If we want to express a quantum mechanical oeprator $ \hat{A}$ in second quantization formalism, it is
$$ \hat{A} = \sum_{\alpha, \beta} \langle \alpha | \hat{A}|\beta \rangle c^{\dagger}_{\alpha}c_{\beta} $$
So if we represent the kinetic operator $ \hat{T} = -\frac{\hbar^2}{2m}\nabla^2 $, it is written formally
$$ \hat{T} = -\frac{\hbar ^2}{2m}\int d\vec{r} ~\Psi_{r}^{\dagger} \nabla^2 \Psi_r $$
But here what does $\nabla^2 $ mean? It should be a number, because it is the matrix element $ \langle \alpha | \hat{A}|\beta \rangle $ in the first equation. But I can't guess what should it be.
In the derivation of kinetic operator in second quantization form, we calculate the matrix element between the two position space eigenfunctions, the delta function. So Actually $\nabla^2 $ means the diagonal component of the delta function, but what is it in terms of number? Just $ \infty $ or 0 like delta function?