Singular perturbations of unbounded selfadjoint operators Reverse approach

Abstract

Let A and A1 be unbounded selfadjoint operators in a Hilbert space $\mathcal H$. Following [3], we call A1 a singular perturbation of A if A and A1 have different domains $\mathcal D(A), D(A_1)$ but $\mathcal D(A) \cap D(A_1)$ is dense in $\mathcal H$ and A = A1 on $\mathcal D(A) \cap D(A_1)$. 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 A1 of a given selfadjoint operator A.

For the special case when A is the standardly defined selfadjoint Laplace operator in L₂( $\mathbb R^3$) we describe using the M.G. Krein resolvent formula a class of singular perturbations A1, which are defined by special selfadjoint boundary conditions on a finite or spaced apart by bounded from below distances infinite set of points in $\mathbb R^3$ and also on a bounded segment of straight line embedded into $\mathbb R^3$ by connecting parameters in the boundary conditions for A1 and the independent on A matrix or operator parameter in the Krein formula for the pair A, A1.