In mean-field approximation we replace the interaction term of the Hamiltonian by a term, which is quadratic in creation and annihilation operators. For example, in the case of the BCS theory, where
$$ \sum_{kk^{\prime}}V_{kk^{\prime}}c_{k\uparrow}^{\dagger}c_{-k\downarrow}^{\dagger}c_{-k^{\prime}\downarrow}c_{k^{\prime}\uparrow}\to\sum_{k}\Delta_{k}c_{k\uparrow}^{\dagger}c_{-k\downarrow}^{\dagger} + \Delta_{k}^{\star}c_{-k\downarrow}c_{k\uparrow}\text{,} $$
with $\Delta_{k}=\sum_{k^{\prime}}V_{kk^{\prime}}\langle c_{-k^{\prime}\downarrow}c_{k^{\prime}\uparrow}\rangle\in\mathbb{C}$. Then, in books, like this by Bruss & Flensberg, there is always a sentence like "the fluactuations around $\Delta_{k}$ are very small", such that the mean-field approximation is a good approximation. But we known for example in the case of the 1D Ising model the mean-field approximation is very bad.
My question: Is there a inequality or some mathematical conditions which says something about the validity of the mean-field approach? Further, is there a mathematical rigoros derivation of the mean-field approximation and the validity of it?