perturbation terms.
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