Singular selfadjoint perturbations of unbounded selfadjoint operators. Reverse approach

Abstract

Let $A$ and $A_{1}$ are unbounded selfadjoint operators in a Hilbert space $\mathcal{H}$. Following \cite{AK} we call $A_{1}$ a \textit{singular} perturbation of $A$ if $A$ and $A_{1}$ have different domains $\mathcal{D}(A),\mathcal{D}(A_{1})$ but $\mathcal{D}(A)\cap\mathcal{D}(A_{1})$ is dense in $\mathcal{H}$ and $A=A_{1}$ on $\mathcal{D}(A)\cap\mathcal{D}(A_{1})$. In this note we specify without recourse to the theory of selfadjoint extensions of symmetric operators the conditions under which a given bounded holomorphic operator function in the open upper and lower half-planes is the resolvent of a singular perturbation $A_{1}$ of a given selfadjoint operator $A$. For the special case when $A$ is the standardly defined selfadjoint Laplace operator in $\mathbf{L}_{2}(\mathbf{R}_{3})$ we describe using the M.G. Krein resolvent formula a class of singular perturbations $A_{1}$, which are defined by special selfadjoint boundary conditions on a finite or spaced apart by bounded from below distances infinite set of points in $\mathbf{R}_{3}$ and also on a bounded segment of straight line embedded into $\mathbf{R}_{3}$ by connecting parameters in the boundary conditions for $A_{1}$ and the independent on $A$ matrix or operator parameter in the Krein formula for the pair $A, A_{1}$.