On Form-Sum Approximations of Singularly Perturbed Positive Self-adjoint Operators

We discuss singular perturbations of a self-adjoint positive operator A in Hilbert space H formally given by AT=A+T, where T is a singular positive operator (singularity means that Ker T is dense in H). We prove the following result: if T is strongly singular with respect to A in the sense that Ker T is dense in the Hilbert space H1(A)=D(A1/2) equipped by the graph-norm, then any suitable approximation by positive operators, Tn→T, gives a trivial result, i.e., ATn→A in the strong resolvent sense, where ATn is defined as a form-sum of A and Tn. A corresponding statement is true for operators T, Tn of finite rank which are not necessarily positive. This can be considered as an abstract version of the well known result for the perturbation by a point interaction of the Laplace operator in L2(R3). In the more general case, where the singular operator T has a nontrivial regular component Tr in H1(A), we prove that ATn→ATr in the strong resolvent sense. We give applications to the case of perturbations of the Laplace operator by a positive Radon measure with a nontrivial singular component.