p-Adic Schrödinger-Type Operator with Point Interactions

Abstract

A $p$-adic Schrödinger-type operator $D^α+V_Y$ is studied. $D^α$ ($α>0$) is the operator of fractional differentiation and $V_Y=\sum_{i,j=1}^nb_{ij}<δ_{x_j}, \cdot>δ_{x_i}$ $(b_{ij}\in\mathbb{C})$ is a singular potential containing the Dirac delta functions $δ_{x}$ concentrated on points $\{x_1,...,x_n\}$ of the field of $p$-adic numbers $\mathbb{Q}_p$. It is shown that such a problem is well-posed for $α>1/2$ and the singular perturbation $V_Y$ is form-bounded for $α>1$. In the latter case, the spectral analysis of $η$-self-adjoint operator realizations of $D^α+V_Y$ in $L_2(\mathbb{Q}_p)$ is carried out.