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We discuss singular perturbations of a self-adjoint positive operator A in Hilbert space H formally given by A T =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 H 1 (A)=D(A 1Â2 ) equipped by the graph-norm, then any suitable approximation by positive operators, T n Ä T, gives a trivial result, i.e., A T n Ä A in the strong resolvent sense, where A T n is defined as a form-sum of A and T n . A corresponding statement is true for operators T, T n 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 L 2 (R 3 ). In the more general case, where the singular operator T has a nontrivial regular component T r in H 1 (A), we prove that A T n Ä A T r 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. Academic Press

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We use the method of self-adjoint extensions to define a self-adjoint operator A, as the singular perturbation of a given self-adjoint operator A by a singular operator T on a Hilbert space.We also find the structure of a singular operator Q such that the singular perturbation of A' by Q satisfies (A2)? = (AT)2. We obtain the explicit form of Q in terms of A and T. A definition of the n-th power for strictly positive symmetric operators is also given. ') Partially supported by KBN grant 4052/PB/IFT/91.

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We prove that for any self-adjoint operator A in a separable Hilbert space H and a given countable set Λ = {λi} i∈N of real numbers, there exist H−2-singular perturbationsà of A such that Λ ⊂ σp`ô. In particular, if Λ = {λ1, . . . , λn} is finite, then the operatorà solving the eigenvalues problem,Ãψ k = λ k ψ k , k = 1, . . . , n, is uniquely defined by a given set of orthonormal vectors {ψ k } n k=1 satisfying the condition span{ψ k } n k=1 ∩ dom`|A| 1/2´= {0}.

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