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We study the quantum dynamics of a single mode/particle interactinginhomogeneously with a large number of particles and introduce an effectiveapproach to find the accessible Hilbert space where the dynamics takes place.Two relevant examples are given: the inhomogeneous Tavis-Cummings model (e.g.,N atomic qubits coupled to a single cavity mode, or to a motional mode intrapped ions) and the inhomogeneous coupling of an electron spin to N nuclearspins in a quantum dot.Comment: 9 pages and 10 figures, new version, accepted in Physical Review

Effective quantum dynamics of interacting systems with inhomogeneous coupling