We develop a systematic theoretical approach to incorporate the effects of a static white-noise disorder into the BCS-BEC crossover near the critical temperature $T_c$ of the superfluid transition. Starting from a functional-integral formulation in momentum-frequency space, we derive an effective thermodynamic potential that fully accounts for Gaussian fluctuations of the order-parameter field and its coupling to the disorder potential. The effective action, expanded to second order in both the disorder potential and the bosonic field, naturally involves third- and fourth-order terms arising from the logarithmic expansion near $T_c$. By providing a controlled description of pairing fluctuations and disorder effects, this formalism correctly recovers the well-established BCS and BEC limits. This ensures a consistent physical foundation for analyzing the entire BCS-BEC crossover, effectively anchoring the intermediate regime between these two analytically robust end points. The approach applies equally to continuum and lattice systems and provides a natural framework for generalizations to multiband models.

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• BCS-BEC crossover
• Disordered systems
• Superfluids