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We revisit the approach proposed by Mukamel and coworkers to describeinteracting excitons through infinite series of composite-boson operators forboth, the system Hamiltonian and the exciton commutator -- which, in thisapproach, is properly kept different from its elementary boson value. Insteadof free electron-hole operators, as used by Mukamel's group, we here work withcomposite-exciton operators which are physically relevant operators for excitedsemiconductors. This allows us to get \emph{all} terms of these infinite seriesexplicitly, the first terms of each series agreeing with the ones obtained byMukamel's group when written with electron-hole pairs. All these terms nicelyread in terms of Pauli and interaction scatterings of the composite-excitonmany-body theory we have recently proposed. However, even if knowledge of theseinfinite series now allows to tackle $N$-body problems, not just 2-bodyproblems like third order nonlinear susceptibility $\chi^{(3)}$, the necessaryhandling of these two infinite series makes this approach far more complicatedthan the one we have developed and which barely relies on just fourcommutators.

Exciton many-body effects through infinite series of composite-exciton operators