  Finite rank point perturbations of the $p$-adic fractional differentiation
operator $D^{\alpha}$ are studied. The main attention is paid to the
description of operator realizations (in $L_2(\mathbb{Q}_p)$) of the heuristic
expression $D^{\alpha}+\sum_{i,j=1}^{n}b_{ij}<\delta_{x_j}, \cdot>\delta_{x_i}$
in a form that is maximally adapted for the preservation of physically
meaningful relations to the parameters $b_{ij}$ of the singular potential.
