p-Adic Schr\"{o}dinger-Type Operator with Point Interactions

TL;DR

This paper investigates a $p$-adic Schrödinger-type operator with point interactions, analyzing its well-posedness, form-boundedness, and spectral properties within the $p$-adic number field for certain parameter ranges.

Contribution

It establishes conditions for well-posedness and form-boundedness of the $p$-adic Schrödinger operator with point interactions, and performs spectral analysis of its self-adjoint realizations.

Findings

Well-posedness for $oxed{ ext{alpha} > 1/2}$.

Form-boundedness of the potential for $oxed{ ext{alpha} > 1}$.

Spectral analysis of self-adjoint realizations conducted.

Abstract

A $p$-adic Schr\"{o}dinger-type operator $D^{\alpha}+V_Y$ is studied. $D^{\alpha}$ ($\alpha>0$) is the operator of fractional differentiation and $V_Y=\sum_{i,j=1}^nb_{ij}<\delta_{x_j}, \cdot>\delta_{x_i}$ $(b_{ij}\in\mathbb{C})$ is a singular potential containing the Dirac delta functions $\delta_{x}$ 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 $\alpha>1/2$ and the singular perturbation $V_Y$ is form-bounded for $\alpha>1$. In the latter case, the spectral analysis of $\eta$-self-adjoint operator realizations of $D^{\alpha}+V_Y$ in $L_2(\mathbb{Q}_p)$ is carried out.