p-Adic Schrödinger-Type Operator with Point Interactions

Abstract

A $p$-adic Schrödinger-type operator $D^{\alpha }+V_Y$ is studied. $D^{\alpha }$ ($\alpha >0$) is the operator of fractional differentiation and $V_Y={\sum }_{i,j=1}^n{b}_{ij}<{\delta }_{x_j},\cdot >{\delta }_{x_i}$ $(b_{ij}\in \mathbb{C})$ 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.

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Cited by in corpus (2)

• p-Adic Mathematical Physics
• $p$-Adic Mathematical Physics: The First 30 Years