ON THE PROBLEM OF THE RIGHT HAMILTONIAN UNDER SINGULAR FORM-SUM PERTURBATIONS

Abstract

Let a perturbation of the self-adjoint operator H0>0 in the Hilbert space ℋ be given by an operator V (or by a quadratic form ν) which is possibly singular and in general nonpositive, so H0+V on [Formula: see text] is only a symmetric operator with nontrivial deficiency indices. The definition of the sum [Formula: see text] in the sense of quadratic forms is extended to cases which are not covered by the well-known KLMN-theorem and conditions are found which ensure the unique self-adjoint realization of H in ℋ. It is also shown that ℋ coincides with the strong resolvent limit of the approximating sequence Hn = H0+Vn, where Vn are bounded self-adjoint operators such that Vn → V in a suitable sense. Essentially that operator V might be strongly singular and acts in the H0-scale of spaces, V:ℋ+→ℋ-.