p-Adic Fractional Differentiation Operator with Point Interactions

Abstract

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