  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 $A, A_{1}$.
