p-Adic Schrödinger-Type Operator with Point Interactions

Details p-Adic Schrödinger-Type Operator with Point Interactions p-Adic Schrödinger-Type Operator with Point Interactions

2007-03-27

arxiv.org/abs/math-ph/0703077v2

J. Math. Anal. Appl. 338 (2008), No. 2, 1267-1281

Physics

Mathematical Physics

Scientific paper

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.

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